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+{"text":"\\section{Introduction}\n\nIn recent years, extended affine Lie algebras (EALAs) have been\nstudied in great detail.  EALAs are  Lie algebras which have a\nnon-degenerate invariant form, a self-centralizing finite\ndimensional ad-diagonalizable abelian subalgebra (i.e., a Cartan\nsubalgebra), a discrete irreducible root system, and ad-nilpotency\nof non-isotropic root spaces (see \\cite{AABGP,BGK} for\ndefinitions and structure theory).\nThere are many EALAs which allow not only the Laurent polynomial\nalgebras as co-ordinate algebras but also quantum tori, Jordan tori\nand Octonion tori as co-ordinate algebras depending on the type of\nalgebras (see\n\\cite{AABGP,BGK,Y1,Y2,Y3}). For\ninstances, EALAs of type $A_{d-1}$ are tied up with the Lie algebra\n$\\mathfrak {gl}_d(\\C_q)$. Quantum tori are important algebras in the theories of algebra and non-commutative geometry, see \\cite{MP} and \\cite{Ma}.\n To get an\nEALA one has to form appropriate central extension of $\\mathfrak\n{gl}_d(\\C_q)$ and add certain outer derivations (just like one\nobtains an affine Kac-Moody Lie algebra from a  loop algebra  by\nforming a one-dimensional central extension and then adding the\ndegree derivation).\nUnlike the affine Lie algebras, the\nrepresentation theory of EALAs is far from well developed.\nSee \\cite{DVF,E,ER,EZ,G,G1,G2,G3,L} for several interesting results on the representation theory for  the extended affine Lie algebras.\n\n\n\nFree field realizations of Lie algebras play important role in representation\ntheory and conformal field theory. For an affine Lie algebra $\\mathfrak{L}$, imaginary Verma modules of $\\mathfrak{L}$ arise from non-standard partitions of the root system of $\\mathfrak{L}$, see \\cite{F}.\nA $q$-version of imaginary Verma modules for the quantum groups of type $U_q(A^{(1)}_1)$ was constructed in \\cite{CFKM}, and was further studied in \\cite{CFM1,CFM2}.\nFree field realizations of imaginary Verma modules over the affine Lie\nalgebra $A^{(1)}_1$ were constructed  in \\cite{JK} for zero central charge and in \\cite{W} for arbitrary\ncentral charge. In \\cite{FGR}, the localization technique was used to construct a new family of free field realizations.\n In particular, the (twisted) localization of imaginary Verma modules, for the Kac-Moody Lie algebra $A^{(1)}_1$, provides new irreducible weight dense modules with infinite weight multiplicities.\n In \\cite{GZ}, the free  field realizations\nof highest weight modules over  the  extended affine Lie algebra $\\widehat{\\frak{gl}_{2}(\\bc_q)}$ were first given. In \\cite{Z}, those  realizations were\ngeneralized to $\\widehat{\\frak{gl}_{d}(\\bc_q)}$ and the simplicity of the corresponding  modules was also given.\nIn the present paper, inspired by the idea of \\cite{FGR},  we will construct  free field realizations for a class of simple weight modules with infinite weight multiplicities over the extended affine Lie algebras.\n\n\nThe organization of the paper  is as follows. In Section 2, we recall the  free  field realizations\nof highest weight modules and their simplicity. In Subsection 3.1, we collect some preliminary results on the twisted localization.\nIn Subsection 3.2, we decide the simplicity of $\\md_\\bm M\/M$. We show that $\\md_\\bm M\/M$ is simple if and only if $\\mu \\notin \\Z$, see Theorem \\ref{Maintheorem1}. In Subsection 3.3, we give the simplicity of $\\md_\\bm^bM$ in the case $b\\not\\in\\Z$.\nWe prove that $\\md_\\bm^bM$ is irreducible if and only if $b+\\mu \\notin \\Z$, see Theorem \\ref{maintheorem2}.\n\nWe denote by $\\mathbb{Z}$, $\\mathbb{Z}_+$, $\\mathbb{N}$ and\n$\\mathbb{C}$ the sets of  all integers, nonnegative integers,\npositive integers and complex numbers, respectively. For any Lie\nalgebra ${L}$, we denote  its\nuniversal enveloping algebra by $U(L)$.\n\\section{Preliminaries}\n\n Let $q$ be a non-zero complex number. Let $\\C_q$ be the\nassociative algebra generated by $t_1^{\\pm1}, t_2^{\\pm1}$ subject to the relation\n$$t_1t_1^{-1}=t_1^{-1}t_1=t_2t_2^{-1}=t_2^{-1}t_2=1, t_2t_1=qt_1t_2.$$\n\nFor any $\\bm=(m_1,m_2)\\in\\Z^2$, denote ${\\bt}^{\\bm}=t_1^{m_1}t_2^{m_2}$.\nThen it is clear that $${\\bt}^{\\bm}{\\bt}^{\\bn}=q^{m_2n_1}\\bt^{\\bm+\\bn}=q^{m_2n_1-m_1n_2}t^{\\bn}t^{\\bm}$$ for any $\\bm,\\bn\\in\\Z^2$.\n\nLet $\\frak{gl}_2(\\C_q):=\\frak{gl}_2(\\C)\\otimes_\\C \\C_q$ be the general linear Lie algebra coordinated by $\\C_q$. Denote $X(\\bm)=X\\otimes t^\\bm$ for\n$X\\in\\frak{gl}_2(\\C), \\bm\\in \\Z^2$. We identify $X$ with $X(0)$.\nThen the  Lie bracket of $\\frak{gl}_2(\\C_q)$ is given by\n\\begin{align*} & [e_{ij}(\\bm), e_{kl}(\\bn)]\n=\\delta_{jk}q^{m_2n_1}e_{il}(\\bm+\\bn)-\\delta_{il}q^{n_2m_1}e_{kj}(\\bm+\\bn)\n\\end{align*}\nfor $\\bm,\\bn\\in\\bz^2$, $1\\leq i, j, k, n\\leq 2$, where\n$e_{ij}$ is the matrix whose $(i, j)$-entry is $1$ and $0$\nelsewhere.\n\nClearly $\\frak{gl}_2(\\C_q)$  is $\\Z^2$-graded and to reflect this fact we add degree derivations.\nLet $\\widehat{\\frak{gl}_{2}(\\bc_q)} ={\\frak{gl}_{2}(\\bc_q)} \\oplus \\bc d_1\n\\oplus \\bc d_2$ and extend the Lie bracket as\n$$[d_1, X(\\bm)]=m_1X(\\bm),\\ \\  [d_2, X(\\bm)]=m_2X(\\bm).$$\n\nThe Lie subalgebra $[{\\frak{gl}_{2}(\\bc_q)},{\\frak{gl}_{2}(\\bc_q)}]  \\oplus \\bc d_1\n\\oplus \\bc d_2$ of $\\mg$ is called an extended\naffine Lie algebra of type $A_{1}$ with nullity $2$. ( See [AABGP] and [BGK]\nfor definitions).\n\nLet $\\frak{h}=\\C e_{11}\\oplus \\C e_{22}\\oplus \\bc d_1\\oplus \\bc d_2$ which is a Cartan subalgebra  of\n$\\widehat{\\frak{gl}_{2}(\\bc_q)}$. A $\\mg$-module  $M$ is called a weight module if $\\mh$ acts diagonally on  $M$, i.e.\n$ M=\\oplus_{\\lambda\\in \\mh^*} M_\\lambda,$\nwhere $M_\\lambda:=\\{v\\in M \\mid xv=\\lambda(x) v, \\forall\\ x\\in \\mh\\}.$  A nonzero element $v\\in M_\\lambda$ is called a weight vector.\n\n\nLet us denote\n$$\\frak{n}_+=e_{12}\\otimes \\bc_q,\\ \\  \\frak{n}_-=e_{21}\\otimes \\bc_q,$$ $$\\mathcal{H}=(e_{11}\\otimes \\C_q) \\oplus (e_{22}\\otimes \\C_q)\\oplus  \\bc d_1\\oplus \\bc d_2.$$\nThen $$\\mg=\\frak{n}_+\\oplus \\mathcal{H}\\oplus \\frak{n}_-.$$\nFor a weight $\\mg$-module $M$, a weight vector $v$ in $M$ is called a highest weight vector if $\\frak{n}_+v=0$.\nThe module $M$ is called a highest weight module if it is generated by  a highest weight vector.\n\nNext, we will recall a class of  highest weight modules over\n$\\widehat{\\frak{gl}_{2}(\\bc_q)}$ defined by free fields, see \\cite{GZ}.\n\n Let\n $$\\bc [\\mathbf{x} ]=\\bc [x_{\\bm}, \\bm\\in\\bz^2]$$ be a polynomial ring with infinitely many\nvariables $x_{\\bm}$.  Denote by $\\mathcal{A}(\\bx)$ the associative algebra of formal power series of differential operators on\n$\\bc [\\mathbf{x} ]$.  By \\cite{GZ}, there is an algebra homomorphism $$\\Phi: U(\\mg)\\rightarrow \\text{End}(\\bc [\\mathbf{x} ])$$ defined by:\n\\begin{eqnarray*}\ne_{21}(\\bn)&\\mapsto&x_\\bn ,\\\\\ne_{12}(\\bn)&\\mapsto&q^{-n_1n_2}\\mu\\p{\\x_{-\\bn}}\n-\\sum_{\\bs,\\br\\in \\bz^2}\nq^{n_2r_1+s_2n_1+s_2r_1}x_{\\bn+\\bs+\\br}\\p{\\x_\\bs}\\p{\\x_{\\br}},\\\\\ne_{22}(\\bn)&\\mapsto&\\sum_{\\bs\\in \\bz^2} q^{s_1n_2}x_{\\bn+\\bs}\\p{\\x_\\bs},\\\\\ne_{11}(\\bn)&\\mapsto&\\mu\\delta_{\\bn,0}-\\sum_{\\bs\\in \\bz^2} q^{s_2n_1}x_{\\bn+\\bs}\\p{\\x_\\bs},\\\\\n D_1&\\mapsto&\\sum_{\\bs\\in \\bz^2}s_1x_\\bs\\p{\\x_\\bs},\\\\\n D_2&\\mapsto&\\sum_{\\bs\\in \\bz^2}s_2x_\\bs\\p{x_\\bs},\n\\end{eqnarray*}\nwhere $\\mu\\in\\C$.\n\nVia the homomorphism $\\Phi$, $\\C [\\mathbf{x} ]$ can be viewed as  a module over $\\mg$.  We can see that $\\bc [\\mathbf{x} ]$ is a module of $\\mg$ generated by $1$, and $e_{12}(\\bn)1 = 0$ for any $\\bn\\in\\Z^2$.\nHence $\\bc [\\mathbf{x} ]$ is a highest weight module of $\\mg$, and $1$ is a highest weight\nvector such that $e_{11}1=\\mu, e_{22}1=0$.\nThe following result was given by Zeng, see Corollary 4.1 in \\cite{Z}.\n\\begin{theorem}The $\\mg$-module $\\bc [\\mathbf{x} ]$ is irreducible if and only if $\\mu\\neq 0$.\n\\end{theorem}\n\n\n\\section{The simplicity of the twisted localization}\n\nIn this section, we will apply the twisted localization functor to the $\\mg$-module $\\bc [\\mathbf{x} ]$ to obtain new irreducible modules with infinite dimensional weight spaces.\nThe twisted localization functor was introduced by O. Mathieu to classify irreducible cuspidal modules over finite dimensional simple Lie algebras, see [M].\n\\subsection{The definition of the twisted localization}\n\nLet $U(\\mg)$ be the universal enveloping algebra of $\\mg$. For a fixed $\\bm\\in \\Z^2$, since $e_{21}(\\bm)$ is an ad nilpotent element of $U(\\mg)$, $F=\\{e_{21}(\\bm)^i\\mid i\\in \\Z^+\\}$ is a left and\nright Ore subset of $U(\\mg)$. We denote  the Ore localization of $U(\\mg)$ with respect to $F$ by $U^F$.\nFor a $\\mg$-module $M$, we denote  by $\\md_\\bm M$ the $F$-localization, i.e., $\\md_\\bm M= U^F\\otimes_U M$.\n\n For $b\\in\\C $ and $u\\in U^F$, we set\n$$\\Theta_b(u)=\\sum_{j\\geq 0}\\binom{b}{j}(\\text{ad}e_{21}(\\bm))^j (u) e_{21}(\\bm)^{-j},$$\nwhere\n$\\binom{b}{j}= \\frac{b(b-1)\\cdots(b-j+1)}{j!}$ .\n Since $\\text{ad}e_{21}(\\bm)$ is locally nilpotent on $U^F$, the sum above\nis actually finite. It is known that $\\Theta_b$ is an automorphism of $U^F$. Note that for $b=k\\in \\Z$, we have $\\Theta_b(u)=e_{21}(\\bm)^kue_{21}(\\bm)^{-k}$.\n\n\\begin{proposition}\nFor $b\\in\\C^*$, we have that\n\\begin{align*}\n\\Theta_b(e_{21}(\\bn))=&\\ e_{21}(\\bn),\\\\\n\\Theta_b(e_{12}(\\bn))=&\\ e_{12}(\\bn)+b\\big(q^{m_2n_1}e_{22}(\\bm+\\bn)-q^{m_1n_2}e_{11}(\\bm+\\bn)\\big)e_{21}(\\bm)^{-1}\\\\\n& -b(b-1)q^{m_2n_1+m_2m_1+m_1n_2}e_{21}(2\\bm+\\bn)e_{21}(\\bm)^{-2},\\\\\n\\Theta_b(e_{11}(\\bn))=&\\ e_{11}(\\bn)+bq^{m_2n_1}e_{21}(\\bm+\\bn)e_{21}(\\bm)^{-1},\\\\\n\\Theta_b(e_{22}(\\bn))=&\\ e_{22}(\\bn)-bq^{m_1n_2}e_{21}(\\bm+\\bn)e_{21}(\\bm)^{-1},\\\\\n\\Theta_b(D_1)=& D_1-bm_1,\\\\\n\\Theta_b(D_1)=& D_2-bm_2,\n\\end{align*}\nfor any $\\bn\\in\\Z^2$.\n\\end{proposition}\n\\begin{proof} First, from  $[e_{21}(\\bm),e_{21}(\\bn)]=0,$\nwe have that $$\\Theta_b(e_{21}(\\bn))=e_{21}(\\bn).$$\nNext, from $(\\text{ad}e_{21}(\\bm))^3(e_{12}(\\bn))=0$, we obtain that\n\\begin{align*}\n\\Theta_b(e_{12}(\\bn))=&\\ e_{12}(\\bn)+b[e_{21}(\\bm),e_{12}(\\bn)]e_{21}(\\bm)^{-1}\\\\\n&+\\frac{1}{2}b(b-1)[e_{21}(\\bm),[e_{21}(\\bm),e_{12}(\\bn)]]e_{21}(\\bm)^{-2}\\\\\n=&\\ e_{12}(\\bn)+b\\big(q^{m_2n_1}e_{22}(\\bm+\\bn)-q^{m_1n_2}e_{11}(\\bm+\\bn)\\big)e_{21}(\\bm)^{-1}\\\\\n&+\\frac{1}{2}b(b-1)[e_{21}(\\bm),[e_{21}(\\bm),e_{12}(\\bn)]]e_{21}(\\bm)^{-2}\\\\\n=&\\ e_{12}(\\bn)+b\\big(q^{m_2n_1}e_{22}(\\bm+\\bn)-q^{m_1n_2}e_{11}(\\bm+\\bn)\\big)e_{21}(\\bm)^{-1}\\\\\n&-b(b-1)q^{m_2n_1+m_2m_1+m_1n_2}e_{21}(2\\bm+\\bn)e_{21}(\\bm)^{-2}.\n\\end{align*}\nFinally, from $(\\text{ad}e_{21}(\\bm))^2(e_{11}(\\bn))=(\\text{ad}e_{21}(\\bm))^2(e_{22}(\\bn))=0$, we get that\n\\begin{align*}\n\\Theta_b(e_{11}(\\bn))=e_{11}(\\bn)+bq^{m_2n_1}e_{21}(\\bm+\\bn)e_{21}(\\bm)^{-1},\n\\end{align*}\n\\begin{align*}\n\\Theta_b(e_{22}(\\bn))=e_{22}(\\bn)-bq^{m_1n_2}e_{21}(\\bm+\\bn)e_{21}(\\bm)^{-1}.\n\\end{align*}\n\\end{proof}\n\nFor a $U^F$-module $M$, by $\\Phi^b_\\bm M$ we denote the $U^F$-module $M$ twisted by the action\n$$ u\\cdot v=\\Theta_b(u)v, $$ where $v\\in M, u\\in U^F$.\n\nIn what follows, for the $\\mg$-module $M=\\C[\\bx]$, we define $\\md_\\bm^bM=\\Phi^b_\\bm \\md_\\bm M$. Restricted to $U(\\mg)$, $\\md_\\bm^bM$\n becomes a $\\mg$-module, which is still denoted by $\\md_\\bm^bM$.\n\n\nFor $\\bm\\in \\Z^2$, set $\\mathcal{E}_\\bm=\\sum\\limits_{\\bs,\\br\\in \\bz^2}\nq^{-m_2r_1-s_2m_1+s_2r_1}x_{-\\bm+\\bs+\\br}\\p{\\x_\\bs}\\p{\\x_{\\br}}$. Then\n$$\\Phi(e_{12}(-\\bm))=q^{-m_1m_2}\\mu\\p{\\x_{\\bm}}-\\mathcal{E}_\\bm.$$\nThe  following lemma is a standard fact following from that $e_{12}(\\bn)$ is an ad-locally nilpotent element in $U(\\mg)$.\n\\begin{lemma} \\label{Nil}If $M$ is an irreducible $\\mg$-module, then the action of $e_{12}(\\bn)$ on $M$ is torsion free or locally  nilpotent, for any $\\bn\\in\\Z^2$.\n\\end{lemma}\n\nIn the module $\\md_\\bm M$, we have the following useful formulas.\n\n\\begin{lemma}\\label{3.5} Let  $v_0:=x_{\\bn_1}^{i_1}\\cdots x_{\\bn_k}^{i_k}$, where $\\bn_1,\\cdots,\\bn_k\\neq \\bm$, $i_1,\\cdots,i_k\\in\\N$. Then\n\\begin{itemize}\n\\item[(1).]$e_{12}(-\\bm)x_\\bm^{-i}v_0=-x_m^{-i} \\mathcal{E}_\\bm(v_0)+(\\sum\\limits_{j=1}^k2i_j-\\mu-i-1)iq^{-m_2m_1}x_\\bm^{-i-1}v_0$,\n   \\item[(2).]$e_{12}(-\\bm)x_\\bm^{-i}=(-\\mu-i-1)iq^{-m_2m_1}x_\\bm^{-i-1}$,\n   \\item[(3).]$\\mathcal{E}_\\bm^{d+1} v_0=0$, where $d=i_1+\\cdots+i_k$.\n  \n\\end{itemize}\n\\end{lemma}\n\\begin{proof}For $\\bn\\in\\Z^2$, we can compute that\n\\begin{align*}& e_{12}(-\\bn)x_\\bm^{-i}x_{\\bn_1}^{i_1}\\cdots x_{\\bn_k}^{i_k}\\\\\n=&(q^{-n_1n_2}\\mu\\p{\\x_{\\bn}}\n-\\sum_{\\bs,\\br\\in \\bz^2}\nq^{-n_2r_1-s_2n_1+s_2r_1}x_{-\\bn+\\bs+\\br}\\p{\\x_\\bs}\\p{\\x_{\\br}})x_\\bm^{-i}x_{\\bn_1}^{i_1}\\cdots x_{\\bn_k}^{i_k}\\\\\n=&-i\\delta_{\\bn,\\bm}q^{-n_1n_2}\\mu x_\\bm^{-i-1}x_{\\bn_1}^{i_1}\\cdots x_{\\bn_k}^{i_k}+x_\\bm^{-i}q^{-n_1n_2}\\mu\\p{\\x_{\\bn}}(x_{\\bn_1}^{i_1}\\cdots  x_{\\bn_k}^{i_k})\\\\\n&+\\sum_{\\bs\\in \\bz^2}iq^{-n_2m_1-s_2n_1+s_2m_1}x_{-\\bn+\\bs+\\bm}\\p{\\x_\\bs}x_\\bm^{-i-1}x_{\\bn_1}^{i_1}\\cdots x_{\\bn_k}^{i_k}\\\\\n& -\\sum_{j=1}^k\\sum_{\\bs\\in \\bz^2}q^{-n_2n_{j1}-s_2n_1+s_2n_{j1}}i_jx_{-\\bn+\\bs+\\bn_j}\\p{\\x_\\bs}x_\\bm^{-i}x_{\\bn_1}^{i_1}\\cdots x_{\\bn_j}^{i_j-1}\\cdots x_{\\bn_k}^{i_k}\\\\\n=&-x_\\bm^{-i-1}i\\delta_{\\bn,\\bm}q^{-n_1n_2}\\mu x_{\\bn_1}^{i_1}\\cdots x_{\\bn_k}^{i_k}\\\\\n&-x_\\bm^{-i-2}i(i+1)q^{-n_2m_1-m_2n_1+m_2m_1}x_{-\\bn+2\\bm}x_{\\bn_1}^{i_1}\\cdots x_{\\bn_k}^{i_k}\\\\\n& +x_\\bm^{-i-1}\\sum_{j=1}^ki_jiq^{-n_2m_1-n_{j2}n_1+n_{j2}m_1}x_{-\\bn+\\bn_j+\\bm}x_{\\bn_1}^{i_1}\\cdots x_{\\bn_j}^{i_j-1}\\cdots x_{\\bn_k}^{i_k}\\\\\n&+x_\\bm^{-i-1}\\sum_{j=1}^kii_jq^{-n_2n_{j1}-m_2n_1+m_2n_{j1}}x_{-\\bn+\\bm+\\bn_j}x_{\\bn_1}^{i_1}\\cdots x_{\\bn_j}^{i_j-1}\\cdots x_{\\bn_k}^{i_k}\\\\\n& +x_\\bm^{-i}e_{12}(-\\bn)(x_{\\bn_1}^{i_1}\\cdots x_{\\bn_k}^{i_k}),\n\\end{align*}\nTake $\\bn=\\bm$ in the above computation, (1) follows. (2), (3) are clear.\n\\end{proof}\n\n\\subsection{The simplicity for the case $b\\in\\Z$}\n\nLet $\\C[x_\\bm^{-1}, \\bx, \\hat x_\\bm]$ be the polynomial ring in the variables $x_{\\bm}^{-1}, x_{\\bn}, \\bn\\neq \\bm$.\n\\begin{lemma} \\label{Generator}Let $\\mu\\notin \\Z$. Then\n the $\\mg$-module $\\md_\\bm M\/M$  is generated by $x_\\bm^{-1}$.\n\\end{lemma}\n\\begin{proof} As vector spaces $\\md_\\bm M\/M\\cong x_\\bm^{-1}\\C[x_\\bm^{-1}, \\bx, \\hat x_\\bm]$. For any $i\\in\\N$, we have that\n\\begin{align*}e_{12}(-\\bm)^ix_\\bm^{-1}=& q^{-im_1m_2}(\\mu\\p{\\x_\\bm}-x_\\bm\\p{\\x_\\bm}\\p{\\x_\\bm})^ix_\\bm^{-1}\\\\\n=& (-1)^iq^{-im_1m_2}i!(\\mu+2)\\cdots (\\mu+1+i)x_\\bm^{-1-i}.\n\\end{align*}\nThe condition that $\\mu\\notin \\Z$ forces that $x_\\bm^{-1-i}$ can be generated by $x_\\bm^{-1}$.\nThen from  $ e_{21}(\\bn)x_\\bm^{-1-i}=x_\\bm^{-1-i}x_\\bn$ for any $\\bn\\in\\Z^2$ with $\\bn\\neq \\bm$, we see that $x_\\bm^{-1}$ can generate  $\\md_\\bm M\/M$.\n\\end{proof}\n\\begin{theorem}\\label{Maintheorem1}Let $\\bm\\in \\Z^2, b\\in \\C$.\n If $b\\in \\Z$, then $\\md_\\bm^bM\\cong \\md_\\bm M$ and $\\md_\\bm M\/M$ is irreducible if and only if $\\mu \\notin \\Z$.\n\\end{theorem}\n\\begin{proof}In the case $b=k\\in \\Z$, define the following linear map\n$$ \\rho: \\md_\\bm M\\rightarrow \\md_\\bm^bM; v\\mapsto e_{21}(\\bm)^k v ,$$ which is clearly a bijection.\nFrom $$\\rho(uv)=e_{21}(\\bm)^k uv= e_{21}(\\bm)^ku e_{21}(\\bm)^{-k}e_{21}(\\bm)^k v = u\\cdot \\rho(v),$$ where $u\\in U^F, v\\in \\md_\\bm M$,\nwe see that $ \\rho$ is a $\\mg$-module isomorphism.\n\n($\\Rightarrow$) Let $\\mu\\in\\Z$.\n\n{\\bf Claim 1:} There exists a  nonzero  $w\\in \\md_\\bm M\/M$ such that $e_{12}(-\\bm)\\cdot w=0$.\n\nChoose a positive integer  $d$ such that $-d+\\mu+1< 0$.  Choose $\\bn\\in\\Z^2$ such that $n_1m_2-n_2m_1\\neq 0$. Let  $l=2d-\\mu-1$ and\n $a_j=q^{-m_1m_2}(\\mu+j+l+1-2d)(j-l)$ which is nonzero by the choice of $d$, for $j: 1\\leq j\\leq d$.\nSet $w_j=\\frac{1}{a_1\\cdots a_j}\\mathcal{E}_\\bm^{j} (x_{\\bn}^d)$ for $j: 0\\leq j\\leq d+1$. Note that  $w_0=x_{\\bn}^d, w_{d+1}=0$, $\\mathcal{E}_\\bm w_j=a_{j+1}w_{j+1}$,\nand $\\sum\\limits_{\\br\\in \\bz^2}x_\\br\\p{\\x_\\br}w_j=(d-j)w_j$, for any $0\\leq j\\leq d-1$. By the choice of $\\bn$, we see that $\\p{\\x_{\\bm}}(w_j)=0$ for $j: 0\\leq j\\leq d$.\n\nLet $w=\\sum_{j=0}^d x^{j-l}_\\bm w_j$. We will show that $w$ is annihilated by\n$ e_{12}(-\\bm)$.\n\nFirstly, we have\n\\begin{align*}\\mathcal{E}_\\bm x^{j-l}_\\bm w_j=&(\\mathcal{E}_\\bm x^{j-l}_m )w_j+  x^{j-l}_\\bm (\\mathcal{E}_\\bm w_j)\\\\\n&+\\sum_{\\bs,\\br\\in \\bz^2}\nq^{-m_2r_1-s_2m_1+s_2r_1}x_{-\\bm+\\bs+\\br}(\\p{\\x_\\bs}x^{j-l}_m)(\\p{\\x_{\\br}}w_j)\\\\\n&+\\sum_{\\bs,\\br\\in \\bz^2}\nq^{-m_2r_1-s_2m_1+s_2r_1}x_{-\\bm+\\bs+\\br}(\\p{\\x_\\br}x^{j-l}_\\bm)(\\p{\\x_{\\bs}}w_j)\\\\\n=& q^{-m_1m_2}(j-l)(j-l-1)x^{j-l-1}_\\bm w_j+a_{j+1}x^{j-l}_\\bm w_{j+1}\\\\\n& +2\\sum_{\\br\\in \\bz^2}q^{-m_2m_1}(j-l)x_\\br x^{j-l-1}_\\bm(\\p{\\x_{\\br}} w_j)\\\\\n=& q^{-m_1m_2}(j-l)(j-l-1)x^{j-l-1}_\\bm w_j+a_{j+1}x^{j-l}_\\bm w_{j+1}\\\\\n& +2(d-j)q^{-m_2m_1}(j-l)x^{j-l-1}_\\bm w_j\\\\\n=& q^{-m_1m_2}(j-l)(-j-l-1+2d)x^{j-l-1}_\\bm w_j+a_{j+1}x^{j-l}_\\bm w_{j+1}.\n\\end{align*}\n\nConsequently,\n\\begin{align*}& e_{12}(-\\bm)w =\\sum_{j=0}^d(q^{-m_1m_2}\\mu\\p{\\x_{\\bm}}-\\mathcal{E}_\\bm) x^{j-l}_\\bm w_j\\\\\n=& \\sum_{j=0}^d q^{-m_1m_2}\\mu\\big((j-l)x^{j-l-1}_\\bm w_j+  x^{j-l}_\\bm( \\p{\\x_{\\bm}} w_j)\\big)\\\\\n& -\\sum_{j=0}^d   q^{-m_1m_2}(j-l)(-j-l-1+2d)x^{j-l-1}_\\bm w_j-\\sum_{j=0}^d a_{j+1}x^{j-l}_\\bm w_{j+1}\\\\\n=& \\sum_{j=0}^d q^{-m_1m_2}(\\mu+j+l+1-2d)(j-l)x^{j-l-1}_\\bm w_j -\\sum_{j=1}^d  a_{j}x^{j-l-1}_\\bm w_{j}\\\\\n=& q^{-m_1m_2}(\\mu+l+1-2d)(-l)x^{-l-1}_\\bm w_0=0,\n\\end{align*}\nin the third equality, we have used the fact that $ \\p{\\x_{\\bm}}(w_j)=0$.\n\nIf $\\md_\\bm M\/M$ is an irreducible $\\mg$-module, then  by Lemma \\ref{Nil}, $ e_{12}(-\\bm)$ acts locally nilpotently on $\\md_\\bm M\/M$. On the other hand,\nfor a enough large   integer $j$,  by (3) in Lemma \\ref{3.5}, we have\n$$e_{12}(-\\bm)^ix_\\bm^{-j}=(-1)^iq^{-im_1m_2}j\\cdots (j+i-1)(\\mu+j+1)\\cdots (\\mu+j+i)x_\\bm^{-j-i}$$ which is a nonzero element in\n$\\md_\\bm M\/M$, for any $i\\in\\N$. This is a contradiction. So  $\\md_\\bm M\/M$ is reducible when $\\mu\\in \\Z$.\n\n($\\Leftarrow$): Let $v$ be any nonzero element in $\\md_\\bm M\/M$.  By Lemma \\ref{Generator}, it suffices to show that there exists $u\\in U(\\mg)$ such\nthat $uv=x_\\bm^{-1}$.\n\nAfter applying some power of  $e_{21}(\\bm)$ to $v$ if necessary, we may assume that $v=x_{\\bm}^{-1}f$, where $f$ is a homogeneous polynomial in $\\C[\\bx, \\hat x_\\bm]$ of degree $d$.\n\n\\noindent{\\bf Claim 2:} For such $v$, there exists $u\\in U(\\mg)$ such that $uv=x_{\\bm}^{-1}f_1\\neq 0$, where $f_1$ is a homogeneous polynomial in $\\C[\\bx, \\hat x_\\bm]$ whose  degree is smaller than $d$.\n\nFirst, we consider the case that $v$ is a monomial, i.e., $v=x_\\bm^{-1}x_{\\bn_1}^{i_1}\\cdots x_{\\bn_k}^{i_k}$. From the proof of Lemma \\ref{3.5},\nwe have that\n\\begin{align*}& e_{12}(-\\bn_1)x_\\bm^{-1}x_{\\bn_1}^{i_1}\\cdots x_{\\bn_k}^{i_k}\\\\\n=&-x_\\bm^{-3}2q^{-n_{12}m_1-m_2n_{11}+m_2m_1}x_{-\\bn_1+2\\bm}x_{\\bn_1}^{i_1}\\cdots x_{\\bn_k}^{i_k}\\\\\n& +x_\\bm^{-2}\\sum_{j=1}^ki_jq^{-n_{12}m_1-n_{j2}n_{11}+n_{j2}m_1}x_{-\\bn_1+\\bn_j+\\bm}x_{\\bn_1}^{i_1}\\cdots x_{\\bn_j}^{i_j-1}\\cdots x_{\\bn_k}^{i_k}\\\\\n& +x_\\bm^{-2}\\sum_{j=1}^ki_jq^{-n_{12}n_{j1}-m_2n_{11}+m_2n_{j1}}x_{-\\bn_1+n_j+\\bm}x_{\\bn_1}^{i_1}\\cdots x_{\\bn_j}^{i_j-1}\\cdots x_{\\bn_k}^{i_k}\\\\\n& +x_\\bm^{-1}e_{12}(-\\bn_1)(x_{\\bn_1}^{i_1}\\cdots x_{\\bn_k}^{i_k})\\\\\n =& x_\\bm^{-3}a_{-3}+x_\\bm^{-2}a_{-2}+x_\\bm^{-1}a_{-1},\n\\end{align*}\nwhere\n\\begin{align*}a_{-1}=&2i_1q^{-n_{11}n_{12}}x_{\\bn_1}^{-1}x_{\\bn_1}^{i_1}\\cdots x_{\\bn_k}^{i_k}+ e_{12}(-\\bn_1)(x_{\\bn_1}^{i_1}\\cdots x_{\\bn_k}^{i_k})\\\\\n=& i_1(3+\\mu-i_1-2i_2\\cdots-2i_k)q^{-n_{11}n_{12}}x_{\\bn_1}^{-1}x_{\\bn_1}^{i_1}\\cdots x_{\\bn_k}^{i_k}\\\\\n& + \\sum_{j=2}^k\\sum_{l=2}^k i_l (i_j-\\delta_{j,l})q^{-n_{12}n_{j1}-n_{l2}n_{11}+n_{l2}n_{j1}}x_{-\\bn_1+\\bn_l+\\bn_j}x_{\\bn_j}^{-1}x_{\\bn_l}^{-1}x_{\\bn_1}^{i_1}\\cdots x_{\\bn_k}^{i_k},\\\\\na_{-2}=&\\sum_{j=2}^ki_j(q^{-n_{12}m_1-n_{j2}n_{11}+n_{j2}m_1}+q^{-n_{12}n_{j1}-m_2n_{11}+m_2n_{j1}})\\\\\n& \\times x_{-\\bn_1+\\bn_j+\\bm}x_{\\bn_1}^{i_1}\\cdots x_{\\bn_j}^{i_j-1}\\cdots x_{\\bn_k}^{i_k},\\\\\na_{-3}=& -2q^{-n_{12}m_1-m_2n_{11}+m_2m_1}x_{-\\bn_1+2\\bm}x_{\\bn_1}^{i_1}\\cdots x_{\\bn_k}^{i_k}.\n\\end{align*}\n\nLet $A=-2(\\sum_{j=1}^k2i_j-\\mu-1)q^{-m_2m_1}$. By Lemma \\ref{3.5}, we have\n\\begin{align*}\n&(e_{12}(-\\bm)e_{21}(\\bm)+A)e_{12}(-\\bn_1)x_\\bm^{-1}x_{\\bn_1}^{i_1}\\cdots x_{\\bn_k}^{i_k}\\\\\n=&2(\\sum_{j=1}^k2i_j-\\mu-1)q^{-m_1m_2}x_\\bm^{-3}a_{-3}-x_\\bm^{-2}\\mathcal{E}_\\bm(a_{-3})\\\\\n & + x_\\bm^{-2}(\\sum_{j=1}^k2i_j-\\mu-2)q^{-m_2m_1}a_{-2}-x_\\bm^{-1}\\mathcal{E}_\\bm(a_{-2})\\\\\n & + Ax_\\bm^{-3}a_{-3}+Ax_\\bm^{-2}a_{-2}+Ax_\\bm^{-1}a_{-1}\\\\\n=&  x_\\bm^{-2}b_{-2}+x_\\bm^{-1}b_{-1},\n\\end{align*}\nwhere\n\\begin{align*}\nb_{-2}\n= & -\\mathcal{E}_\\bm(a_{-3})+ (\\sum_{j=1}^k2i_j-\\mu-2)q^{-m_2m_1}a_{-2} +Aa_{-2},\\\\\nb_{-1}\n= &-\\mathcal{E}_\\bm(a_{-2})+Aa_{-1}.\n\\end{align*}\n\nLet $ B=-(\\sum_{j=1}^k2i_j-\\mu-2)q^{-m_1m_2}$. Then  we have that\n\\begin{align*}\n&(e_{12}(-\\bm)e_{21}(\\bm)+B)(e_{12}(-\\bm)e_{21}(\\bm)+A)e_{12}(-\\bn_1)x_\\bm^{-1}x_{\\bn_1}^{i_1}\\cdots x_{\\bn_k}^{i_k}\\\\\n=&  (e_{12}(-\\bm)e_{21}(\\bm)+B)(x_\\bm^{-2}b_{-2}+x_\\bm^{-1}b_{-1})\\\\\n=&e_{12}(-\\bm)x_\\bm^{-1}b_{-2}+Bx_\\bm^{-2}b_{-2}+Bx_\\bm^{-1}b_{-1}\\\\\n=& -x_\\bm^{-1}\\mathcal{E}_\\bm(b_{-2})+(\\sum_{j=1}^k2i_j-\\mu-2)q^{-m_1m_2}x_\\bm^{-2}b_{-2}+Bx_\\bm^{-2}b_{-2}+Bx_\\bm^{-1}b_{-1}\\\\\n=& x_\\bm^{-1}(Bb_{-1}-\\mathcal{E}_\\bm(b_{-2}))+((\\sum_{j=1}^k2i_j-\\mu-2)q^{-m_1m_2}+B)x_\\bm^{-2}b_{-2}\\\\\n=& x_\\bm^{-1}(Bb_{-1}-\\mathcal{E}_\\bm(b_{-2})),\n\\end{align*}\nwhere \\begin{align*}Bb_{-1}-\\mathcal{E}_\\bm(b_{-2})=& -B\\mathcal{E}_\\bm(a_{-2})+BAa_{-1}+ \\mathcal{E}_\\bm^2(a_{-3})\\\\\n & -(\\sum_{j=1}^k2i_j-\\mu-2)q^{-m_2m_1}\\mathcal{E}_\\bm(a_{-2}) -A\\mathcal{E}_\\bm(a_{-2})\\\\\n =&-A\\mathcal{E}_\\bm(a_{-2})+BAa_{-1}+ \\mathcal{E}_\\bm^2(a_{-3}).\n\\end{align*}\n\nWe can see that the coefficients of $x_{\\bn_1}^{-1}x_{\\bn_1}^{i_1}\\cdots x_{\\bn_k}^{i_k}$ in $\\mathcal{E}_\\bm(a_{-2})$ and $\\mathcal{E}_\\bm^2(a_{-3})$ are zero.\nThus the coefficient of of $x_{\\bn_1}^{-1}x_{\\bn_1}^{i_1}\\cdots x_{\\bn_k}^{i_k}$ in $Bb_{-1}-\\mathcal{E}_\\bm(b_{-2})$ is\n$ABi_1(3+\\mu-i_1-2i_2\\cdots-2i_k)q^{-n_{11}n_{12}}$ which is nonzero,\nsince $\\mu\\not\\in\\Z$, $A\\neq 0, B\\neq 0$. Hence $Bb_{-1}-\\mathcal{E}_\\bm(b_{-2})\\neq 0$.  Therefore, the claim 2 is true in this case.\n\n\nFor arbitrary nonzero $v\\in\\md_\\bm M\/M$, by the action of $e_{21}(\\bm)$, we assume that\n$$v=x_{\\bm}^{-1}\\sum\\limits_{j_1,\\dots,j_k\\in\\Z_+}a_{j_1,\\dots,j_k}x_{\\bn_1}^{j_1}\\cdots x_{\\bn_k}^{j_k},$$ where  $x_{\\bn_1}^{j_1}\\cdots x_{\\bn_k}^{j_k}\\in\\C[\\bx, \\hat x_\\bm]$,\n$j_1+\\cdots+j_k=d$. Suppose that the maximal degree of $x_{\\bn_1}$ in $v$ is $i_1$. By the above arguments, we see that\nthe coefficient of $x_{\\bm}^{-1}x_{\\bn_1}^{i_1-1}\\sum\\limits_{j_2+\\dots + j_k =d-i_1}a_{i_1,\\dots,j_k}x_{\\bn_2}^{j_2}\\cdots x_{\\bn_k}^{j_k}$ in  $(e_{12}(-\\bm)e_{21}(\\bm)+B)(e_{12}(-\\bm)e_{21}(\\bm)+A)e_{12}(-\\bn_1)v$\nis $ABi_1(3+\\mu+i_1-2d)q^{-n_{11}n_{12}}$ which is nonzero. So the claim 2 is true in general.\n\nBy induction on the degree of $f$ in $v=x_{\\bm}^{-1}f$, we can complete the proof.\n\n\\end{proof}\n\\subsection{The case $b\\not\\in\\Z$}\n\n\\begin{theorem}\\label{maintheorem2}Let $m\\in \\Z^2, b\\in \\C$.\nIf $b\\notin \\Z$, then  $\\md_\\bm^bM$ is irreducible if and only if $b+\\mu \\notin \\Z$.\n\\end{theorem}\n\\begin{proof}($\\Rightarrow$): Let $\\mu+b\\in\\Z$.\n\nFrom \\begin{align*}e_{12}(-\\bm)\\cdot x_\\bm^{j}\n=& \\Big(q^{-m_1m_2}\\mu\\p{\\x_{\\bm}}\n-\\sum_{\\bs,\\br\\in \\bz^2}\nq^{-m_2r_1-s_2m_1+s_2r_1}x_{-\\bm+\\bs+\\br}\\p{\\x_\\bs}\\p{\\x_{\\br}}\\\\\n& +2b\\sum_{\\bs\\in \\bz^2} q^{-m_1m_2}x_{\\bs}\\p{\\x_\\bs}x_\\bm^{-1}-b(b+\\mu-1)q^{-m_1m_2} x_\\bm^{-1}\\Big) x_\\bm^{j}\\\\\n=& -(j-b)(j-b-1-\\mu)q^{-m_1m_2}x_\\bm^{j-1},\n\\end{align*}\nwe see that $e_{12}(-\\bm)x_\\bm^{b+1+\\mu}=0$ and $e_{12}(-\\bm)x_\\bm^{j}\\neq 0$ for any $j<b+1+\\mu$.\nBy Lemma \\ref{Nil}, $\\md_\\bm^bM$ is reducible.\n\n\n($\\Leftarrow$): Suppose that $b+\\mu \\notin \\Z$.  Let $V$ be nonzero submodule of $\\md_\\bm^bM$.\n\n\\noindent{\\bf Claim :} There is an integer $p$ and some nonzero $w\\in V$ such that $$\\Big(e_{21}(\\bm)e_{12}(-\\bm)-(b-p)(2d-\\mu-b-p-1)\\Big)w=a x_\\bm^{d}$$ for some $a\\in\\C$, where\n$d$ is a positive integer.\n\nLet $v\\in V$ be a nonzero homogeneous polynomial in $\\C[x_\\bm^{-1}, \\bx, \\hat x_\\bm]$ of degree $d$. We assume that\n$v=\\sum\\limits_{z\\leq i\\leq l}x_{\\bm}^i f_i$, where each $f_{i}$ is a homogeneous polynomial in $\\C[\\bx, \\hat x_\\bm]$ of degree $d-i$.\nWe define $d-z$ as the ${\\hat x_\\bm}$-degree of $v$. Note that $(e_{11}-e_{22})\\cdot v=(\\mu-2d+2b)v $.\n\nFrom \\begin{align*}&e_{21}(\\bm)\\cdot e_{12}(-\\bm)\\cdot x_\\bm^{-i}x_{\\bn_1}^{i_1}\\cdots x_{\\bn_k}^{i_k}\\\\\n=& e_{21}(\\bm)\\cdot \\Big(-x_\\bm^{-i} \\mathcal{E}_\\bm(x_{\\bn_1}^{i_1}\\cdots x_{\\bn_k}^{i_k})+(\\sum\\limits_{j=1}^k2i_j-\\mu-i-1)iq^{-m_2m_1}x_\\bm^{-i-1}x_{\\bn_1}^{i_1}\\cdots x_{\\bn_k}^{i_k}\\\\\n & +2b\\sum_{\\bs\\in \\bz^2} q^{-m_1m_2}x_{\\bs}\\p{\\x_\\bs}(x_\\bm^{-i-1}x_{\\bn_1}^{i_1}\\cdots x_{\\bn_k}^{i_k})-b(b+\\mu-1)q^{-m_1m_2} x_\\bm^{-i-1}x_{\\bn_1}^{i_1}\\cdots x_{\\bn_k}^{i_k}\\Big)\\\\\n=& -x_\\bm^{-i+1}\\mathcal{E}_\\bm(x_{\\bn_1}^{i_1}\\cdots x_{\\bn_k}^{i_k})+(i+b)(\\sum\\limits_{j=1}^k2i_j-b-\\mu-i-1)q^{-m_1m_2} x_\\bm^{-i}x_{\\bn_1}^{i_1}\\cdots x_{\\bn_k}^{i_k}\n\\end{align*}\nwe see that $\\Big(e_{21}(\\bm)e_{12}(-\\bm)-(b-z)(2d-\\mu-b-z-1)\\Big)\\cdot v$ has smaller ${\\hat x_\\bm}$-degree than $v$. By induction  on the ${\\hat x_\\bm}$-degree of $v$,\nwe know that the above claim is true.\n\nIf $a\\neq 0$, from that  $e_{12}(-\\bm)\\cdot x_\\bm^{d}$ is a nonzero multiple of $ x_\\bm^{d-1}$ and $e_{21}(\\bm)\\cdot x_\\bm^{d}=x_\\bm^{d+1}$, we have that $\\C[x_\\bm]\\subset V$. Consequently,  $V=\\md_\\bm^bM$.\n\nIf $a=0$, then $e_{12}(-\\bm)\\cdot w= (b-p)(2d-\\mu-b-p-1)e_{21}(\\bm)^{-1}w$.\n\nLet $B_i=(i+b-p)(2d-b-\\mu-1-i-p)q^{-m_1m_2}$ which is nonzero for any $i\\in \\Z_+$.\nUsing  the following formula:\n$$e_{21}(\\bm)^{-1}e_{12}(-\\bm)-e_{12}(-\\bm)e_{21}(\\bm)^{-1}=q^{-m_1m_2}e_{21}(\\bm)^{-1}(e_{11}-e_{22})e_{21}(\\bm)^{-1},$$ we obtain that\n\\begin{align*} &e_{12}(-\\bm)\\cdot e_{21}(\\bm)^{-k} \\cdot w\\\\\n=& \\sum_{i=0}^{k-1}e_{21}(\\bm)^{-i}\\cdot[e_{12}(-\\bm),e_{21}(\\bm)^{-1}]\\cdot e_{21}(\\bm)^{-(k-i-1)} \\cdot w+ B_0e_{21}(\\bm)^{-k-1}\\cdot w\\\\\n=& -q^{-m_1m_2}\\sum_{i=0}^{k-1}e_{21}(\\bm)^{-i-1}\\cdot (e_{11}-e_{22})\\cdot e_{21}(\\bm)^{-(k-i)}\\cdot w+ B_0e_{21}(\\bm)^{-k-1}\\cdot w\\\\\n=&\\Big(B_0- q^{-m_1m_2}\\sum_{i=0}^{k-1}(\\mu-2d+2b+2(k-i))\\Big) e_{21}(\\bm)^{-k-1} \\cdot w\\\\\n=&\\Big(B_0- q^{-m_1m_2}k(\\mu-2d+2b+k+1)\\Big)e_{21}(\\bm)^{-k-1}\\cdot w\\\\\n=& B_ke_{21}(\\bm)^{-k-1}\\cdot w.\n\\end{align*}\nBy induction on $k$, we have that\n\\begin{align*} e_{12}(-\\bm)^k \\cdot w= B_0B_1\\cdots B_{k-1}e_{21}(\\bm)^{-k}w.\\end{align*}\nMore precisely \\begin{align*} e_{12}(-\\bm)^{k+1} \\cdot w= &B_0B_1\\cdots B_{k-1}e_{12}(-\\bm)\\cdot e_{21}(\\bm)^{-k}\\cdot w\\\\\n=&B_0B_1\\cdots B_{k-1}B_ke_{21}(\\bm)^{-k-1} \\cdot w.\n\\end{align*}\n\nFrom the action of $e_{21}(\\bm)$, we can assume that $w\\in \\C[\\bx]$. By the simplicity of the $\\mg$-module $\\C[\\bx]$, there a $u_1\\in U(\\mg)$ such that\n$u_1 w=1$. Since $\\Theta_b$ is an automorphism of $U^F$, there are  $u_2\\in U(\\mg)$ and $k\\in \\Z_+$ such that $ \\Theta(u_2 e_{21}(\\bm)^{-k})=u_1$.\n\nThen $$1=u_1 w=\\Theta(u_2 e_{21}(\\bm)^{-k})w=u_2 \\cdot e_{21}(\\bm)^{-k}\\cdot w=C u_2  \\cdot e_{12}(-\\bm)^{k}\\cdot w\\in V,$$\nwhere $C=\\frac{1}{B_0B_1\\cdots B_{k-1}}$. Since $\\md_\\bm^bM$ is generated by $1$, $V=\\md_\\bm^bM$. Now we complete the proof.\n\\end{proof}\n\n\n\\section*{Acknowledgment}\nG.L. is partially supported by NSF of China\n(Grant 11301143, 11771122) and the grant at Henan University (yqpy20140044).\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe problem of the pair creation in the external field has been investigated since historical works of Schwinger \\cite{schwinger} and Euler and Heisenberg \\cite{euler}. However, this process hasn't been observed experimentally due to huge strengths of fields needed. The pair production rate is exponentially\nsuppressed when the strength of the electrical field $E$ is lower than $\\frac {m_e^2}e$, which is much larger than any available one. The natural way to observe the pair creation in weaker fields is to consider the pair creation induced by external particles, for example by hard photons in the field of high-energy lasers \\cite{schutzhold}, \\cite{dunne}. This phenomenon is referred to as induced Schwinger process.\n\nThere are various types of Schwinger-like processes. They all are closely related to the decay of false vacuum. The decay of the false vacuum in case of scalar fields was studied in \\cite{okun}, \\cite{callan}, \\cite{coleman}, \\cite{SelivanovVoloshin},\n\\cite{Kiselev}. These results can be generalized to the case of electron-positron pair creation \\cite{electron}. Monopole decay in constant electrical field appears to be quite similar to induced Schwinger process \\cite{gorsky},\n\\cite{monin_lagr}, \\cite{zayakin}, \\cite{mo-za}. Schwinger type processes also arise in the weak coupling limit of the string theory \\cite{gorsky}, \\cite{gorsky2}.\n\nIn this paper we consider the Schwinger process induced by several external particles in two-dimensional electrodynamics and two-dimensional scalar theory. We mostly use the techniques developed in \\cite{scalar}, \\cite{electron}. First we consider the process induced by two external particles. To find the rate of the process we calculate the imaginary part of the four-fold correlator in the momentum representation and see that the result can be generalized to the case of arbitrary number of external particles. Then we use this result to compute the correction to the action caused by the interaction of the wall of the bounce with itself. We see that the consideration of the Coulomb and Yukawa interaction changes the radius of the bounce.\n\n\\section{Cross-section of the process enhanced by two particles}\n\\subsection{Scalar field}\nIn this paper we focus on two examples of the two-dimensional interacting field theories which allow the processes of the false vacuum decay, namely the theory of two scalar fields, one of which is in quartic potential, and electrodynamics in presence of constant electric field. Both theories are considered in the limit of the thin-wall approximation, so that the wall of the bounce can be represented as a $\\delta$-function. The main goal of the analysis is to study the scattering of the particles on the background of the bounce of the false vacuum in the semiclassical limit. We claim that the interaction with external particles significantly changes the rate of the process even when the momenta of the external particles are small.\n\nFirst we consider the decay of the false vacuum in the scalar theory and reproduce the result of \\cite{scalar}. In that work the cross-section of the scattering of the scalar particles at the background of the bubble of the false vacuum was computed. The authors have used the thermal bath approach. We will calculate the same cross-section directly and show that this calculation can be straightforwardly generalized to any number of interacting particles.\n\nWe consider the theory of two interacting scalar fields, namely the free massive field $\\chi$ and the field $\\phi$ in the quartic potential. In the following we study the scattering of the $\\chi$-particles.\n\n\\begin{equation}\n\\label{lagrangian}\n\\mathcal L=\\frac 12 \\left(\\partial_\\mu \\phi\\right)^2+\\frac 12 \\left(\\partial_\\mu\n\\chi\\right)^2-\\frac 12 m^2 \\chi^2 -V(\\phi)- V_{int}(\\phi, \\chi)=\\mathcal L_\\chi+\\mathcal L_\\phi+\\mathcal L_{int}.\n\\end{equation}\n\nThe quartic potential $V(\\phi)$ has two minima which are referred to as the \"false\" and the \"true\" vacua,\n\n\\begin{equation}\nV(\\phi)=\\frac 14 \\lambda \\left(\\phi^2-\\nu^2\\right)^2-\\frac{\\varepsilon}{2\\nu}\\left(\\phi+\\nu\\right).\n\\end{equation}\n\nWe consider the mass of the $\\chi$ field to be much less than the mass of the $\\phi$ field and the wall of the bounce to be thin compared to the radius,\n\n\\begin{equation}\n\\label{thin-wall}\nm \\ll M=\\nu\\sqrt{\\lambda}, \\qquad R \\gg 1\/M.\n\\end{equation}\n\nIn the absence of the $\\chi$ field the action of the bounce can be divided into the internal part and the boundary part,\n\n\\begin{equation}\n\\label{action_0}\n S= S_{wall}+S_{int}=\\mu L - \\varepsilon A, \\qquad \\mu = \\nu^3 \\frac{\\sqrt{8\\lambda}}{3}.\n\\end{equation}\n\nHere $L$ is the length of the wall of the bounce and $A$ is the area of the bounce. The extremum of the (\\ref{action_0}) is reached when the bounce is circular and its radius is given by:\n\n\\begin{equation}\nR=\\frac{\\mu}{\\varepsilon},\n\\end{equation}\n\nand the extremized action becomes:\n\n\\begin{equation}\nS_0=\\frac {\\pi \\mu^2} \\varepsilon.\n\\end{equation}\n\nThe rate of the spontaneous decay can be expressed in terms of the path integral \\cite{voloshin},\n\n\\begin{equation}\n\\frac \\Gamma L=\\frac 2{LT} \\Im \\int \\mathcal D \\phi e^{-S[\\phi]}=\\frac \\varepsilon {2\\pi} \\exp\\left(-\\frac {\\pi \\mu^2}\\varepsilon\\right).\n\\end{equation}\n\n\nWe consider the interaction term in (\\ref{lagrangian}) to be linear on $\\chi$,\n\n\\begin{equation}\n\\label{interaction}\nV_{int}=g \\rho(\\phi) \\chi.\n\\end{equation}\n\nThe exact form of the function $\\rho(\\phi)$ is not very important, but it must significantly differ from zero only when the field $\\phi$ is far from vacuum. The interaction is non-zero at the boundary of the bounce, so the function $\\rho$ has the shape of the $\\delta$-function of the radial coordinate,\n\n\\begin{equation}\n\\label{rho_def}\n\\rho^B({\\bf r})=\\frac g{2\\pi R}\\delta\\left(r-R\\right).\n\\end{equation}\n\nTo find the rate of the process induced by two $\\chi$ particles we consider the four-fold correlator,\n\n\\begin{equation}\n\\Im \\int \\mathcal D \\chi \\mathcal D \\phi \\frac {\\delta^4}{\\delta \\chi^4} \\exp \\left( -S[\\phi,\\chi]\\right)=\\Im \\int \\mathcal D \\phi \\mathcal D \\chi \\rho^4(\\phi)e^{-S[\\phi,\\chi]},\n\\end{equation}\n\nwhich in the leading semiclassical approximation is given by\n\n\\begin{multline}\n\\label{Pi}\n\\Im \\Pi^{(4)}({\\bf r_1},{\\bf r_2},{\\bf r_3})=\\Im \\int \\mathcal Dr \\rho^B({\\bf r_1})  \\rho^B({\\bf r_2})  \\rho^B({\\bf r_3})  \\rho^B(0,0) e^{-S[{\\bf r}]}=\\\\\n\\frac \\Gamma {2L} \\int  \\rho^B({\\bf r_1}-{\\bf r})  \\rho^B({\\bf r_2}-{\\bf r})  \\rho^B({\\bf r_3}-{\\bf r})  \\rho^B({\\bf r}) d^2{\\bf r}.\n\\end{multline}\n\nGeometrically the support of each density $\\rho^B$ is a circle with fixed radius. In the first quantized picture the centers of the circles must belong to the boundary of the bounce (see fig. \\ref{corr}). This consideration might give an additional condition on relative locations of ${\\bf r_i}$; however, this condition is satisfied automatically because the centers of four circles with equal radii intersecting in a single point necessarily belong to the circle of the same radius. This feature of the geometry allows to forget about relative position of the densities and to study the problem in the momentum representation.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=150pt]{1corr}\n\\end{center}\n\\caption{The bold circle represents the wall of the bounce, the other four circles represent the densities of $\\chi$ particles. The centers of the four circles lie on the bold circle, because the $\\delta$-functions force the four circles to intersect in one point.}\n\\label{corr}\n\\end{figure}\n\nFirst we consider the correlator (\\ref{Pi}) in the coordinate form. The integration over ${\\bf r}$ gives\n\n\\begin{equation}\n\\label{step1}\n\\Im \\Pi ({\\bf r_1}, {\\bf r_2}, {\\bf r_3})=\\frac{\\Gamma}{2L}\\frac {g^4}{(2\\pi R)^4}\\frac{4R^2}{r_1\\sqrt{4R^2-r_1^2}} \\delta\\left(r_2-R\\right)\\delta\\left(r_3-R\\right).\n\\end{equation}\n\nSwitching to the momentum representation we obtain a simple expression for the four-fold correlator. ${\\bf q_i}$ are the momenta of the $\\chi$ particles, so that $|{\\bf q_i}|=q_i=m$.\n\n\\begin{multline}\n\\label{step2}\n\\Im \\Pi ({\\bf q_1},{\\bf q_2},{\\bf q_3})=\\frac{\\Gamma}{2L}\\int \\frac{g^4\\delta\\left(r_2-R\\right)\\delta\\left(r_3-R\\right)}{\\pi^2(2\\pi R)^2r_1\\sqrt{4R^2-r_1^2}} e^{i({\\bf q_1 r_1})+i({\\bf q_2 r_2})+i({\\bf q_3 r_3})}d^2{\\bf r_1}d^2{\\bf r_2}d^2{\\bf r_3}=\\\\\n-\\frac {g^4}{(2\\pi R)^2}I_0^2(mR)\\int \\delta\\left(r_2-R\\right)\\delta\\left(r_3-R\\right)e^{i({\\bf q_2 r_2})+i({\\bf q_3 r_3})}d^2{\\bf r_2}d^2{\\bf r_3}=-\\frac \\Gamma {2L}g^4I_0^4(mR).\n\\end{multline}\n\nHere the integral representation of the Bessel function was used,\n\n\\begin{equation}\n\\label{bessel_1}\n\\int\\delta\\left(r-R\\right)e^{i({\\bf q r})}d^2{\\bf r}=\\int e^{iqR\\cos\\theta}Rd\\theta=2\\pi R\nJ_0(qR)=2\\pi RI_0(mR),\n\\end{equation}\n\\begin{equation}\n\\label{bessel_2}\n\\int\\frac{e^{i({\\bf q r})}}{r\\sqrt{4R^2-r^2}}d^2{\\bf r}=\\int \\frac{e^{iqr\\cos\n\\theta}}{\\sqrt{4R^2-r^2}}drd\\theta=-\\pi^2I_0^2(mR).\n\\end{equation}\n\nWe see that the geometry of the problem does not impose additional restrictions on the positions of ${\\bf r_i}$. Hence we can first switch to the momentum form of the densities (\\ref{rho_def}), and then compute the correlator. This significantly simplifies the calculations.\n\nThe density in the momentum representation is as follows,\n\n\\begin{equation}\n\\label{rho_q}\n\\rho^B({\\bf r_1}-{\\bf r})\\rightarrow\\rho^B({\\bf q_1},{\\bf r})=\\int\\rho^B({\\bf r_1}-{\\bf r})e^{i({\\bf q_1 r_1})}d^2{\\bf r_1}=I_0(mR)e^{i({\\bf q_1 r})}.\n\\end{equation}\n\nThe calculation of the correlator in terms of densities (\\ref{rho_q}) reduces to the computation of an integral of type (\\ref{bessel_1}),\n\n\\begin{multline}\n\\label{momentum}\n\\Im \\Pi ({\\bf q_1},{\\bf q_2},{\\bf q_3})=\\frac{\\Gamma}{2L} \\frac{g^4}{(2\\pi R)^4}\\int\\rho^B({\\bf q_1},{\\bf r})\\rho^B({\\bf q_2},{\\bf r})\\rho^B({\\bf q_3},{\\bf r})\\rho^B({\\bf r})d^2{\\bf r}=\\\\\\frac \\Gamma {2L}\\frac {g^3I_0^3(mR)}{2\\pi\nR}\\int\\delta\\left(r-R\\right)e^{i\\left(({\\bf q_1}+{\\bf q_2}+{\\bf q_3}),{\\bf r}\\right)}d^2{\\bf r}=-\\frac \\Gamma {2L}g^4I_0^4(mR),\n\\end{multline}\n\nand can be easily generalized to a calculation of an $n$-fold correlator,\n\n\\begin{equation}\n\\label{n-fold}\n\\Im \\Pi^{(n)}=-\\frac \\Gamma {2L} \\left(g I_0(mR)\\right)^n.\n\\end{equation}\n\nIn the expression (\\ref{momentum}) the momentum conservation law was used, $\\sum{\\bf q_i}=0$ , $|{\\bf q_4}|=m$. To reproduce the result of \\cite{scalar}, we write the cross-section in terms of the imaginary part of the correlator,\n\n\\begin{equation}\n\\sigma=-\\frac{\\Im \\Pi^{(4)}}{2I}=\\frac{\\varepsilon}{4\\pi}\\frac1{2I}(gI_0(mR))^4\\exp\\left(-\\frac{\n\\pi\\mu^2}{\\varepsilon}\\right),\n\\end{equation}\n\nwhere $I=\\sqrt{({\\bf q_1q_2})^2-m^4}$ is a kinematical invariant. Using (\\ref{n-fold}), we can calculate the rate of the process induced by any number of the external particles.\n\n\\subsection{Electrodynamics}\n\nNow we move to our second example, namely the electrodynamics in two Euclidean dimensions in presence of a constant electric field. Our goal is to calculate the rate of the pair creation enhanced by two photons. We proceed just the same way as before, i.e. we calculate the imaginary part of the four-fold correlator and argue that this technique allows to calculate the polarization operator with any number of external photons.\n\nThe creation of an electron-positron pair in external field can be thought of as the creation of a bubble of the true vacuum in the false vacuum. The action of the configuration consists of a surface term and a boundary term, just as in the scalar case (\\ref{action_0}), with the following identification of the parameters:\n\n\\begin{equation}\n\\mu = m,\\qquad \\varepsilon = eE,\\qquad R=\\frac{m}{eE},\n\\end{equation}\n\nwhere $m$ is the mass of electron. The rate of the spontaneous process is,\n\n\\begin{equation}\n\\label{gamma2}\n\\frac {\\Gamma} {L}=\\frac {eE}{2\\pi}\\exp\\left(-\\frac{\\pi m^2}{eE}\\right).\n\\end{equation}\n\nApplying the technique of calculation of the imaginary part of the four-fold correlator, we find the polarization operator in leading order on $\\frac {eE}{m^2}$ (neglecting the spinorial structure of the electron current). Then the electron current is of the form,\n\n\\begin{equation}\n\\label{current}\nj^{\\mu}({\\bf r})=en^{\\mu}\\delta \\left(r-R\\right)=e\\frac {\\epsilon^{\\mu\\nu}r_\\nu}r \\delta (r-R),\\qquad {\\bf n} =\\begin{pmatrix}\\sin \\phi\\\\-\\cos\\phi\\end{pmatrix},\\qquad \\mu,\\nu=1,2,\n\\end{equation}\n\nwhere $\\bf n$ is a unit vector tangential to the boundary of the bounce.\n\nWe calculate two-fold and four-fold polarization operators as the imaginary parts of the corresponding correlators,\n\n\\begin{equation}\n\\label{ImPi2}\n\\pi^{\\mu \\nu}({\\bf r_1})=\\Im \\Pi^{\\mu \\nu}({\\bf r_1})=\\frac {\\Gamma} {2L} \\int\nj^{\\mu}\\left({\\bf r_1}-{\\bf r}\\right)j^{\\nu}\\left(-{\\bf r}\\right)d^2{\\bf r},\n\\end{equation}\n\n\\begin{multline}\n\\label{ImPi}\n\\pi^{\\mu \\nu\\kappa\\rho}({\\bf r_1}, {\\bf r_2}, {\\bf r_3})=\\Im \\Pi^{\\mu \\nu \\kappa \\rho}({\\bf r_1}, {\\bf r_2}, {\\bf r_3})=\\\\\n\\frac {\\Gamma} {2L} \\int j^{\\mu}\\left({\\bf r_1}-{\\bf r}\\right)j^{\\nu}\\left({\\bf r_2}-{\\bf r}\\right)j^{\\kappa}\\left({\\bf r_3}-{\\bf r}\\right)j^{\\rho}\\left(-{\\bf r}\\right)d^2{\\bf r}.\n\\end{multline}\n\nWe switch to the momentum form of the current (\\ref{current}),\n\n\\begin{multline}\n\\label{currentOrt}\nj^\\mu ({\\bf r_1}-{\\bf r})\\rightarrow j^{\\mu}({\\bf q_1},{\\bf r})=e^{i({\\bf q_1 r})}\\int e \\begin{pmatrix} \\sin \\phi \\\\ -\\cos \\phi \\end{pmatrix}\\delta(r-R) e^{iq_1r\\cos(\\theta_1-\\phi)}rd\\phi dr=\\\\\n-2\\pi ieRJ_1(q_1R)e^{i({\\bf q_1 r})}\\begin{pmatrix}\\sin\\theta_1 \\\\ -\\cos\\theta_1\\end{pmatrix}=-2\\pi ie \\frac {\\epsilon^{\\mu \\nu}{q_1^\\nu}}{q_1}RJ_1(q_1R)e^{i({\\bf q_1 r})}\n\\end{multline}\n\nNote that here $q=\\sqrt {q_x^2+q_t^2}$ is the euclidean absolute value of ${\\bf q}$ and is nonzero though the photon is massless. The calculation similar to (\\ref{momentum}) leads to the following expressions for the polarization operators:\n\n\\begin{equation}\n\\label{pop_2}\n\\pi^{\\mu\\nu}({\\bf q})=\\frac {\\Gamma} {2L} \\left(2\\pi e R J_1(qR)\\right)^2\n\\left(\\delta^{\\mu\\nu}-\\frac{q^\\mu q^\\nu} {q^2}\\right),\n\\end{equation}\n\n\\begin{equation}\n\\label{pop_4}\n\\pi^{\\mu\\nu\\kappa\\rho}({\\bf q_1},{\\bf q_2},{\\bf q_3},{\\bf q_4})=\\frac {\\Gamma}{2L} \\left(\\prod_{i=1}^4 \\frac\n{2\\pi e R J_1(q_iR)}{q_i}\\right)\\left(q_2^\\mu q_1^\\nu - ({\\bf q_1 q_2})\\delta^{\\mu \\nu}\\right)\\left(q_4^\\kappa q_3^\\rho - ({\\bf q_3 q_4})\\delta^{\\kappa \\rho}\\right).\n\\end{equation}\n\nThe latter expression is subject to the condition of the momentum conservation, $\\sum {\\bf q_i}=0$. As it was expected from (\\ref{currentOrt}), it is transverse to respective momenta.\nThe result (\\ref{pop_4}) directly generalizes to any even number of external photons:\n\n\\begin{equation}\n \\label{pop_n}\n\\pi^{\\mu_1 \\mu_2 \\ldots \\mu_{2n}} ({\\bf q_1},{\\bf q_2}, \\ldots ,{\\bf q_{2n}}) = \\frac {\\Gamma}{2L} \\left( \\prod_{i=1}^{2n} \\frac {2\\pi e R J_1(q_iR)}{q_i} \\right) \\prod_{i=1}^{n} ( q_{2i}^{\\mu_{2i-1}} q_{2i-1}^{\\mu_{2i}} - ({\\bf q_{2i} q_{2i-1}})\\delta^{\\mu_{2i-1}\\mu_{2i}} )\n\\end{equation}\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=200pt]{2shape}\n\\end{center}\n\\caption{The shape of the bounce changes considerably when the interaction with hard photons takes place.}\n\\label{shape}\n\\end{figure}\n\n All the above results were obtained in the limit of soft photons $q\\ll m$. Generally speaking, the interaction with external photons considerably changes the form of the bounce, because it acquires cusps at the points where the interaction takes place (see fig. \\ref{shape}). This leads to an exponential correction to the rate of the process. However consideration of processes with hard photons results in significant complication of the problem and lies beyond the scope of the current paper.\n\nIn the next section we calculate another exponential correction to the action which is caused by interaction of the particles of the bounce with each other. The results (\\ref{n-fold}) and (\\ref{pop_4}) allow to calculate the rate of the processes in which electrons exchange photons and $\\phi$ particles interact by means of $\\chi$ field. This corrections are referred to as Coulomb and Yukawa corrections.\n\n\\section{Coulomb and Yukawa corrections}\n\nIn this section we compute the exponential corrections to the rates of the processes caused by the interaction of the wall of the bounce with itself. A similar quantity was calculated in \\cite{affleck} for four-dimensional electrodynamics. Their result for the corrected effective action reads as:\n\n\\begin{equation}\nS_C^{4d}=\\frac {\\pi m^2}{eE}-\\frac {e^2}4.\n\\end{equation}\n\nThe correction is small in the considered limit of small charges and is independent of the radius of the bounce. Hence in the four-dimensional electrodynamics the Coulomb correction does not change the radius of the bounce. We see below that in the two-dimensional case this is not so.\n\nWe calculate analogous corrections using our results (\\ref{n-fold}) and (\\ref{pop_4}) for correlators. Using the fact that the correlators in the momentum representation are products of uniform multiples we can calculate contribution of any number of particles interacting with bounce.\n\nWe begin with the Coulomb correction for the rate of processes in two-dimensional electrodynamics. To calculate the $n$-photon contribution, we consider the $(2n+l)$-fold correlator, where $l$ is the number of external photons. Then we integrate over the momenta of $n$ internal photons with their propagators. To find the full Coulomb correction we have to sum up the contributions of any number of photons taking into account combinatorial factors (see fig. \\ref{coulomb}).\n\n\\begin{figure}\n\\begin{tabular}{ccccccccc}\n\\begin{tabular}{c} \\includegraphics[scale=.3]{3.pdf} \\end{tabular} & + & \\begin{tabular}{c} \\includegraphics[scale=.3]{4.pdf} \\end{tabular} & + & \\begin{tabular}{c} \\includegraphics[scale=.3]{5.pdf} \\end{tabular} & + & \\begin{tabular}{c}  \\includegraphics[scale=.3]{6.pdf} \\end{tabular}&+&\\ldots\\\\ $\\pi^{\\mu \\nu}_{0}$ & \\qquad & $\\pi^{\\mu \\nu}_{1}$ & \\qquad & $\\pi^{\\mu \\nu}_{2}$ & \\qquad & $\\pi^{\\mu \\nu}_{3}$&\\qquad\n\n\\end{tabular}\n\\caption{To obtain the Coulomb correction for polarization operator, the interaction with arbitrarily many photons is to be studied.}\\label{coulomb}\n\\end{figure}\n\nFirst we find one-photon contribution. For simplicity we consider the width of the spontaneous decay, though the case of induced decay is absolutely analogous. We integrate two-photon polarization operator (\\ref{pop_2}) with the propagator of the photon.\n\n\\begin{multline}\n\\label{coul}\n\\frac{\\Gamma_1}{2L}= \\frac 12 \\int_0^\\infty \\pi^{\\mu\\nu}({\\bf q}) \\frac{\\delta^{\\mu\\nu}}{q^2}{\\frac {d^2q}{(2\\pi)^2}} = \\\\\n= \\frac 12 \\frac{ \\Gamma_{}} {2L} \\int_0^\\infty \\left(2 \\pi e R J_1(qR)\\right)^2\n\\left(\\delta^{\\mu\\nu}-\\frac{q^\\mu q^\\nu} {q^2}\\right)\\frac\n{\\delta^{\\mu\\nu}}{q^2}{\\frac {d^2q}{(2\\pi)^2}}=\\frac {\\Gamma} {2L} \\frac {\\pi R^2e^2}2.\n\\end{multline}\n\nHere $\\Gamma_n$ is the $n$-photon correction to the width. The calculation (\\ref{coul}) contains a certain subtlety. As we have argued above our results are valid only in case of small photon momenta $q$, but the integration in (\\ref{coul}) is over all values of $q$. However the integrand significantly differs from zero only in area $qR \\sim 1$, so the contribution from hard photons is negligible.\n\nCorrections $\\Gamma_n$ for $n\\ge 2$ are calculated  in a similar fashion with $n$-photon polarization operator (\\ref{pop_n}) and $n$ photon propagators. As we see on example of $\\Gamma_2$,\nintegrals over momenta of different photons are independent from each other:\n\n\\begin{multline}\n \\frac{\\Gamma_2}{2L} = \\frac 14 \\int_0^\\infty \\int_0^\\infty \\pi^{\\mu\\nu\\kappa\\rho}({\\bf q},{\\bf -q},{\\bf p},{\\bf -p}) \\frac {\\delta^{\\mu\\nu}}{q^2} \\frac {\\delta^{\\kappa\\rho}}{p^2}\n \\frac {d^2q}{(2\\pi)^2} \\frac {d^2p}{(2\\pi)^2} = \\\\\n= \\frac 14  \\frac{ \\Gamma_{}} {2L} \\left( \\int_0^\\infty \\left(2 \\pi e R J_1(qR)\\right)^2\n\\left(\\delta^{\\mu\\nu}-\\frac{q^\\mu q^\\nu} {q^2}\\right)\\frac\n{\\delta^{\\mu\\nu}}{q^2}{\\frac {d^2q}{(2\\pi)^2}} \\right)^2 = \\frac {\\Gamma} {2L} \\left(\\frac {\\pi R^2e^2}2\\right)^2.\n\\end{multline}\n\nThe same thing happends for arbitrary $n$, resulting in $n$-th order correction\n\n\\begin{equation}\n \\Gamma_n = \\Gamma \\left(\\frac{\\pi R^2 e^2}2\\right)^n.\n\\end{equation}\n\n\nSumming up the contributions of any number of photons and taking into account the combinatorial factor $\\frac 1 {n!}$ we find the corrected width:\n\n\\begin{equation}\n\\Gamma_{C}=\\sum_{n=0}^{\\infty} \\frac 1{n!} \\Gamma_n= \\Gamma \\exp \\left(\\frac{\\pi R^2 e^2}2\\right).\n\\end{equation}\n\nAll the polarization operators acquire the same exponential factor; for example, the corrected polarization operator corresponding to the process induced by a single photon reads as:\n\n\\begin{equation}\n\\pi^{\\mu\\nu}_{C}(q)=\\pi^{\\mu\\nu}(q)\\exp \\left(\\frac{\\pi R^2 e^2}2\\right).\n\\end{equation}\n\nThe exponential factor can be interpreted as an additional term in the effective action:\n\n\\begin{equation}\nS_C=2\\pi mR - \\pi eE R^2-\\frac 12 \\pi R^2 e^2.\n\\end{equation}\n\nThe extremization of the action gives the radius of the process with the Coulomb interaction present:\n\n\\begin{equation}\nR_C=\\frac {2m R}{e(2E+e)}\n\\end{equation}\n\nWe see that the Coulomb interaction decreases the radius of the bounce; however the change is small in the limit $e \\ll E$.\n\nIn the same way we find the Yukawa correction for the false vacuum decay in scalar theory. The two-fold correlator integrated with the propagator of the $\\chi$ particle reads as:\n\n\\begin{equation}\n\\label{yuk}\n\\Gamma_{1}=\\frac 12 \\frac \\Gamma {2L}\\int_0^\\infty g^2 J_0(qR)^2\n\\frac{qdq}{q^2+m^2}=\\frac \\Gamma {2L} \\frac {g^2}2 I_0(mR)K_0(mR).\n\\end{equation}\n\nThe width acquires exponential correction which again depends on the radius:\n\n\\begin{equation}\n\\label{gamma_yuk}\n\\Gamma_{Y}= \\frac{\\varepsilon}{4\\pi} \\exp\\left(-2\\pi\\mu R + \\pi \\epsilon R^2+ \\frac{g^2}2 I_0(mR)K_0(mR)\\right).\n\\end{equation}\n\nThe exponential correction $I_0\\left(mR\\right)K_0\\left(mR\\right)$ behaves like a logarithm for small argument. However, this is not an unusual property for a two-dimensional theory to have infrared logarithmic divergences. Bessel functions often appear in answers for two dimensions (see e.g. \\cite{Gorsky_diver}). The correction is significant in the area of small $m$ where the distortion of the shape of the bounce is negligible. When $m > \\mu$ the change in the shape is the main source of the correction.\n\nThe correlators for the induced processes acquire the same exponential factor. This factor can be interpreted as an additional term in the action too,\n\n\\begin{equation}\nS_Y=-2\\pi\\mu R + \\pi \\epsilon R^2+ \\frac{g^2}2 I_0(mR)K_0(mR),\n\\end{equation}\n\nand this means that the Yukawa interaction changes the radius of the bounce too. The exact value of the radius can be found as the solution of the extremization condition.\n\n\\section{Conclusions}\n\nIn this work the induced Schwinger process was studied by purely semicalssical means. We calculated the cross-section for two-photon interaction at the background of the bounce as leading term in $\\frac {eE}{m^2}$ expansion which is applicable for the electric field strength much smaller than $E_{crit}$. The process was studied for soft photons ($\\omega \\ll m_e$). In case of hard photons the cross-section acquires exponential correction because of the change in the shape of the bounce. We do not investigate this correction because the increase of the momenta of the photons leads to considerable complication of the setting.\n\nThe calculation of the rate of the two-photon interaction easily generalizes to the case of the process induced by arbitrarily many photons. Extremely simple structure of the corresponding correlator allowed us to compute the correction to the action which is caused by interaction of the wall of the bounce with itself. This correction depends on the radius and therefore results in the change of the radius of the bounce. So we have seen that the Coulomb interaction decreases the radius of the bounce. Similar calculation was carried out for the scalar fields. In both cases the corrections are significant in the limit of soft particles where the distortion of the bounce is negligible.\n\nThe authors are especially grateful to A.\\,Gorsky for initiating the work and for useful and productive discussions. The work was supported by grants RFBR 12-02-00284-a and RFBR-CNRS 12-02-91052 as well as Dynasty Foundation stipend program.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nLet $\\Omega\\subset \\mR^d$ be a domain, $d\\geq 1$, and let $p,q\\in (1,\\infty)$. Let $K\\colon \\mR^d \\times \\mR^d \\to (0,\\infty]$ be a homogeneous, radial kernel, i.e. $K(x,y) = k(|x-y|)$, satisfying $\\int_{\\mR^d}  (1\\wedge|y|^q)K(0,y) \\, dy <\\infty$. We define the (generalized) Triebel--Lizorkin space on $\\Omega$ as\n\\begin{equation}\\label{eq:tldef}\n\\fpqom := \\Big\\{f\\in L^p(\\Omega): \\intom\\bigg(\\intom |f(x) - f(y)|^q K(x,y) \\,\\red{dy}\\bigg)^{\\frac pq} \\, \\red{dx} < \\infty \\Big\\}.\n\\end{equation}\nThe space $\\fpqom$ obviously depends on $K$, however we skip it in the notation for clarity.\n\n$\\fpqom$ is endowed with the norm\n$$\\| f\\|_{\\fpqom} = \\|f\\|_{L^p(\\Omega)} + \\bigg(\\intom\\Big(\\intom |f(x) - f(y)|^q K(x,y) \\, \\red{dy}\\Big)^{\\frac pq} \\, \\red{dx} \\bigg)^{\\frac 1p}.$$\n\nWe are interested in the Gagliardo-type seminorm\n\\begin{equation}\\label{eq:fullsemi}\n\\bigg(\\intom\\Big(\\intom |f(x) - f(y)|^q K(x,y) \\, \\red{dy}\\Big)^{\\frac pq} \\, \\red{dx} \\bigg)^{\\frac 1p},\n\\end{equation}\nwhich will be called the \\textit{full seminorm}. Let \\red{$\\theta \\in (0,1]$} and let $\\delta(x) = d(x,\\partial \\Omega)$. Our main goal is to establish the comparability of the full seminorm and the \\textit{truncated seminorm}\n\\begin{equation}\\label{eq:truncated}\n\\bigg(\\intom\\Big(\\int_{B(\\red{x},\\theta\\delta(\\red x))} |f(x) - f(y)|^q K(x,y) \\, \\red {dy}\\Big)^{\\frac pq} \\, \\red {dx} \\bigg)^{\\frac 1p}\n\\end{equation}\nfor sufficiently regular $K$ and $\\Omega$. Later on, such occurrence will be called a \\textit{comparability result}.\n\nHere is our first comparability result.\n\\begin{theorem}\\label{th:main}\n\tAssume that $\\Omega$ is a uniform domain, and that $K$ satisfies \\AJ, \\AD\\ and \\AT\\ \\red{formulated in Subsection \\ref{sec:a} below}. Assume that $1<q\\leq p<\\infty$. Then for every $0<\\theta\\leq 1$\n\t\\begin{align*}\\bigg(\\intom\\Big(\\intom |f(x) - f(y)|^q K(x,y) \\, \\red {dy}\\Big)^{\\frac pq} \\, \\red {dx} \\bigg)^{\\frac 1p} \\approx \\bigg(\\intom\\Big(\\int_{B(\\red x,\\theta\\delta(\\red x))} |f(x) - f(y)|^q K(x,y) \\, \\red{dy}\\Big)^{\\frac pq} \\, \\red{dx} \\bigg)^{\\frac 1p}.\\end{align*}\n\tThe comparability constant depends on $p,q,\\theta,\\Omega$, and the constants in assumptions \\AD, \\AT.\n\\end{theorem}\n\n\n\nThis is a generalization of the result of Prats and Saksman \\cite[Theorem 1.6]{PS} who prove it for the kernels of the form: $K(x,y) = |x-y|^{-d-qs}$ for $s\\in (0,1)$. Recently, we were informed that this classical case can also be resolved using the much earlier result of Seeger \\cite[Corollary 2]{MR1097208}. \n\nAnother result of this flavor was established by Dyda \\cite[(13)]{DYDA2006564} and was used to obtain Hardy inequalities for nonlocal operators. \\red{More recent results on reduction of integration domain in fractional Sobolev spaces include Bux, Kassmann, and Schulze \\cite{2017arXiv170709277B} who consider certain cones with apex at $x$ instead of $B(x,\\delta(x))$, and Chaker and Silvestre \\cite{2019arXiv190413014C}.} \n\nHere we mention that, independently of our work, Kassmann and Wagner \\cite{2018arXiv181012289K} have also proved comparability results which extend the ones from \\cite{PS}, allowing for kernels with scaling conditions \\red{for $p=q=2$}. However, their overall aim and scope are different than ours.\n\nIn Theorem \\ref{th:main} we adapt the method of proof from \\cite{PS} for a wide class of kernels of the form $K(x,y) = |x-y|^{-d}\\phi(|x-y|)^{-q}$. The most technical assumption \\textbf{A2} is tailored for the key Lemmas \\ref{lemMaximal}, \\red{\\ref{lem:phil}}, however in Subsection \\ref{sec:zal} we argue that it amounts to \\red{at least a power-type decay at 0, and for unbounded $\\Omega$ at least a power-type growth at $\\infty$, of $\\phi$.} For the formulation of the assumptions, see Subsection \\ref{sec:a}.\n\n\nNotably, we go beyond the uniform domains, where the methods used by Prats and Saksman are no longer available. Namely, we prove that the comparability may hold for the fractional Sobolev spaces in strip domains. \n\n\\begin{theorem}\\label{th:stripmd}\n\tAssume that $p=q=2$. Let $\\Omega = \\mR^k\\times (0,1)^l \\subseteq \\mR^{k+l}$ with $k,l>0$. For $d=k+l$ let $K(x,y) = |x-y|^{-d-\\alpha}$ with $\\alpha\\in (0,2)$. If \\red{$k-l - \\alpha < -1$}, then the seminorms \\eqref{eq:fullsemi} and \\eqref{eq:truncated} are comparable.\n\\end{theorem}\n\nWe also construct a counterexample for $\\alpha < 1$ and $k=l=1$. This shows an intriguing interplay between the kernel and the width of the domain. \\red{Heuristically speaking, it can be seen both in Theorem \\ref{th:stripmd} and in Subsection \\ref{sec:zal} that the comparability holds if the stochastic process corresponding to the jump kernel $K\\cdot \\textbf{1}_{\\Omega\\times \\Omega}$ and the shape of the domain $\\Omega$ favor small jumps over large jumps. We remark that the connection between the jump kernel and the stochastic process is a very delicate matter. In Section \\ref{sec:proces} we present a short discussion of this subject and we place our comparability results in this context}.\n\nAnother object of our studies is the 0-order kernel $K(x,y) \\approx |x-y|^{-d}$. We provide examples showing that the comparability does not hold in this case. In an attempt to repeat the proof of Theorem \\ref{th:main} we obtain an estimate of \\eqref{eq:fullsemi} by a truncated seminorm with a slightly more singular kernel, \\red{see Theorem \\ref{th:0order} below.}\n\n\n\n\n\nThe classical Triebel--Lizorkin spaces were introduced independently by Lizorkin \\cite{Liz74} and Triebel \\cite{triebel1973}. The original definition is formulated using Paley--Littlewood theory and is widely used in analysis and applications, see e.g. \\cite{doi:10.1002\/mana.201600315,bu2009,grafakos2014modern}. For various cases of $p,q,d$ and $\\Omega$ the classical definition was proved to be equivalent to $\\eqref{eq:tldef}$ with $K(x,y) = |x-y|^{-d-sq}$, where $s\\in(0,1)$, see \\cite{PS,stein1961,triebel2010theory}.\n\nThe definition \\eqref{eq:tldef} seems more natural if the starting point is $p=q=2$, e.g., fractional Sobolev spaces in nonlocal PDEs \\cite{DINEZZA2012521, MR3318251, 2018AR}, or Dirichlet forms for Hunt processes \\cite{MR2006232, MR2778606}. It is also a suitable definition for considering kernels $K$ more general than $|x-y|^{-d-sq}$ which is also of interest in the field of nonlocal operators and stochastic processes. In this paper we will not attempt to characterize the definition \\eqref{eq:tldef} in the spirit of classical definitions by Triebel and Lizorkin in the full generality. However, we \\red{use} the Fourier methods in Section 5, where we compare spaces with kernels which are only slightly different from each other. \n\n\nAs we argue further in the article, the comparability results can be used to study a class of stochastic processes, whose jumps from \\red{$x$} are restricted to the ball $B(\\red x,\\theta \\delta(\\red x))$. The truncated seminorms may also prove useful in peridynamics, as $B(\\red x,\\theta\\delta(\\red x))$ may be understood as the \\textit{variable horizon}, see e.g. \\cite{varhor2,varhor}, and in particular Du and Tian \\cite{doi:10.1137\/16M1078811} where horizons depending on the distance from the boundary are studied.\n\nThe paper is organized as follows. Section 2 contains notions, assumptions, and basic facts used further in our work. Section 3 is devoted to proving Theorem \\ref{th:main}. In Section 4 we present positive and negative examples of kernels concerning the comparability results. Section 5 contains the analysis of 0-order kernels. In Section 6 we consider strip domains, in particular we prove Theorem \\ref{th:stripmd}. Section 7 presents the connection of our development with the theory of Hunt processes.\n\\section*{Acknowledgements}\nI am grateful to Bart\\l omiej Dyda for introduction to the subject, many hours of helpful discussions and for reading the manuscript. I thank Tomasz Grzywny and Dariusz Kosz for stimulating discussions. I also thank Mart\\'i Prats for pointing out a flaw in the proof of Theorem \\ref{th:main} in the previous version of the manuscript.\n\\red{I express my gratitude to the anonymous referees for numerous essential remarks and for raising important questions concerning the proofs and the main assumptions of the paper. Research was supported by the grant 2015\/18\/E\/ST1\/00239 of the National Science Center (Poland).}\n\\section{Preliminaries and assumptions}\n\\subsection{Assumptions on the kernel}\\label{sec:a}\n\\red{In the sequel we will consider the exponents $1<q\\leq p<\\infty$ and the assumptions are subordinated to them. As usual, $p' = \\frac p{p-1}$ is the H\\\"older conjugate of $p$. We also let\n\t$$N(r) = \\inf\\{k\\in \\mathbb{N}: 2^kr > \\diam (\\Omega)\\},\\quad r>0.$$\n\tWe have $N(r) = \\infty$ for every $r>0$ if and only if $\\Omega$ is unbounded.}\\\\\nWe assume that the kernel $K$ is of the form $K(x,y) = |x-y|^{-d}\\phi(|x-y|)^{-q}$, where $\\phi\\colon (0,\\infty) \\to (0,\\infty)$ satisfies\n\\begin{enumerate}\n\t\\item[\\AJ] $(1\\wedge|y|^q)|y|^{-d}\\phi(|y|)^{-q}\\in L^1(\\mR^d)$,\n\t\\item[\\AD] $\\phi$ is increasing \\red{and there exists $C_2>0$ such that for $t_1 = \\min(q,p - \\frac pq)$, $t_2= \\frac 1{q-1}$, and for every $0<r<\\diam(\\Omega)$, we have $$\\sum\\limits_{k=1}^{N(r)} \\frac{\\phi(r)^{t_1}}{\\phi(2^kr)^{t_1}} \\leq C_2$$ and $$\\sum\\limits_{k=1}^\\infty \\frac{\\phi(2^{-k}r)^{t_2}}{\\phi(r)^{t_2}} \\leq C_2.$$}\n\n\t\\item[\\AT] \\red{There exists $C_3\\geq 1$ such that for every $0<r<3\\diam(\\Omega)$, we have $\\phi(2r) \\leq C_3 \\phi (r)$.}\n\\end{enumerate}\nIn particular, we allow unbounded domains in which the scaling conditions \\AD,\\ \\AT\\ become global. Note that \\AJ\\ is a L\\'evy \\red{measure}-type condition\\red{, which assures the finiteness of \\eqref{eq:fullsemi} for smooth, compactly supported~$u$}. If $q=2$ and $\\phi(r) = r^{s}$, $s\\in (0,1)$, then $K$ corresponds to the fractional Laplacian of order $s$ and all the assumptions are satisfied. The conditions \\AD \\ and \\AT\\ imply certain \\red{scaling} for $K$, see Subsection \\ref{sec:zal} for the details. \\red{The exponents $t_1$ and $t_2$ in \\AD\\ stem from the five instances of usage of Lemma \\ref{lemMaximal} and \\ref{lem:phil} in the proof of Theorem \\ref{th:main}. Since $\\phi$ is increasing, the bounds in \\AD\\ hold for all larger exponents in place of $t_1$ and $t_2$. We note the following, frequently used below, consequence of \\AT\\ and the monotonicity of $\\phi$: if $x \\lesssim y$, then $\\phi(y)^{-1} \\lesssim \\phi(x)^{-1}$.}\n\\subsection{Whitney decomposition and uniform domains}\nFor cubes $Q, R$ in $\\mR^d$ we consider $l(R)$ --- the length of the side of $R$, and the long distance between $Q$ and $R$: $D(Q,R) = l(Q) + d(Q,R) + l(R)$, where $d$ is the Euclidean distance. The scaling of the cube is done from its center $x_Q$.\n\n\nWe say that a family of (closed) dyadic cubes $\\mathcal{W}$ is a Whitney decomposition of $\\Omega$ if for every $Q, S \\in \\mW$\n\\begin{itemize}\n\t\\item if $Q\\neq S$, then  $\\intr(Q)\\cap\\intr(S) = \\emptyset;$\n\t\\item if $Q\\cap S \\neq \\emptyset$, then $l(Q)\\leq 2 l(S)$;\n\t\\item if $Q\\subseteq 5S$, then $l(S) \\leq 2l(Q)$;\n\t\\item there is a constant $C_{\\mW}$ such that $C_{\\mW}l(Q) \\leq d(Q,\\partial\\Omega)\\leq 4C_{\\mW}l(Q)$.\n\\end{itemize}\nA sequence of cubes $(Q,R_1,\\ldots,R_n,S)$ is a chain connecting $Q$ and $S$, if every cube \\red{is a neighbor of} its successor and predecessor (if it has one), \\red{by which we mean that their boundaries have non-empty intersection}. We will denote the chain as $[Q,S]$ and the sum of the lengths of its cubes as $l([Q,S])$. We let $[Q,S)= [Q,S]\\setminus \\{S\\}$.\n\nThe Whitney decomposition is admissible, if there exists $\\eps > 0$ such that for every pair of cubes $Q,S$, there exists an $\\eps$-admissible chain $[Q,S] = (Q_1,Q_2,\\ldots Q_n)$, i.e.\n\\begin{itemize}\n\t\\item $l([Q,S]) \\leq \\frac 1\\eps D(Q,S)$,\n\t\\item there exists $j_0 \\in \\{1,\\ldots, n\\}$ for which $l(Q_{j}) \\geq \\eps D(Q,Q_j)$ for every $1\\leq j \\leq j_0$, and $l(Q_{j}) \\geq \\eps D(Q_j,S)$ for every $j_0\\leq j\\leq n$. $Q_{j_0}$ will be denoted as $Q_S$ --- the central cube of the chain $[Q,S]$.\n\\end{itemize}\nA domain which has an admissible Whitney decomposition is called a uniform domain. Unless we state otherwise, $[Q,S]$ is an arbitrary ($\\eps$-)admissible chain connecting $Q$ and $S$. \n\nThe shadow of a cube is $\\Sh_\\rho(Q) = \\{S\\in \\mW: S\\subseteq B(x_Q,\\rho l(Q))\\}$, $\\rho > 0$. We also denote $\\SH_\\rho(Q) = \\bigcup \\Sh_\\rho(Q)$. Note that we can take a sufficiently large $\\rho_\\eps$ so that\n\\begin{itemize}\n\n\t\\item for every $\\eps$-admissible chain $[Q,S]$, and every $P\\in [Q,Q_S]$, we have $Q \\in \\Sh_{\\rho_\\eps}(P)$,\n\t\\item if $[Q,S]$ is $\\eps$-admissible, then every cube from it belongs to $\\Sh_{\\rho_\\eps}(Q_S)$,\n\t\\item for every $Q\\in \\mW$, $5Q \\subseteq \\SH_{\\rho_\\eps}(Q)$.\n\\end{itemize}\nFrom now on we fix $\\rho_\\eps$ and write $\\Sh(Q) = \\Sh_{\\rho_\\eps}(Q)$ and $\\SH(Q) = \\SH_{\\rho_\\eps}(Q)$.\n\\begin{remark}\n\t\\red{The proofs appearing throughout the paper involve a lot of '$\\lesssim$', and '$\\gtrsim$' signs. We would like to stress that any comparability for $\\phi$ stems from \\AD\\ and \\AT. In particular, for fixed $p,q$ the constants can be chosen to depend only on the geometry of $\\Omega$ (including the dimension) and the constants in \\AD\\ and \\AT\\ wherever $\\phi$ is used.}\n\\end{remark}\n\nThe next lemma provides some inequalities for the \\red{non-centered} Hardy--Littlewood maximal operator (denoted by $M$) with connection to the kernel $K$. \\red{It is inspired by the results of \\cite[Section 2]{PS} and Prats and Tolsa \\cite[Section 3]{PRATS20152946}.}\n\\begin{lemma}\\label{lemMaximal}\n\tLet $\\Omega$ be a domain with Whitney covering $\\mW$, and let $\\phi$ satisfy \\AJ ,\\ \\AD\\ and \\AT. Assume that $g\\in L^1_{loc}(\\mR^d)$ \\red{is nonnegative} and $0<r<3\\diam(\\Omega)$. For every \\red{$\\eta \\geq \\min(q,p - \\frac pq)$}, $Q\\in\\mW$ and $x\\in \\red{\\Omega}$, we have\n\t\\begin{align}\\label{eqMaximalFar}\n\t\\int_{\\red{\\Omega \\cap \\{|x-y|>r\\}}} \\frac{g(y) \\, dy}{|x-y|^d\\phi(|x-y|)^\\eta}&\\lesssim \\frac{Mg(x)}{\\phi(r) ^\\eta},\\\\\n\t\\sum_{S:D(Q,S)>r}  \\frac{\\int_S g(y) \\, dy}{D(Q,S)^d\\phi(D(Q,S))^\\eta}&\\lesssim \\frac{\\inf_{x\\in Q} Mg(x)}{\\phi(r) ^\\eta},\\label{eqMaximalfar2}\n\t\\end{align}\n\n\n\n\n\n\tand\n\t\\begin{equation}\\label{eqMaximalAllOver}\n\t\\sum_{S\\in\\mathcal{W}} \\frac{l(S)^d}{D(Q,S)^d\\phi(D(Q,S))^{\\eta}} \\lesssim \\frac{1}{\\phi(l(Q))^\\eta}.\n\t\\end{equation}\n\n\n\n\n\\end{lemma}\n\\begin{proof}\n\tLet us look at \\eqref{eqMaximalFar}. For clarity, assume that $\\Omega \\ni x=0$. Since $1\/\\phi$ is decreasing, we get\n\t\\begin{align*}\n\t\\int_{\\red{\\Omega \\cap \\{|y|>r\\}}} \\frac{\\phi(r)^\\eta g(y) \\, dy}{|y|^d\\phi(|y|)^\\eta} &\\red{\\leq} \\sum\\limits_{k=1}^{\\red{N(r)}} \\int_{2^{k-1}r<|y|<2^kr} \\frac {g(y)}{|y|^d}\\frac {\\phi(r)^\\eta}{\\phi(|y|)^\\eta} \\, dy\\\\\n\t&\\lesssim \\sum\\limits_{k=1}^{\\red{N(r)}} \\frac {\\phi(r)^\\eta}{\\phi(2^{k-1}r)^\\eta}\\frac 1 {|B_{2^kr}|}\\int_{2^{k-1}r<|y|<2^kr} g(y) \\, dy \\\\\n\t&\\leq \\sum\\limits_{k=1}^{\\red{N(r)}} \\frac {\\phi(r)^\\eta}{\\phi(2^{k-1}r)^\\eta} Mg(0).\n\t\\end{align*}\n\tThe sum is bounded with respect to $r$ thanks to \\AD.  \n\tIn order to prove \\eqref{eqMaximalfar2} note that if $D(Q,S)>r$, then for every $x\\in Q$, $y\\in S$, we have $|x-y| + r \\lesssim D(Q,S)$. Therefore, by \\AT\\ \\red{and the fact that $\\phi$ is increasing,}\n\n\tfor every $x\\in Q$ we have\n\t\\begin{align*}\n\t&\\sum_{S:D(Q,S)>r} \\frac {\\phi(r)^\\eta \\int_S g(y) \\, dy}{D(Q,S)^d\\phi(D(Q,S))^\\eta} \\lesssim \\int_{\\red{\\Omega}} \\frac{\\red{\\phi(r)^\\eta g(y)} \\, dy}{(|x-y|+ r)^d\\phi(|x-y| + r)^\\eta}\\\\\n\t&\\leq \\int_{\\red{\\Omega \\cap\\{|x-y| > r\\}}} \\frac{\\phi(r)^\\eta g(y) \\, dy}{|x-y|^d\\phi(|x-y|)^\\eta} +  \\int_{|x-y|<r} \\frac {\\phi(r)^\\eta g(y) \\, dy }{\\red{r^d\\phi(r)^\\eta}}\\\\\n\t&\\red{\\lesssim} \\int_{\\red{\\Omega \\cap \\{|x-y| > r\\}}} \\frac{\\phi(r)^\\eta g(y) \\, dy}{|x-y|^d\\phi(|x-y|)^\\eta} + \\frac 1 {|B_r|}\\int_{|x-y|<r}g(y)\\, dy.\n\t\\end{align*}\n\tThe claim follows from the previous estimate. Since the constants in the inequalities \\red{do} not depend on $x$, the same holds for the infimum.\n\t\n\t\\red{Inequality }\\eqref{eqMaximalAllOver} can be obtained by taking $g\\equiv 1$ and $r = l(Q)$ in \\eqref{eqMaximalfar2}. In that case $D(Q,S) > r$ for every $S$, including $Q$.\n\\end{proof}\n\\red{The following lemma is an extension of \\cite[(2.7),(2.8)]{PS}.}\n\\begin{lemma}\\label{lem:phil}\n\t\\red{Let $\\eta \\geq \\min(q,p-\\frac pq)$, $\\kappa\\geq \\frac 1{q-1}$, assume that \\AD\\ and \\AT\\ hold, and assume that $\\mW$ is admissible. Then}\n\t\\begin{equation}\\label{eq:insh}\n\t\\sum_{R: P \\in \\Sh_\\rho(R)} \\phi(l(R))^{-\\eta} \\lesssim \\phi(l(P))^{-\\eta}.\n\t\\end{equation}\n\tFurthermore, if $S\\in\\Sh_{\\red{\\rho}}(R)$, then \n\t\\begin{equation}\\label{eq:chain}\n\t\\sum_{P\\in [S,R]}\\phi(l(P))^{\\kappa} \\lesssim \\phi(l(R))^{\\kappa}.\n\t\\end{equation}\n\\end{lemma}\n\\begin{proof}\n\t\\red{Since the cubes are dyadic, we may and do assume in \\eqref{eq:insh} that $l(P) = 2^{p_0}$ for some $p_0\\in \\mathbb{Z}$. Every $R$ which satisfies $P\\in \\Sh_\\rho(R)$ must be at a distance from $P$ smaller than a multiple of $l(R)$, therefore there can only be a bounded number $K$ of such cubes $R$ with a given side length. Furthermore, the considered cubes must be sufficiently large to contain $P$ in its shadow, that is $l(R) \\geq 2^{p_0-l_0}$ with $l_0\\in \\mathbb{N}_0$ independent of $p_0$. We also obviously have $l(R) < \\diam (\\Omega)$. Thus, the sum in the first assertion can be bounded from above as follows:\n\t\t\\begin{align*}\n\t\t\\sum_{R: P \\in \\Sh_\\rho(R)} \\phi(l(R))^{-\\eta} \\leq K\\sum_{k=p_0-l_0}^{p_0+N(2^{p_0})} \\phi(2^k)^{-\\eta}= K\\sum_{k=p_0-l_0}^{p_0} \\phi(2^k)^{-\\eta}+ K\\sum_{k=p_0+1}^{p_0+N(2^{p_0})} \\phi(2^k)^{-\\eta}.\n\t\t\\end{align*}\n\t\tThe sums are estimated by a multiple of $\\phi(2^{p_0})^{-\\eta}$ using \\AT\\ and \\AD\\ respectively, which proves \\eqref{eq:insh}.}\n\n\n\n\t\n\t\\red{As in the proof of \\cite[(2.8)]{PS} we may deduce that if $S\\in \\Sh_\\rho(R)$, then there is a bounded number $L$ of cubes $P\\in [S,R]$ of a given side length. Furthermore, for every $P\\in[S,R]$ we have $l(P)\\leq 2^{r_0+l_0}$, where $l(R) = 2^{r_0}$ and $l_0$ is a fixed natural number independent of $S$ and $R$. Therefore we estimate \\eqref{eq:chain} as follows:\n\t\t\\begin{align*}\n\t\t\\sum_{P\\in [S,R]}\\phi(l(P))^{\\kappa} \\leq L\\sum_{k=-\\infty}^{r_0+l_0} \\phi(2^k)^\\kappa = L\\sum_{k=-\\infty}^{r_0}\\phi(2^k)^\\kappa + L\\sum_{k=r_0+1}^{r_0+l_0} \\phi(2^k)^\\kappa.\n\t\t\\end{align*}\n\t\tThe first sum is bounded from above by a multiple of $\\phi(2^{r_0})^\\kappa$ because of the second assertion of \\AD\\ and the second is handled by using \\AT. This ends the proof.}\n\\end{proof}\n\\section{Proof of Theorem \\ref{th:main}}\n\\begin{proof}[Proof of Theorem \\ref{th:main}]\n\tObviously it suffices to show that the truncated seminorm dominates the full one up to a multiplicative constant.\n\t\n\tWe will work with dual norms, namely\n\t\\begin{equation}\\label{eq:dual}\n\t\\sup\\limits_{\\substack{g\\geq 0\\\\ \\|g\\|_{L^{p'}(L^{q'}(\\Omega))}\\leq 1}} \\intom\\intom |f(x) - f(y)| |x-y|^{-\\frac dq} \\phi(|x-y|)^{-1} g(x,y) \\, dy \\, dx.\n\t\\end{equation}\n\tFrom now on, $g$ will be like in formula \\eqref{eq:dual}.\n\t\n\tFirst let us take care of the case when $x$ and $y$ are close to each other. By the H\\\"older's inequality, we get\n\t\\begin{align*}\n\t&\\sum_{Q\\in\\mW}\\int_Q\\int_{2Q} \\frac{|f(x)-f(y)|g(x,y)}{|x-y|^{\\frac dq}\\phi(|x-y|)}\\, dy \\, dx\\\\ \\leq &\\sum_{Q\\in\\mW}\\int_Q\\Big(\\int_{2Q} \\frac{|f(x) - f(y)|^q}{|x-y|^d\\phi(|x-y|)^q} \\, dy\\Big)^{\\frac 1q} \\Big(\\int_{2Q} g(x,y)^{q'} \\, dy\\Big)^{\\frac 1{q'}} \\, dx\\\\\n\t\\leq &\\bigg(\\sum_{Q\\in\\mW} \\int_Q\\Big(\\int_{2Q} \\frac{|f(x) - f(y)|^q}{|x-y|^d \\phi(|x-y|)^q} \\, dy\\Big)^{\\frac pq}\\, dx\\bigg)^{\\frac 1p}.\n\t\\end{align*}\n\t\n\tWhat remains is the integral over $(\\Omega\\times\\Omega) \\setminus \\bigcup_{Q\\in\\mW} Q\\times 2Q = \\bigcup_{Q\\in\\mW}Q\\times(\\Omega\\setminus 2Q) = \\bigcup_{Q,S\\in\\mW}Q\\times(S\\setminus 2Q)$. We claim that in this case $|x-y| \\approx D(Q,S)$. Indeed, since $y \\notin 2Q$, we immediately get $l(Q) \\red{\\leq} |x-y|$. Furthermore, if $l(S)\\geq l(Q)$, and $|x-y|\\leq 2l(S)$, then $Q\\subseteq 5S$, and by the definition of the Whitney decomposition $l(Q) \\geq \\frac 12 l(S)$ which proves the claim. Therefore, by \\AT\\\n\n\twe get\n\t\\begin{align}\\nonumber\n\t&\\sum_{Q,S}\\int_Q\\int_{S \\setminus 2Q} \\frac{|f(x) - f(y)|g(x,y)}{|x-y|^{\\frac dq}\\phi(|x-y|)} \\, dy \\, dx\\\\\n\t \\lesssim &\\sum_{Q,S}\\int_Q\\int_S \\frac{|f(x) - f(y)|g(x,y)}{D(Q,S)^{\\frac dq}\\phi(D(Q,S))} \\, dy \\, dx.\\label{eq:podd}\n\t\\end{align}\n\tLet $f_Q = \\frac 1{|Q|} \\int_Q f(x) \\, dx$. By the triangle inequality \\eqref{eq:podd} does not exceed\n\t\\begin{align}\n\n\t&\\sum_{Q,S}\\int_Q\\int_S \\frac{|f(x) - f_Q|g(x,y)}{D(Q,S)^{\\frac dq}\\phi(D(Q,S))}\\, dy \\, dx\\tag{\\textbf{A}}\\label{A}\\\\\n\t+&\\sum_{Q,S}\\int_Q\\int_S \\frac{|f_Q - f_{Q_S}|g(x,y)}{D(Q,S)^{\\frac dq}\\phi(D(Q,S))}\\, dy \\, dx.\\tag{\\textbf{B}}\\label{B}\\\\\n\t+&\\sum_{Q,S}\\int_Q\\int_S \\frac{|f_{Q_S} - f(y)|g(x,y)}{D(Q,S)^{\\frac dq}\\phi(D(Q,S))}\\, dy \\, dx.\\tag{\\textbf{C}}\\label{C}\n\t\\end{align}\n\tUsing H\\\"older's inequality and \\eqref{eqMaximalAllOver} we can estimate \\eqref{A} from above by\n\t\\begin{align}\n\t&\\sum_Q \\int_Q |f(x) - f_Q| \\Big(\\intom g(x,y)^{q'} \\, dy\\Big)^\\frac 1{q'}\\Big(\\sum_S \\frac {l(S)^d}{D(Q,S)^d\\phi(D(Q,S))^q}\\Big)^{\\frac 1q} \\, dx\\nonumber\\\\\n\t\\lesssim &\\sum_Q \\int_Q |f(x) - f_Q|\\Big(\\intom g(x,y)^{q'} \\, dy\\Big)^{\\frac 1{q'}}\\frac 1{\\phi(l(Q))} \\, dx\\label{eq:repA}\\\\\n\t\\lesssim &\\bigg(\\sum_Q \\int_Q \\Big(\\frac{|f(x) - f_Q|}{\\phi(l(Q))}\\Big)^p \\, dx\\bigg)^{\\frac 1p}.\\nonumber\n\t\\end{align}\n\tNow, by the definition of $f_Q$, Jensen's inequality, and \\red{\\AT}\\ we get\n\t\\begin{align*}\n\t\\AAA\\, &\\red{\\lesssim} \\bigg(\\sum_Q \\int_Q \\Big(\\int_Q\\frac{|f(x) - f(y)|^q}{l(Q)^d\\phi(l(Q))^q}\\, dy\\Big)^{\\frac pq} \\, dx\\bigg)^{\\frac 1p}\\\\ &\\red{\\lesssim} \\bigg(\\sum_Q \\int_Q \\Big(\\int_Q\\frac{|f(x) - f(y)|^q}{|x-y|^d\\phi(|x-y|)^q}\\, dy\\Big)^{\\frac pq} \\, dx\\bigg)^{\\frac 1p}.\n\t\\end{align*}\n\tLet us consider \\eqref{B}. If we denote the successor of \\red{$P$} in a chain $[Q,S\\red{)}$ as \\red{$\\mN(P)$}, then by the triangle inequality\n\t\\begin{align*}\n\t\\BB \\leq \\sum_{Q,S}\\Bigg(\\int_Q\\int_S \\frac{g(x,y)}{D(Q,S)^{\\frac dq}\\phi(D(Q,S))}\\, dy \\, dx \\sum_{P\\in[Q,Q_S)} |f_P - f_{\\mN(P)}|\\Bigg).\n\t\\end{align*}\n\tRecall that $\\mN(P)\\subseteq 5P$, and for every $P\\in [Q,Q_S]$, $Q\\in \\Sh(P)$. For such $P$ it is also true that $D(P,S) \\approx D(Q,S)$, see \\cite[(2.6)]{PS}. Therefore, by \\AT\\ we estimate \\BB\\ from above by a multiple of\n\t\\begin{align*}\n\t\\red{\\sum_P\\int_P\\int_{5P} \\frac{|f(\\xi) - f(\\zeta)|}{|P||5P|}\\, d\\xi \\, d\\zeta}\\sum_{Q\\in \\Sh(P)}\\int_Q\\sum_S\\int_S\\frac {g(x,y)}{D(P,S)^{\\frac dq}\\phi(D(P,S))} \\, dy \\, dx.\n\t\\end{align*}\n\tBy the H\\\"older's inequality and \\eqref{eqMaximalAllOver} this expression approximately less than or equal to\n\t\\begin{align}\n\t\\sum_P\\int_P\\int_{5P} \\frac{|f(\\xi) - f(\\zeta)|}{|P||5P|}\\, d\\xi \\, d\\zeta \\int_{\\SH(P)}\\Big(\\int_\\Omega g(x,y)^{q'} \\, dy\\Big)^{\\frac 1{q'}}\\frac 1{\\phi(l(P))} \\, dx.\\label{eq:repB} \n\t\\end{align}\n\tLet $G(x) = \\big(\\int_\\Omega g(x,y)^{q'} \\, dy\\big)^{\\frac 1{q'}}$. By \\cite[Lemma 2.7]{PS} we have $\\int_{\\SH (P)} G(x) \\, dx \\hspace{-2.07pt}\\lesssim \\inf\\limits_{y\\in P} MG(y) l(P)^d$. Using this, the Jensen's inequality, the H\\\"older's inequality, and the fact that the maximal operator is continuous in $L^{p'}$, $p'>1$, we obtain\n\t\\begin{align*}\n\t\\BB &\\lesssim \\sum_P\\frac 1{|P||5P|}\\frac {l(P)^d}{\\phi(l(P))}\\int_P\\int_{5P} |f(\\xi) - f(\\zeta)|MG(\\zeta)\\, d\\xi \\, d\\zeta\\nonumber\\\\\n\t&\\lesssim \\sum_P\\int_P\\frac {MG(\\zeta)}{l(P)^{\\frac dq}\\phi(l(P))}\\bigg(\\int_{5P} |f(\\xi) - f(\\zeta)|^q \\, d\\xi\\bigg)^{\\frac 1q} \\, d\\zeta\\nonumber\\\\\n\t&\\lesssim \\Bigg(\\sum_P\\int_P\\bigg(\\int_{5P}\\frac{|f(\\xi) - f(\\zeta)|^q}{l(P)^d\\phi(l(P))^q} \\, d\\xi\\bigg)^{\\frac pq} \\, d\\zeta\\Bigg)^{\\frac 1p}.\n\t\\end{align*}\n\tSince $|\\xi - \\zeta| \\leq 5l(P)$, \\BB\\ is estimated.\n\t\n\tNow \\red{we will work on \\eqref{C}. Since $D(Q,S) \\approx l(Q_S)$, by \\AT\\ we obtain\n\t\n\t\t\\begin{equation*}\n\t\t\\CC \\lesssim \\sum\\limits_{Q,S}\\int_Q\\int_S \\frac{|f_{Q_S} - f(y)|g(x,y)}{l(Q_S)^{\\frac dq}\\phi(l(Q_S))}\\, dy\\, dx.\n\t\t\\end{equation*}\n\t\n\t\n\t\n\t\tFurthermore, for every admissible chain we have $Q,S\\in \\Sh(Q_S)$, therefore for every $Q,S \\in \\mW$ we have\n\t\t\\begin{equation*}\n\t\t(Q_S,Q,S)\\in \\bigcup\\limits_{R\\in \\mW}\\{(R,P,P') :  P,P' \\in \\Sh (R)\\}.\n\t\t\\end{equation*}\n\t\tConsequently, the following estimate holds:}\n\n\n\t\\begin{align}\n\t\\label{Cstart}\\CC&\\lesssim \\sum\\limits_{R\\in \\mW}\\sum\\limits_{Q\\in\\Sh(R)}\\sum\\limits_{S\\in\\Sh(R)} \\int_Q\\int_S \\frac{|f_{R} - f(y)|g(x,y)}{l(R)^{\\frac dq} \\phi(l(R))}\\, dy \\, dx.\n\t\\end{align}\n\tBy H\\\"older's inequality the above expression does not exceed\n\t\\begin{align*}\n\t&\\sum_{R\\in\\mW}\\frac{\\Big(\\int_{\\SH(R)} |f_R - f(y)|^q \\, dy\\Big)^{\\frac 1q}}{l(R)^{\\frac dq}\\phi(l(R))}\\int_{\\SH(R)}\\Bigg(\\int_{\\SH(R)} g(x,y)^{q'} \\, dy\\Bigg)^{\\frac {1}{q'}} \\, dx\\\\\n\t\\leq &\\sum_{R\\in\\mW}\\frac{\\Big(\\int_{\\SH(R)} |f_R - f(y)|^q \\, dy\\Big)^{\\frac 1q}}{l(R)^{\\frac dq}\\phi(l(R))}\\int_{\\SH(R)} G(x) \\, dx.\n\t\\end{align*}\n\t\\red{By the last estimate of \\cite[Lemma 2.7]{PS}}, the fact that $\\inf\\limits_R MG\\leq \\frac 1{l(R)^d}\\int_R MG$, and the H\\\"older's inequality we get that\n\t\\begin{align*}\n\t\\CC &\\lesssim \\sum_{R\\in\\mW}\\frac{1}{l(R)^{\\frac {d}q}\\phi(l(R))}\\Bigg(\\int_{\\SH(R)} |f_R - f(y)|^q \\, dy\\Bigg)^{\\frac 1q} \\int_R MG(\\xi) \\, d\\xi\\\\\n\t&\\leq \\Bigg(\\sum_{R\\in\\mW}\\int_R\\frac{1}{l(R)^{\\frac {dp}q}\\phi(l(R))^p}\\Big(\\int_{\\SH(R)} |f_R - f(y)|^q \\, dy \\Big)^{\\frac pq}\\, d\\xi \\Bigg)^{\\frac 1p}\\hspace{-3pt} \\|MG\\|_{L^{p'}(\\Omega)}\\\\\n\t&\\leq \\Bigg(\\sum_{R\\in\\mW}\\frac{l(R)^d}{l(R)^{\\frac {dp}q}\\phi(l(R))^p}\\Big(\\sum_{S\\in\\Sh(R)}\\int_{S} |f_R - f(y)|^q \\, dy \\Big)^{\\frac pq}\\Bigg)^{\\frac 1p}.\n\t\\end{align*}\n\tLet $[S,R]$ be an admissible chain between $S$ and $R$. Then, after using the inequality $|f_R - f(y)|^q \\lesssim |f_R - f_S|^q + |f_S - f(y)|^q$, we get\n\t\\begin{align*}\n\t\\CC^p &\\lesssim \\sum_{R\\in\\mW} \\frac{l(R)^d}{l(R)^{\\frac {dp}q}\\phi(l(R))^p}\\Bigg(\\sum_{S\\in\\Sh(R)}\\Big|\\sum_{P\\in [S,R)}f_P - f_{\\mN(P)}\\Big|^{q} l(S)^d\\Bigg)^{\\frac pq}\\\\\n\t&+ \\sum_{R\\in\\mW} \\frac{l(R)^d}{l(R)^{\\frac {dp}q}\\phi(l(R))^p}\\Bigg(\\sum_{S\\in\\Sh(R)}\\int_S |f_S - f(y)|^q \\, dy\\Bigg)^{\\frac pq} = \\CJ + \\CD.\n\t\\end{align*}\n\tIf we write $f_P - f_{\\mN(P)} = (f_P - f_{\\mN(P)}) \\frac{\\phi(l(P))^{\\frac 1q}}{\\phi(l(P))^{\\frac 1q}}$, then by H\\\"older's inequality we estimate \\CJ\\ from above by\n\t\\begin{align*}\n\t&\\sum_{R\\in\\mW} \\frac{l(R)^d}{l(R)^{\\frac {dp}q}\\phi(l(R))^p}\\Bigg(\\sum_{S\\in\\Sh(R)}\\sum_{P\\in \\red{[S,R)}}\\frac{|f_P - f_{\\mN(P)}|^ql(S)^d}{\\phi(l(P))} \\Big(\\sum_{P\\in \\red{[S,R)}}\\phi(l(P))^{\\frac {q'}{q}}\\Big)^{\\frac {q}{q'}}\\Bigg)^{\\frac pq}.\n\t\\end{align*}\n\tBy Lemma \\ref{lem:phil}\n\t\\begin{equation*}\n\t\\CJ \\lesssim \\sum_{R\\in\\mW} \\frac{l(R)^d}{l(R)^{\\frac {dp}q}}\\phi(l(R))^{\\frac pq -p}\\bigg(\\sum_{S\\in\\Sh(R)}\\sum_{P\\in \\red{[S,R)}}\\frac{|f_P - f_{\\mN(P)}|^q}{\\phi(l(P))} l(S)^d\\bigg)^{\\frac pq}.\n\t\\end{equation*}\n\tLet us take $\\rho_2$ large enough for $S\\in\\Sh^2(\\red{P}) := \\Sh_{\\rho_2}(\\red{P})$, and $P\\in \\Sh^2(R)$ to hold. Then $\\sum_{S\\in\\Sh(R)}\\sum_{P\\in \\red{[S,R)}} \\lesssim \\sum_{P\\in \\Sh^2(R)}\\sum_{S\\in\\Sh^2(P)}$. We denote the sum of the neighbors of $P$ as $U_P$. Since $\\sum_{S\\in \\Sh^2(P)} l(S)^d\\lesssim l(P)^d$, we get that, up to a multiplicative constant, \\CJ\\ does not exceed\n\t\\begin{equation*}\n\t\\sum_{R\\in\\mW} \\frac{l(R)^d}{l(R)^{\\frac {dp}q}}\\phi(l(R))^{\\frac pq -p}\\Bigg(\\sum_{P\\in\\Sh^2(R)}\\frac{\\red{(l(P)^{-d}\\int_{U_P} |f_P - f(\\xi)|\\, d\\xi)}^q}{\\phi(l(P))} l(P)^d\\Bigg)^{\\frac pq}.\n\t\\end{equation*}\n\tSince $p\\geq q$, we can use the H\\\"{o}lder's inequality with exponent $\\frac pq$ to estimate this expression from above by\n\t\\begin{align*}\n\t&\\sum_{R\\in\\mW} \\frac{l(R)^d}{l(R)^{\\frac {dp}q}}\\phi(l(R))^{\\frac pq -p}\\Bigg(\\sum_{P\\in\\Sh^2(R)}\\frac{\\red{(l(P)^{-d}\\int_{U_P} |f_P - f(\\xi)|\\, d\\xi)}^p}{\\phi(l(P))^{\\frac pq}} l(P)^d\\Bigg)\\Bigg(\\sum_{P\\in\\Sh^2(R)} l(P)^d\\Bigg)^{(1 - \\frac qp)\\frac pq}\\\\\n\t\\lesssim &\\sum_{R\\in\\mW}\\sum_{P\\in\\Sh^2(R)} \\phi(l(R))^{\\frac pq -p}\\frac{\\red{(l(P)^{-d}\\int_{U_P} |f_P - f(\\xi)|\\, d\\xi)}^pl(P)^d}{\\phi(l(P))^\\frac pq}\\\\\n\t\\lesssim &\\sum_{P\\in \\mW}\\frac{\\red{(l(P)^{-d}\\int_{U_P} |f_P - f(\\xi)|\\, d\\xi)}^pl(P)^d}{\\phi(l(P))^\\frac pq}\\sum_{R: P\\in\\Sh^2 (R)} \\phi(l(R))^{\\frac pq -p}.\n\t\\end{align*}\n\tFurthermore, Lemma \\ref{lem:phil} and Jensen's inequality give\n\t\\begin{align}\n\t\\CJ &\\lesssim \\sum_{P\\in \\mW}\\frac{\\red{(l(P)^{-d}\\int_{U_P} |f_P - f(\\xi)|\\, d\\xi)}^pl(P)^d}{\\phi(l(P))^p}\\nonumber\\\\ &\\lesssim \\sum_{P\\in \\mW}\\int_{U_P}\\frac{|f_P - f(\\xi)|^p}{\\phi(l(P))^p}\\, d\\xi\\label{same}\\\\\n\t&\\leq \\sum_{P\\in \\mW}\\int_{U_P}\\Bigg(\\int_P\\frac{|f(\\zeta) - f(\\xi)|^q}{l(P)^d\\phi(l(P))^q}\\, d\\zeta\\Bigg)^{\\frac pq}\\, d\\xi.\\nonumber\n\t\\end{align}\n\tSince $U_P \\subseteq 5P$ we have finished estimating $\\CJ$.\n\t\n\tNow we proceed with \\CD. By H\\\"{o}lder's inequality\n\t\\begin{align*}\n\t\\CD &= \\sum_{R\\in\\mW} \\frac{l(R)^{d(1-\\frac pq)}}{\\phi(l(R))^p}\\Bigg(\\sum_{S\\in\\Sh(R)}\\int_S |f_S - f(\\xi)|^q \\, d\\xi\\frac{l(S)^{d(1 - \\frac qp)}}{l(S)^{d(1 - \\frac qp)}}\\Bigg)^{\\frac pq}\\\\\n\t&\\leq \\sum_{R\\in\\mW} \\frac{l(R)^{d(1-\\frac pq)}}{\\phi(l(R))^p}\\Bigg(\\sum_{S\\in\\Sh(R)} l(S)^d\\Bigg)^{\\frac pq - 1}\\sum_{S\\in\\Sh(R)}\\frac{(\\int_S|f_S - f(\\xi)|^q \\, d\\xi)^{\\frac pq}}{l(S)^{d(\\frac pq - 1)}}\\\\\n\t&\\lesssim \\sum_{R\\in\\mW}\\sum_{S\\in\\Sh(R)} \\frac{(\\int_S|f_S - f(\\xi)|^q \\, d\\xi)^{\\frac pq}}{l(S)^{d(\\frac pq - 1)}\\phi(l(R))^p}.\n\t\\end{align*}\n\tBy rearranging and using Lemma \\ref{lem:phil} we obtain \n\t\\begin{align*}\n\t\\CD &\\lesssim\\sum_{S\\in \\mW}\\frac{(\\int_S|f_S - f(\\xi)|^q \\, d\\xi)^{\\frac pq}}{l(S)^{d(\\frac pq - 1)}} \\sum_{R: S\\in\\Sh(R)} \\phi(l(R))^{-p}\\\\ &\\lesssim \\sum_{S\\in \\mW}\\Bigg(\\int_S\\frac{|f_S - f(\\xi)|^q }{l(S)^d}\\, d\\xi\\Bigg)^{\\frac pq} \\frac {l(S)^d}{\\phi(l(S))^p}.\n\t\\end{align*}\n\tHence, by Jensen's inequality,\n\t\\begin{equation*}\n\t\\CD \\lesssim \\sum_{S\\in\\mW}\\frac {l(S)^d}{\\phi(l(S))^p} \\int_S\\frac {|f_S - f(\\xi)|^p}{l(S)^d} \\, d\\xi  = \\sum_{S\\in\\mW}\\int_S \\frac {|f_S - f(\\xi)|^p}{\\phi(l(S))^p}\\, d\\xi.\n\t\\end{equation*}\n\tThus we have arrived at the same situation as in \\eqref{same} and the proof is finished (we may need to enlarge the constant $C_\\mW$ which can be done by diminishing the cubes in the Whitney decomposition).\n\\end{proof}\n\\section{Examples of $\\phi$}\n\\subsection{Positive examples}\nWe will present some examples of kernels which satisfy \\AD\\ and \\AT.\n\\begin{example}\n\tStable scaling is more than enough for \\AD\\ to hold. Indeed, if we assume that \\red{there exist $\\beta_1,\\beta_2 \\in (0,1)$ for which we have\n\t\t$$\\lambda^{\\beta_1} \\lesssim \\frac{\\phi(\\lambda r)}{\\phi(r)} \\lesssim \\lambda^{\\beta_2},\\quad r>0,\\ \\lambda \\leq 1,$$\n\t\tthen by the first inequality we get \\AT\\ and by the second inequality the series in \\AD\\ are geometric and independent of $r$.}\n\t\n\tLet us examine the constant $C_2$ in \\AD\\ for $p=q=2$, $\\alpha\\in (0,2)$, and the kernels of the form $K(x,y) = (2-\\alpha)|x-y|^{-d-\\alpha}$, i.e. $\\phi (t) = (2-\\alpha)t^{\\alpha\/2}$. \\red{In this case $\\frac 1{q-1} = \\min(q,p-\\frac pq) = 1$ and} for every $r>0$ we have\n\t$$\\sum\\limits_{k=1}^\\infty \\frac{\\phi(r)}{\\phi(2^kr)} = \\red{\\sum\\limits_{k=1}^\\infty \\frac{\\phi(2^{-k}r)}{\\phi(r)} =} \\sum\\limits_{k=1}^\\infty \\frac 1{(2^{\\alpha\/2})^k} = \\frac{1}{2^{\\alpha\/2} - 1}.$$\n\tThis quantity is bounded as $\\alpha\\to 2^-$. Since the constant in \\AT\\ is also bounded in this case, we get that the comparability in Theorem \\ref{th:main} is uniform  for $\\alpha \\in (\\eps,2)$ for every $\\eps>0$.\n\\end{example}\n\\begin{example}\n\tAssume that $\\Omega$ is bounded. Let \\red{$\\gamma\\in (0,1)$, $\\phi(r) = [\\log(1+r)]^\\gamma$} and let $R=\\diam(\\Omega)$. \\red{Note that for $r > 0$ we have\n\t\t$$1\\leq \\frac {\\log(1+2r)}{\\log(1+r)}\\leq 2.$$\n\t\tIndeed, by looking at the derivative we see that the ratio is decreasing thus the inequalities result from its limits at $0^+$ and at $\\infty$. Therefore, $\\phi$ satisfies \\AT. Furthermore for $r<R$ the lower bound can be replaced with a constant $C = C(R) >1$, hence both series in \\AD\\ become geometric thus it is satisfied.}\n\n\n\n\n\n\n\\end{example}\n\\subsection{O-regularly varying functions}\\label{sec:zal}\n\\begin{definition}\n\tWe say that $\\phi$ is O-regularly varying at infinity if there exist $a, b\\in \\mR$ and $A,B,R>0$ such that\n\t\\begin{equation}\n\tA\\bigg(\\frac{r_2}{r_1}\\bigg)^a \\leq \\frac{\\phi(r_2)}{\\phi(r_1)} \\leq B\\bigg(\\frac{r_2}{r_1}\\bigg)^b\\label{eq:oreg}\n\t\\end{equation}\n\tholds whenever $R<r_1<r_2$. Analogously, $\\phi$ is O-regularly varying at zero if \\eqref{eq:oreg} holds for $0<r_1<r_2<R$. The supremum of $a$ and the infimum of $b$ for which \\eqref{eq:oreg} is satisfied are called lower, respectively upper, Matuszewska indexes (or lower\/upper indexes). \n\\end{definition}\n\\red{A nice short review of the O-regularly varying functions can be found in the work of Grzywny and Kwa\\'s{}nicki \\cite[Appendix A]{GRZYWNY20181}, for further reading we refer to the book by Bingham, Goldie, and Teugels \\cite{bingham}.}\n\nAssume \\textbf{A2} and \\textbf{A3}. \\red{We will show that the assumptions enforce O-regular variation with positive lower index at 0 and, for unbounded $\\Omega$, at infinity by using Proposition A.1 of \\cite{GRZYWNY20181}.  Note that by \\AT\\ for $r>0$, $k\\in \\mathbb{Z}$, and $z\\in [2^{k-1}r,2^{k}r]$ we have $\\phi(z) \\approx \\phi(2^{k}r)$.}\n\n\\red{We first consider the regular variation at zero using \\cite[Proposition A.1 (c)]{GRZYWNY20181}. Let $R = \\diam(\\Omega)$ and $t_2 = \\frac 1{q-1}$. Then, for every $r\\in (0,R)$ and $\\eta \\in \\mR$ we have\n\t\\begin{align*}\n\t\\int_0^r z^{-\\eta}\\phi(z)^{t_2}\\frac{dz}{z} \\approx \\sum_{k=1}^\\infty \\phi(2^{-k}r)^{t_2} (2^{-k}r)^{-\\eta} = r^{-\\eta}\\phi(r)^{t_2}\\sum_{k=1}^\\infty \\frac{\\phi(2^{-k}r)^{t_2}}{\\phi(r)^{t_2}} 2^{k\\eta}.\n\t\\end{align*}\n\tBy \\AD\\ the latter sum is finite for $\\eta \\leq 0$, it is also bounded away from 0 because of \\AT. Therefore we obtain that $\\phi^{t_2}$ (and thus, also $\\phi$) has to be O-regularly varying at 0 with some lower index $a_0>0$, that is\n\t$$\\frac{\\phi(r_2)}{\\phi(r_1)} \\gtrsim \\bigg(\\frac {r_2}{r_1}\\bigg)^{a_0\/t_2},\\quad 0<r_1\\leq r_2\\leq R.$$\n\tThe above condition yields a power-type decay at 0 for $\\phi$. This could also be obtained using the other summation condition from \\AD\\ by applying \\cite[Proposition A.1 (d)]{GRZYWNY20181}.}\n\n\\red{The behavior of $\\phi$ at infinity only comes into play when $\\Omega$ is unbounded, thus we assume that $\\diam (\\Omega) = \\infty$ for the remainder of this subsection. Let $r>0$, $\\eta \\in \\mR$, and $t_1 = \\min(q,p - \\frac pq)$. We have \n\t\\begin{align*}\n\t\\int_r^\\infty z^{-\\eta} \\phi(z)^{-t_1} \\frac{dz}{z} \\approx \\sum_{k=1}^\\infty \\phi(2^kr)^{-t_1}(2^kr)^{-\\eta} = r^{-\\eta}\\phi(r)^{-t_1}\\sum_{k=1}^\\infty \\frac{\\phi(r)^{t_1}}{\\phi(2^kr)^{t_1}}2^{-k\\eta}.\n\t\\end{align*}\n\tBy \\AD\\ and \\AT\\ the sum is finite and bounded away from 0 if $\\eta \\geq 0$. Thus $\\phi^{-t_1}$ is O-regularly varying at infinity with upper index $-a_\\infty<0$, which is equivalent to the O-regular variation with lower index $a_\\infty$ for $\\phi^{t_1}$:\n\t$$\\frac{\\phi(r_2)}{\\phi(r_1)} \\gtrsim \\bigg(\\frac {r_2}{r_1}\\bigg)^{a_\\infty\/t_1}, \\quad R<r_1\\leq r_2<\\infty.$$}\n\n\n\\subsection{Negative examples}\nWe will show some examples for which the seminorms \\eqref{eq:fullsemi} and \\eqref{eq:truncated} are not comparable. Assume for clarity that $p=q=2$.\n\\begin{example}\n\tLet $\\Omega = (0,1)\\subset \\mR$, and let $K(x,y)  \\equiv 1$. Consider the function $f(x) = x^{-\\gamma}$ with $\\gamma\\in (0,\\frac 12)$. A direct calculation shows that\n\t\\begin{equation}\\label{eq:exwhole}\\int_0^1\\int_0^1 (f(x)-f(y))^2 \\, dy \\, dx = 2\\bigg(\\frac 1{1-2\\gamma} - \\frac 1{(1-\\gamma)^2}\\bigg).\n\t\\end{equation}\n\tIn particular, $f$ belongs to the corresponding Sobolev space (actually the \"Sobolev space\" is $L^2(\\Omega)$ in this case).\n\tLet $\\eps \\in (0,1)$. We have\n\n\n\n\n\n\n\n\n\n\n\n\n\t\\begin{align}\\nonumber\n\t&\\int_0^1\\int_{x-\\eps\\delta(x)}^{x+\\eps\\delta(x)}(f(x) - f(y))^2\\, dy \\, dx\\leq \\int_0^1\\int_{x(1-\\eps)}^{x(1+\\eps)}(f(x) - f(y))^2\\, dy \\, dx \\\\\n\t= &\\red{\\frac {\\eps}{1-\\gamma} - \\frac{(1+\\eps)^{1-\\gamma} - (1-\\eps)^{1-\\gamma}}{(1-\\gamma)^2} + \\frac {(1+\\eps)^{1-2\\gamma} - (1-\\eps)^{1-2\\gamma}}{(1-2\\gamma)(2-2\\gamma)} .}\\label{eq:excut}\n\t\\end{align}\n\tAs $\\gamma \\to \\frac 12^-$ the ratio of the right hand side of \\eqref{eq:exwhole} and \\eqref{eq:excut} goes to infinity which shows that in this case the result from Theorem \\ref{th:main} does not hold.\n\\end{example} \n\\begin{example}\n\tThe preceding example gives an idea on how to show an analogous fact for any nonzero $K$ such that $K(0,\\cdot)\\in L^1([0,1])$.\n\tOn the restricted domain of integration we have $x\\approx y$. Therefore $|\\frac 1{x^\\gamma} - \\frac 1{y^\\gamma}| \\lesssim \\frac 1{x^\\gamma}$, hence\n\t\\begin{align} \\nonumber\n\t&\\int_0^1\\int_{B(x,\\varepsilon \\delta(x))} \\Big(\\frac 1{x^\\gamma} - \\frac 1{y^{\\gamma}}\\Big)^2 K(x,y)\\, dy \\, dx\\\\\\label{eq:trl1} \\lesssim &\\int_0^1 \\frac 1{x^{2\\gamma}} \\int_{B(x,\\eps\\delta(x))} K(x,y) \\, dy \\, dx.\n\t\\end{align}\n\tOn the other hand, since $K$ is nontrivial, there exists $\\eta>0$ such that for every $x\\in (0,\\eta)$ we have $\\int_\\eta^1 K(x,y) \\, dy \\geq C > 0$. Therefore\n\t\\begin{align}\n\t\\int_0^1 \\int_0^1 \\Big(\\frac 1{x^\\gamma} - \\frac 1{y^{\\gamma}}\\Big)^2 K(x,y) \\, dy \\, dx &\\geq \\int_0^{\\eta\/2} \\int_\\eta^1 \\Big(\\frac 1{x^\\gamma} - \\frac 1{\\eta^{\\gamma}}\\Big)^2 K(x,y) \\, dy \\, dx\\nonumber\\\\ &\\gtrsim \\int_0^{\\eta\/2} \\frac 1 {x^{2\\gamma}} \\int_{\\eta}^{1} K(x,y) \\, dy \\, dx\\nonumber\\\\\n\t&\\gtrsim \\int_0^{\\eta\/2} \\frac 1{x^{2\\gamma}} \\, dx.\\label{eq:whl1}\n\t\\end{align}\n\tNote that \\eqref{eq:trl1} is of the form $\\int_0^1 \\frac {f(x)}{x^{2\\gamma}} \\, dx$ with $f(x)$ bounded and $\\lim\\limits_{x\\to 0^+} f(x) = 0$. Let us fix an arbitrarily small $\\xi > 0$, and let $\\rho$ be sufficiently small so that $f(x)\\leq \\xi$ for $x\\in(0,\\rho)$. If we separate $\\int_0^1 = \\int_0^\\rho + \\int_\\rho^1$, then we see that the ratio of \\eqref{eq:trl1} and \\eqref{eq:whl1} tends to 0 as $\\gamma \\to \\frac 12$.\n\\end{example}\n\\begin{remark}\n\tIn previous examples the kernel was integrable. This means that\n\t\\begin{align*}\\int_\\Omega\\int_\\Omega (f(x) - f(y))^2 K(x,y) \\, \\red{dy \\, dx} \\leq 2\\int_\\Omega \\int_\\Omega f(x)^2 K(x,y) \\, dy \\, dx \\leq 2\\|f\\|_{L^2(\\Omega)}^2\\|K(0,\\cdot)\\|_{L^1(\\mR^d)}.\n\t\\end{align*}\n\tTherefore, even though the quadratic forms \\eqref{eq:fullsemi} and \\eqref{eq:truncated} are incomparable, the Triebel--Lizorkin norm $\\|\\cdot\\|_{F_{p,q}(\\Omega)}$ is comparable when we replace the full seminorm with the truncated one.\n\\end{remark}\n\\begin{example}\n\tFor $K(x,y) = |x-y|^{-1}$ on $\\Omega=(0,1)$ the seminorms also fail to be comparable. Consider the functions $f_n(x) = n \\wedge \\frac 1x$. Since $$\\int_0^1\\int_0^{\\red{x}} (f(x) - f(y))^2K(x,y) \\red{\\, dy \\, dx} = \\frac 12 \\int_0^1\\int_0^1 (f(x) - f(y))^2K(x,y) \\red{\\, dy \\, dx},$$\n\twe will assume that \\red{$y<x$}, and work only with the integral on the left hand side. Note that for $f_n$, the integral over $(0,\\frac 1n)^{2}$ vanishes. We split as follows\n\t\\begin{align}\n\t\\int_0^1 \\int_0^{\\red{x}} (f_n(x) - f_n(y))^2 K(x,y) \\red{\\, dy \\, dx} &= \\int_{1\/n}^{1}\\int_{1\/n}^{\\red{x}} \\bigg(\\frac 1x - \\frac 1y\\bigg)^2 K(x,y)\\, \\red{dy \\, dx}\\label{eq:hil1}\\\\\n\t&+ \\int_{1\/n}^1\\int_0^{1\/n}\\bigg(n - \\red{\\frac 1x}\\bigg)^2K(x,y) \\red{\\, dy \\, dx}. \\label{eq:hil2}\n\t\\end{align}\n\tWe first compute the right hand side of \\eqref{eq:hil1}. Note that the integrand is equal to $\\frac {(\\red{x}-\\red{y})^2}{\\red{y}^2\\red{x}^2}\\cdot\\frac{1}{\\red{x}-\\red{y}} = \\frac {\\red{x}-\\red{y}}{\\red{y}^2\\red{x}^2}$.\n\t\\begin{align*}\n\t\\int_{1\/n}^1 \\int_{1\/n}^{\\red{x}} \\frac{\\red{x}-\\red{y}}{\\red{y}^2\\red{x}^2} \\, d\\red{y} \\, d\\red{x} = \t\\int_{1\/n}^1 \\int_{1\/n}^{\\red{x}} \\frac{1}{\\red{y}^2\\red{x}} \\, d\\red{y} \\, d\\red{x} - \t\\int_{1\/n}^1 \\int_{1\/n}^{\\red{x}}\\frac{1}{\\red{y}\\red{x}^2} \\, d\\red{y} \\, d\\red{x}= n\\log n - 2n + \\log n +2.\n\t\\end{align*}\n\tFor \\eqref{eq:hil2} we only show the asymptotics.\n\t\\begin{align*}\n\t&\\int_{1\/n}^1\\int_0^{1\/n}\\bigg(n - \\frac 1{\\red{x}}\\bigg)^2K(x,y) \\, d\\red{y} \\, d\\red{x}\\\\ = &\\int_{1\/n}^1 \\bigg(n - \\frac 1{\\red{x}}\\bigg)^2\\bigg(\\log \\red{x} - \\log \\bigg(\\red{x}-\\frac 1n\\bigg)\\bigg)\\, d\\red{x}\\\\\n\t= &-n^2\\int_{1\/n}^1\\bigg(1 - \\frac 1{n\\red{x}}\\bigg)^2\\log\\bigg(1 - \\frac 1{n\\red{x}}\\bigg) \\, d\\red{x}\\\\ = &-n\\int_0^{1-1\/n}\\frac{t^2}{(1-t)^2}\\log t\\, dt.\\\\\n\t\\end{align*}\n\tFor $n>2$ we split the latter integral: $\\int_0^{1-1\/n} = \\int_0^{1\/2} + \\int_{1\/2}^{1-1\/n}$. The first one converges, i.e. it is a (negative) constant. In the second one $t^2 \\approx 1$, and $\\frac{\\log t}{1-t} \\approx -1$, therefore\n\t\\begin{equation}\n\t-n\\int_0^{1-1\/n}\\frac{t^2}{(1-t)^2}\\log t\\, dt \\approx n\\bigg(1 + \\int_{1\/2}^{1-1\/n} \\frac {dt}{1-t}\\bigg) = n(1 + \\log n - \\log 2).\\label{eq:t1mt}\n\t\\end{equation}\n\tThus we get the asymptotics\n\t\\begin{align}\n\t\\int_0^1 \\int_0^1 (f_n(x) - f_n(y))^2 K(x,y) \\, d\\red{y} \\, d\\red{x} \\approx n\\log n. \\label{eq:fulhil}\n\t\\end{align}\n\tNow consider the truncated case. For clarity, assume that $\\epsilon = \\frac 12$. \n\t\\begin{align}\n\t\\int_0^1 \\int_{\\red{x}\/2}^{\\red{x}} (f_n(x) - f_n(y))^2 K(x,y) \\, d\\red{y} \\, d\\red{x} &= \\int_{2\/n}^{1}\\int_{\\red{x}\/2}^{\\red{x}} \\bigg(\\frac 1x - \\frac 1y\\bigg)^2 K(x,y)\\, d\\red{y} \\, d\\red{x}\\label{eq:trhil1}\\\\\n\t&+ \\int_{1\/n}^{2\/n}\\int_{1\/n}^{\\red{x}}\\bigg(\\frac 1x - \\frac 1y\\bigg)^2K(x,y) \\, d\\red{y} \\, d\\red{x} \\label{eq:trhil2}\\\\\n\t&+ \\int_{1\/n}^{2\/n}\\int_{\\red{x}\/2}^{1\/n}\\bigg(n - \\frac 1{\\red{x}}\\bigg)^2K(x,y) \\, d\\red{y} \\, d\\red{x}.\\label{eq:trhil3}\n\t\\end{align}\n\tFor the right hand side of \\eqref{eq:trhil1} and \\eqref{eq:trhil2} we note that\n\t\\begin{equation*}\n\t\\int_{2\/n}^{1}\\int_{\\red{x}\/2}^{\\red{x}} \\bigg(\\frac 1x - \\frac 1y\\bigg)^2 K(x,y)\\, d\\red{y} \\, d\\red{x} \\leq \\int_{2\/n}^1\\int_{\\red{x}\/2}^{\\red{x}} \\frac 1{\\red{y}^2\\red{x}} \\, d\\red{y} \\, d\\red{x} = \\frac n2 - 1,\n\t\\end{equation*}\n\tand\n\t\\begin{equation*}\n\t\\int_{1\/n}^{2\/n}\\int_{1\/n}^{\\red{x}}\\bigg(\\frac 1x - \\frac 1y\\bigg)^2K(x,y) \\, d\\red{y} \\, d\\red{x} \\leq \\int_{1\/n}^{2\/n}\\int_{1\/n}^{\\red{x}} \\frac 1{\\red{y}^2\\red{x}} \\, d\\red{y} \\, d\\red{x} = n\\log 2 - \\frac n2.\n\t\\end{equation*}\n\tThe last integral \\eqref{eq:trhil3} is estimated as follows\n\t\\begin{align*}\n\t&\\int_{1\/n}^{2\/n}\\int_{\\red{x}\/2}^{1\/n}\\bigg(n - \\frac 1{\\red{x}}\\bigg)^2K(x,y) \\, d\\red{y} \\, d\\red{x}\\\\\n\t = &\\int_{1\/n}^{2\/n} \\bigg(n - \\frac 1{\\red{x}}\\bigg)^2\\bigg(\\log \\frac {\\red{x}}2 - \\log \\bigg(\\red{x} - \\frac 1n\\bigg)\\bigg) \\, d\\red{x}\\\\\n\t= &-n^2 \\int_{1\/n}^{2\/n} \\bigg(1 - \\frac 1{n\\red{x}}\\bigg)^2 \\bigg(\\log \\bigg(1 - \\frac 1{n\\red{x}}\\bigg) + \\log 2 \\bigg) \\, d\\red{x} \\\\\\leq\n\t&-n\\int_0^{1\/2}\\frac{t^2}{(1-t)^2}\\log t\\, dt \\approx n.\n\t\\end{align*}\n\tTo conclude, we get\n\t\\begin{equation}\\label{eq:trhil}\n\t\\int_0^1\\int_{B(\\red{x},\\delta(\\red{x})\/2)} (f_n(x) - f_n(y))^2 K(x,y) \\, d\\red{y} \\, d\\red{x} \\lesssim n.\n\t\\end{equation}\n\tSince the ratio of the right hand sides of \\eqref{eq:fulhil} and \\eqref{eq:trhil} diverges as $n\\to\\infty$, our claim is proven.\n\\end{example}\n\\section{The 0-order kernel}\n\\begin{theorem}\\label{th:0order}\n\tLet $\\Omega$ be a bounded uniform domain. Then, if $1< q\\leq p<\\infty$, then for every $0<\\theta\\leq 1$\n\t\\begin{align}\n\t&\\bigg(\\int_{\\Omega}\\bigg(\\int_{\\Omega} \\frac{|f(x) - f(y)|^q}{|x-y|^d}\\, dy\\bigg)^{\\frac pq}\\, dx\\bigg)^{\\frac 1p}\\label{eq:0order}\\\\ \\lesssim &\\,\\bigg(\\int_{\\Omega}\\bigg(\\int_{B(x,\\theta\\delta(x))} \\frac{|f(x) - f(y)|^q}{|x-y|^d}(\\big|\\hspace{-1pt}\\log|x-y|\\big|\\,\\vee\\, 1)^{\\red{q}}\\, dy\\bigg)^{\\frac pq}\\, dx\\bigg)^{\\frac 1p}.\\label{eq:0orderlog}\n\t\\end{align}\n\t\\red{The constant in the inequality depends only on $p,q,\\theta,\\Omega$.}\n\\end{theorem}\nIn order to obtain this result we first prove an analogue of Lemma \\ref{lemMaximal} for $K(x,y) = |x-y|^{-d}$, i.e. $\\phi \\equiv 1$. For now every integral is restricted to $\\Omega$ by default.\n\\begin{lemma}\\label{lem:max2}\n\tLet $\\Omega$ be a bounded domain with Whitney covering $\\mW$. Assume that $g\\in L^1_{loc}(\\mR^d)$, and $0<r<\\diam(\\Omega)$. Then for every $Q\\in\\mW$ and $x\\in\\Omega$ we have\n\t\\begin{align}\\label{eq:max2far}\n\t\\int_{|y-x|>r} \\frac{g(y) \\, dy}{|y-x|^d}&\\lesssim Mg(x)(|\\log r|\\red{\\vee 1}),\\\\\n\t\\sum_{S:D(Q,S)>r}  \\frac{\\int_S g(y) \\, dy}{D(Q,S)^d}&\\lesssim \\inf_{x\\in Q} Mg(x)(|\\log r|\\red{\\vee 1}).\\label{eq:max2far2}\n\t\\end{align}\n\tand\n\t\\begin{equation}\\label{eq:max2allover}\n\t\\sum_{S\\in\\mathcal{W}} \\frac{l(S)^d}{D(Q,S)^d} \\lesssim |\\log \\red{l(Q)}|\\red{\\vee 1}.\n\t\\end{equation}\n\\end{lemma}\n\\begin{proof}\n\tLet $x\\in \\Omega$. If we take $R = \\diam(\\Omega)$, then proceeding as in Lemma \\ref{lemMaximal} we get\n\t\\begin{align*}\n\t\\int_{|y-x|\\red{>}r} \\frac{g(y)\\, dy}{|y-x|^d} \\leq \\sum_{k=1}^{\\lceil\\log_2(R\/r)\\rceil} \\int_{2^{k-1}r\\leq|y-x|<2^k r} \\frac{g(y)\\,  dy}{|x-y|^d} \\lesssim Mg(x) \\lceil\\log_2(R\/r)\\rceil \\lesssim Mg(x)(|\\log r|\\red{\\vee 1}).\n\t\\end{align*}\n\tAs in the proof of Lemma \\ref{lemMaximal}, in order to prove \\eqref{eq:max2far2} we use \\eqref{eq:max2far}, and we are left with\n\t\\begin{equation*}\n\t\\int_{|x-y|<r} \\frac {g(y)\\, dy}{(|x-y| + r)^d} \\lesssim \\frac 1{|B(x,r)|} \\int_{B(x,r)} g(y) \\, dy \\leq Mg(x)(|\\log r|\\red{\\vee 1}).\n\t\\end{equation*}\n\tFinally, \\eqref{eq:max2allover} is obtained by taking $r = l(Q)$ and $g\\equiv 1$.\n\\end{proof}\n\\red{We will also use the following result similar to Lemma \\ref{lem:phil}.\n\t\\begin{lemma}\\label{lem:phil0}\n\t\tLet $\\Omega$ be a bounded uniform domain with admissible Whitney decomposition $\\mW$ and let $\\rho >0$ and $\\eta>1$. Then, for every $S\\in \\mW$ we have\n\t\t\\begin{equation}\\label{eq:phil1}\n\t\t\\sum_{R: S\\in \\Sh_\\rho(R)} 1 \\lesssim |\\log l(S)|\\vee 1.\n\t\t\\end{equation}\n\t\tIf $S\\in \\Sh_\\rho(R)$, then\n\t\t\\begin{equation}\\label{eq:phil2}\n\t\t\\sum_{P\\in [S,R)} (|\\log l(P)|\\vee 1)^{-\\eta} \\lesssim (|\\log l(R)|\\vee 1)^{1-\\eta}.\n\t\t\\end{equation}\n\t\tFurthermore, for every $P\\in \\mW$\n\t\t\\begin{equation}\\label{eq:phil3}\n\t\t\\sum_{R: P\\in \\Sh_\\rho(R)} (|\\log l(R)|\\vee 1)^{-\\eta} \\lesssim 1.\n\t\t\\end{equation}\n\\end{lemma}}\n\\begin{proof}\n\t\\red{Throughout the proof we let $l(S) = 2^{s_0}$, $l(R) = 2^{r_0}$, $l(P) = 2^{p_0}$, whenever the cubes are fixed.}\n\t\n\t\\red{Arguing as in the proof of Lemma \\ref{lem:phil} we get that there is a limited number of cubes of a given side length contributing to the sum in \\eqref{eq:phil1} and the smallest of these cubes must have side length at least $2^{s_0 - l_0}$ for some fixed natural number $l_0 \\geq 0$. Therefore, if we let $2^{m_0}$ be the side length of the largest cube in $\\mW$, then we have\n\t\t\\begin{equation*}\n\t\t\\sum_{R: S\\in \\Sh_\\rho(R)} 1 \\lesssim \\sum_{k=s_0-l_0}^{{m_0}} 1 = {m_0} - s_0 + l_0 + 1 \\approx |\\log l(S)| \\vee 1.\n\t\t\\end{equation*}\n\t\tAs in Lemma \\ref{lem:phil}, in \\eqref{eq:phil2} we have limited number of cubes of the same length and the cube length cannot be larger than $2^{r_0+l_0}$ and smaller than $2^{s_0 - l_0}$ ($l_0$ may be different than above, but it does not depend on $S$ and $R$). Therefore we estimate the sum in \\eqref{eq:phil2} as follows:\n\t\t\\begin{align*}\n\t\t\\sum_{P\\in [S,R)} (|\\log l(P)|\\vee 1)^{-\\eta} \\lesssim \\sum_{k=s_0-l_0}^{r_0 + l_0} (|k|\\vee 1)^{-\\eta} \\leq \\sum_{k=-\\infty}^{r_0+l_0} (|k|\\vee 1)^{-\\eta}.\n\t\t\\end{align*}\n\t\tSince $\\eta > 1$, the latter series is finite and it is of order $(|r_0|\\vee 1)^{1-\\eta}$, which proves \\eqref{eq:phil2}.}\n\t\n\t\\red{In order to prove \\eqref{eq:phil3} we argue as above in terms of the numbers of the cubes, and because of $\\eta>1$ we get\n\t\t\\begin{align*}\n\t\t\\sum_{R: P\\in \\Sh_\\rho(R)} (|\\log l(R)|\\vee 1)^{-\\eta} \\lesssim \\sum_{k = p_0 - l_0}^{{m_0}} (|k|\\vee 1)^{-\\eta} \\leq \\sum_{k=-\\infty}^{{m_0}} (|k|\\vee 1)^{-\\eta} = C.\n\t\t\\end{align*}}\n\\end{proof}\n\\begin{proof}[Proof of Theorem \\ref{th:0order}]\n\n\t\\red{We proceed as in Theorem \\ref{th:main} starting with $1$ in place of $\\phi$. The integrals over $Q\\times 2Q$ are trivially estimated, because the kernel in \\eqref{eq:0orderlog} is larger than the one in \\eqref{eq:0order}.}\n\t\n\t\\red{In \\AAA\\ and \\BB\\ the modification is quite straightforward. Lemma \\ref{lemMaximal} is used in \\eqref{eq:repA} and \\eqref{eq:repB} respectively. Using Lemma \\ref{lem:max2} instead, we get respectively $(|\\log l(Q)|\\vee 1)^{\\frac 1q}$ and $(|\\log l(P)|\\vee 1)^{\\frac 1q}$. The remaining arguments are conducted with $(|\\log r|\\vee 1)^{-\\frac 1q}$ in place of $\\phi(r)$. Note that $(|\\log r|\\vee 1)^{-\\frac 1q} \\approx (|\\log 2r|\\vee 1)^{-\\frac 1q}$. We remark that this yields estimates for \\AAA\\ and \\BB\\ which are better than the ones in the statement, in fact both expressions are bounded from above by\n\t\t\\begin{equation}\n\t\t\\bigg(\\int_{\\Omega}\\bigg(\\int_{B(x,\\theta\\delta(x))} \\frac{|f(x) - f(y)|^q}{|x-y|^d}(\\big|\\hspace{-1pt}\\log|x-y|\\big|\\,\\vee\\, 1)\\, dy\\bigg)^{\\frac pq}\\, dx\\bigg)^{\\frac 1p}.\\label{eq:log0}\n\t\t\\end{equation}\n\t\tNotice the lack of exponent $q$ in the logarithmic term. At this point we distinguish between the case $p=q$ and $p\\neq q$. In the former case the test functions $g$ from \\eqref{eq:dual} are defined by the condition $\\int_\\Omega\\int_\\Omega g(x,y)^{p'}\\, dy\\, dx \\leq~1$, therefore \\CC\\ can be estimated exactly as \\AAA\\ and \\BB\\ because we can interchange the roles of $Q,S$ and $x,y$ using Tonelli's theorem. Thus, in this case we in fact obtain an estimate better than postulated, as the whole expression in \\eqref{eq:0order} is approximately bounded from above by \\eqref{eq:log0}.}\n\n\n\n\n\n\n\n\t\n\t\\red{For the remainder of the proof we assume that $p > q$. The procedure for \\CC\\ is also similar to the one in the proof of Theorem \\ref{th:main}, but the computations are slightly different in terms of the exponents, therefore we give the details. There are no essential changes up to the moment of splitting into \\CJ\\ and \\CD, thus we make it our starting point. As in the proof of Theorem \\ref{th:main} we get\n\t\t\\begin{align*}\n\t\t\\CD &= \\sum_{R\\in\\mW} l(R)^{d(1-\\frac pq)}\\bigg(\\sum_{S\\in\\Sh(R)}\\int_S |f_S - f(\\xi)|^q\\, d\\xi \\frac{l(S)^{d(1-\\frac qp)}}{l(S)^{d(1-\\frac qp)}}\\bigg)^{\\frac pq}\\\\\n\t\t&\\lesssim \\sum_{R\\in\\mW}\\sum_{S\\in\\Sh(R)} l(S)^{d(1-\\frac pq )}\\bigg(\\int_S|f_S - f(\\xi)|^q\\,d\\xi\\bigg)^{\\frac pq}.\n\t\t\\end{align*}\n\t\tWe rearrange, use \\eqref{eq:phil1} and then Jensen's inequality twice to obtain:\n\t\t\\begin{align*}\n\t\t\\CD &\\lesssim \\sum_{S\\in \\mW}l(S)^{d(1-\\frac pq)}\\bigg(\\int_S|f_S - f(\\xi)|^q\\,d\\xi\\bigg)^{\\frac pq}\\bigg(\\sum_{R: S\\in \\Sh(R)} 1\\bigg)\\\\\n\t\t&\\lesssim \\sum_{S\\in \\mW} l(S)^{d}(|\\log l(S)|\\vee 1)\\bigg(\\frac{1}{l(S)^d}\\int_S|f_S - f(\\xi)|^q\\, d\\xi\\bigg)^{\\frac pq}\\\\\n\t\t&\\leq \\sum_{S\\in \\mW} (|\\log l(S)|\\vee 1)\\int_S |f_S - f(\\xi)|^p\\, d\\xi\\\\\n\t\t&\\leq \\sum_{S\\in\\mW}\\int_S \\bigg(\\int_S \\frac{|f(\\zeta) - f(\\xi)|^q}{l(S)^d}(|\\log l(S)|\\vee 1)^{\\frac qp}\\, d\\zeta\\bigg)^{\\frac pq}\\, d\\xi,\n\t\t\\end{align*}\n\t\tand thus \\CD\\ is estimated, since $\\frac qp < 1 < q$.}\n\t\n\t\\red{In order to estimate \\CJ\\ we write $|f_P - f_{\\mN(P)}| = |f_P - f_{\\mN(P)}|\\frac{|\\log l(P)|\\vee 1}{|\\log l(P)|\\vee 1}$ and we use H\\\"older's inequality with exponent $q$ and \\eqref{eq:phil2}:\n\t\t\\begin{align*}\n\t\t\\CJ \\leq &\\hspace{-2.04pt}\\sum_{R\\in \\mW} l(R)^{d(1-\\frac pq)}\\bigg[\\sum_{S\\in\\Sh(R)}\\bigg(\\sum_{P\\in [S,R)}|f_P - f_{\\mN(P)}|^q(|\\log l(P)|\\vee 1)^q l(S)^d\\bigg)\\bigg(\\sum_{P\\in [S,R)}(|\\log l(P)|\\vee 1)^{-q'}\\bigg)^{\\frac q{q'}}\\bigg]^{\\frac pq}\\\\\n\t\t\\lesssim &\\hspace{-2.04pt}\\sum_{R\\in \\mW}l(R)^{d(1-\\frac pq)}(|\\log l(R)|\\vee 1)^{-\\frac pq}\\bigg(\\sum_{S\\in\\Sh(R)} \\sum_{P \\in [S,R)} |f_P - f_{\\mN(P)}|^q(|\\log l(P)|\\vee 1)^{q}l(S)^d\\bigg)^{\\frac pq}.\n\t\t\\end{align*}\n\t\tBy rearranging as in the proof of Theorem \\ref{th:main} and by using H\\\"older's and Jensen's inequalities we further estimate \\CJ\\ from above by a multiple of\n\t\t\\begin{align*}\n\t\t &\\sum_{R\\in\\mW} l(R)^{d(1 - \\frac pq)} (|\\log l(R)|\\vee 1)^{-\\frac pq} \\bigg(\\sum_{P\\in\\Sh^2(R)}\\sum_{S\\in \\Sh^2(P)} \\big(\\int_{U_P} \\frac{|f_P - f(\\xi)|}{l(P)^d}\\, d\\xi\\big)^q(|\\log l(P)|\\vee 1)^q l(S)^d\\bigg)^{\\frac pq}\\\\\n\t\t\\lesssim &\\sum_{R\\in\\mW} l(R)^{d(1 - \\frac pq)} (|\\log l(R)|\\vee 1)^{-\\frac pq}\\bigg(\\sum_{P\\in\\Sh^2(R)} \\big(\\int_{U_P} \\frac{|f_P - f(\\xi)|}{l(P)^d}\\, d\\xi\\big)^q(|\\log l(P)|\\vee 1)^q l(P)^d\\bigg)^{\\frac pq}\\\\\n\t\t\\leq &\\sum_{R\\in\\mW}\\sum_{P\\in\\Sh^2(R)} (|\\log l(R)|\\vee 1)^{-\\frac pq} (|\\log l(P)|\\vee 1)^p\\int_{U_P} |f_P - f(\\xi)|^p\\, d\\xi.\n\t\t\\end{align*}\n\t\tWe rearrange once more and use \\eqref{eq:phil3} (recall that $p>q$) and Jensen's inequality to get that, up to a multiplicative constant, \\CJ\\ does not exceed\n\t\t\\begin{align*}\n\t\t&\\sum_{P\\in \\mW}(|\\log l(P)|\\vee 1)^p\\int_{U_P} |f_P - f(\\xi)|^p\\, d\\xi\\bigg(\\sum_{R: P\\in \\Sh^2(R)} (|\\log l(R)|\\vee 1)^{-\\frac pq}\\bigg)\\\\\n\t\t\\lesssim &\\sum_{P\\in \\mW}(|\\log l(P)|\\vee 1)^p\\int_{U_P} |f_P - f(\\xi)|^p\\, d\\xi\\\\\n\t\t\\lesssim &\\sum_{P \\in \\mW} \\int_{U_P}\\bigg(\\int_P\\frac{|f(\\zeta) - f(\\xi)|^q}{l(P)^d}(|\\log l(P)|\\vee 1)^q\\,d\\zeta\\bigg)^{\\frac pq}\\,d\\xi.\n\t\t\\end{align*}\n\t\tThis finishes the proof.}\n\\end{proof}\nSince the kernel in \\eqref{eq:0orderlog} is significantly larger than the one in \\eqref{eq:0order}, it is plausible that the converse inequality is not true. We will show the existence of a counterexample when $\\Omega = (0,1)$, $p=q=2$. For an open interval $I\\subseteq \\mR$ we let $$F_0(I) = \\bigg\\{f \\in L^2(I): \\int_I\\int_I \\frac{(f(x)-f(y))^2}{|x-y|}\\, dy \\, dx < \\infty\\bigg\\},$$\n$$F_{\\log}(I) = \\bigg\\{f\\in L^2(I):\\int_I\\int_I\\frac{(f(x)-f(y))^2}{|x-y|} (|\\log|x-y|| \\vee 1) \\, dy \\, dx <\\infty \\bigg\\}.$$\n\\red{We note that in $F_{\\log}(I)$ the logarithm is in power 1. This suffices for our present purpose, because $q>1$ in Theorem \\ref{th:0order}.}\n\\begin{theorem}\n\tFor every $\\theta\\in (0,1]$, there exists $f \\in F_0(0,1)\\cap L^\\infty(0,1)$ such that\n\t\\begin{equation}\\label{eq:lognorm}\n\t\\int_0^1 \\int_{B(x,\\theta \\delta(x))} (f(x) - f(y))^2 |x-y|^{-1}(|\\log|x-y||\\vee 1) \\, dy \\, dx = \\infty.\n\t\\end{equation}\n\\end{theorem}\n\\begin{proof}$ $\n\t\n\t\\textbf{Step 1.} First, note that the finiteness of the left hand side of \\eqref{eq:lognorm} implies that $f\\in F_{\\log}(\\frac {n}{2n+1}, \\frac{n+1}{2n+1})$ for a sufficiently large $n\\in \\mathbb{N}$. Indeed, if $\\theta \\geq \\frac 1n$ for some natural number $n \\geq 2$, then \\begin{align}\\label{eq:ciag}\\int_0^1\\int_{B(x,\\theta\\delta(x))} (\\ldots)\\geq \\int_0^1 \\int_{B(x,\\delta(x)\/n)} (\\ldots) \\geq \\int_{\\frac {n}{2n+1}}^{\\frac {n+1}{2n+1}}\\int_{B\\big(x,\\frac 1{2n+1}\\big)} (\\ldots) \\geq \\int_{\\frac {n}{2n+1}}^{\\frac {n+1}{2n+1}}\\int_{\\frac {n}{2n+1}}^{\\frac {n+1}{2n+1}} (\\ldots).\\nonumber\\end{align}\n\tWe fix a number $n$ for which \\eqref{eq:ciag} is satisfied.\n\n\n\n\n\n\n\n\n\n\n\n\n\t\n\t\\textbf{Step 2.} In order to construct the counterexample we will use the asymptotics of the Fourier expansions of functions in $F_0(I)$ and $F_{\\log}(I)$. \\red{We adopt the following convention for the Fourier coefficients of an integrable function $f$ on an interval $(a,b)$:\n\t\t\\begin{equation*}\n\t\t\\widehat{f}(m) = \\frac{1}{b-a}\\int_a^b f(x) e^{-\\frac{2\\pi imx}{b-a}}\\, dx,\\quad m\\in \\mathbb{Z}.\n\t\t\\end{equation*}\n\t\tBelow, by $\\widehat{f}(m)$ we mean the Fourier coefficient on $(0,1)$.} Let $f$ satisfy $f(x+1) = f(x)$ for $x\\in \\mR$. Let $K(x,y)$ be equal to $|x-y|^{-1}$ (resp. $|x-y|^{-1}(|\\log|x-y||\\vee 1)$). \\red{We claim that given $f\\in L^\\infty(0,1)$, it belongs to $F_0(0,1)$} (resp. $F_{\\log} (0,1)$) if and only if\n\t\\begin{align*}\n\t\\int_0^1\\int_0^1 (f(x) - f(x-h))^2 K(0,h) \\, dh \\, dx < \\infty.\n\t\\end{align*}\n\tIndeed, we have\n\t\\begin{align*}\n\t\\int_0^1\\int_0^1 (f(x) - f(y))^2 K(x,y) \\, dy \\, dx =\\, &2\\int_0^1 \\int_0^x (f(x) - f(y))^2 K(x,y) \\, \\red{dy \\, dx}\\\\\n\t=\\, & \\red{2}\\int_0^1\\int_0^x (f(x) - f(x-h))^2 K(0,h) \\, dh \\, dx.\n\t\\end{align*}\n\tTherefore, it suffices to verify that $\\int_0^1 \\int_x^1 (f(x) - f(x-h))^2 K(0,h) \\, dh \\, dx <\\infty$ for bounded $f$. Clearly we can assume that $K(x,y) = |x-y|^{-1}(|\\log|x-y||\\vee 1)$.\n\t\\begin{align*}&\\int_0^1\\int_x^1 (f(x) - f(x-h))^2 K(0,h) \\, dh \\, dx\\lesssim \\int_0^1\\int_x^1 \\frac{(-\\log h) \\vee 1}{h} \\, dh \\, dx\\\\\n\t= &\\int_0^{1\/e} \\int_x^{\\red{1\/e}} \\frac{-\\log h}{h} \\, dh \\, dx + \\red{\\int_0^{1\/e} \\int_{1\/e}^{1} \\frac{1}{h} \\, dh \\, dx+} \\int_{1\/e}^1 \\int_x^1 \\frac{1}{h} \\, dh \\, dx.\n\t\\end{align*}\n\t\\red{All the} integrals are finite, therefore the claim is proved.\n\t\n\tBy Parseval's identity and Tonelli's theorem we get\n\t\\begin{align*}\n\t&\\int_0^1K(0,h)\\int_0^1 (f(x) - f(x-h))^2 \\, \\red{dx} \\, \\red{dh} \\\\= &\\int_0^1 K(0,h) \\sum_{m \\in \\mathbb{Z}} |\\widehat{f}(m)|^2 |1 - e^{2\\pi i m h}|^2 \\, dh\\\\\n\t=&\\sum_{m\\in\\mathbb{Z}} |\\widehat{f}(m)|^2 \\int_0^1 |1 - e^{2\\pi i m h}|^2 K(0,h) \\, dh\\\\ = &2\\sum_{m\\in\\mathbb{Z}} |\\widehat{f}(m)|^2 \\int_0^1 (1-\\cos(2\\pi m h)) K(0,h) \\, dh.\n\t\\end{align*}\n\tNow let us inspect the remaining integrals for both cases of $K$. For $m\\neq 0$ we have\n\t\\begin{align*}\n\t\\int_0^1 \\frac{1-\\cos (2\\pi m h)}{h} \\, dh = \\int_0^{|m|} \\frac{1-\\cos (2\\pi h)}{h} \\, dh \\approx \\log|m|.\n\t\\end{align*}\n\tIn the logarithmic case\n\t\\begin{align*}\n\t\\int_0^1 \\frac{1-\\cos(2\\pi m h)}{h} (-\\log h \\vee 1) \\, dh = \\int_0^{|m|}\\frac{1-\\cos(2\\pi h)}{h} (-\\log \\frac {h}{|m|} \\vee 1) \\, dh\\approx \\log^2|m|.\n\t\\end{align*}\n\tTo summarize, for bounded functions we can characterize $F_0(0,1)$ by\n\t\\begin{equation}\\label{eq:0char}\n\t\\sum_{m\\in \\mathbb{Z}, m \\neq 0} |\\widehat{f}(m)|^2 \\log|m| < \\infty\n\t\\end{equation}\n\tand $F_{\\log}(0,1)$ by \n\t\\begin{equation}\\label{eq:logchar}\n\t\\sum_{m\\in \\mathbb{Z}, m\\neq 0} |\\widehat{f}(m)|^2 \\log^2|m| < \\infty.\n\t\\end{equation}\n\tThe same characterizations hold for $I = (\\frac {n}{2n+1}, \\frac{n+1}{2n+1})$ and the respective Fourier expansion.\n\t\n\t\\textbf{Step 3.} We give an example of $f\\in F_0(0,1)\\cap L^\\infty(0,1)$ for which \\eqref{eq:0char} is satisfied and \\eqref{eq:logchar} is not. For $m = (2n+1) 2^l$, $l=1,2,\\ldots$, we put $\\widehat{f}(m) = \\frac 1{l^{3\/2}}$. For other $m$ we let $\\widehat{f}(m) = 0$. Note that $f$ is $\\frac 1{2n+1}$--periodic. Therefore the $j$-th Fourier coefficient of $f$ on $(\\frac{n}{2n+1},\\frac{n+1}{2n+1})$ is equal to its $(2n+1)\\cdot j$-th Fourier coefficient on $(0,1)$. Since $(\\widehat{f}(m))_{m\\in\\mathbb{Z}}$ is summable, $f$ is bounded. Furthermore $l^{-3}\\log[ (2n+1)2^l] = l^{-2}\\log 2 + l^{-3}\\log (2n+1)$ and $l^{-3} \\log^2(2^l) \\approx l^{-1}$. Therefore \\eqref{eq:0char} is satisfied and \\eqref{eq:logchar} is not. By \\eqref{eq:ciag}, the proof is finished.\n\t\n\\end{proof}\n\\section{Uniformity is not a sharp condition}\\label{sec:pasy}\nIn this section we examine the strip $\\mR\\times (0,1)$ which is a non-uniform domain. We will show that the comparability fails for fractional Sobolev spaces with $\\alpha < 1$. Then we \\red{prove} that for $\\alpha >1$ and slightly more general kernels the comparability holds. Later, we present a higher-dimensional case \\red{in which} the comparability may also hold for $\\alpha < 1$ in non-uniform domains. For clarity of the presentation, we assume that $p=q=2$.\n\\begin{example}\n\tLet $\\Omega = \\mR \\times (0,1)$ and let $K(x,y) = |x-y|^{-2-\\alpha}$. Note that $\\Omega$ is not uniform --- if we take two cubes far from each other we will fail to find a sufficiently large central cube in any chain connecting them.\n\t\n\tWe will show for $\\alpha \\in (0,1)$ the comparability does not hold. Consider a sequence of functions $(f_n)$ given by the formula $f_n(x_1,x_2) = (1 - \\frac{|x_1|}{n})\\vee 0$. Since $f_n$ are constant on the second variable, for every $\\xi\\in (0,1)$ we have\n\t\\begin{align*}\n\t\\int_{\\Omega}\\int_\\Omega \\frac{(f_n(x) - f_n(y))^2}{|x-y|^{2+\\alpha}} \\red{\\, dy \\, dx} = \\int_{\\mR}\\int_{\\mR}(f_n(x_1,\\xi) - f_n(y_1,\\xi))^2\\int_0^1\\int_0^1 |x-y|^{-2-\\alpha} \\red{\\, dy_2 \\, dx_2 \\, dy_1 \\, dx_1}.\n\t\\end{align*}\n\tLet the integral over $(0,1)\\times(0,1)$ be called $\\kappa(x_1,y_1)$. We claim that $\\kappa(x_1,y_1)$ is comparable with $|x_1-y_1|^{-2-\\alpha}$ if $|x_1 - y_1| \\geq 1$ and with $|x_1-y_1|^{-1-\\alpha}$ otherwise. Indeed, we have $|x-y| \\approx |x_1 - y_1| + |x_2 - y_2|$. If $|x_1 - y_1| \\geq 1$, then\n\t\\begin{equation*}\n\t\\int_0^1\\int_0^1 |x-y|^{-2-\\alpha} \\red{\\, dy_2\\, dx_2} \\approx |x_1-y_1|^{-2-\\alpha}\\int_0^1\\int_0^1 \\, \\red{dy_2\\, dx_2} = |x_1 - y_1|^{-2-\\alpha}.\n\t\\end{equation*}\n\tFor $|x_1 - y_1|<1$ note that for fixed $a>0$\n\t\\begin{align*}\n\t&a^{1+\\alpha}\\int_0^1 \\int_0^1 (a + |x_2-y_2|)^{-2-\\alpha} \\, \\red{dy_2 \\, dx_2}\\\\ \\approx\\, &a^{1+\\alpha}\\int_0^1\\int_0^{\\red{x}_2} (a + \\red{x_2 - y_2})^{-2-\\alpha} \\, \\red{dy_2 \\, dx_2}\\\\\n\t=\\, &\\frac{a^{1+\\alpha}}{\\red{1+\\alpha}}\\int_0^1 (a^{-1-\\alpha} - (a+\\red{x_2})^{-1-\\alpha}) \\, \\red{dx_2}\\\\ = &\\red{\\frac 1{1+\\alpha}} - \\red{\\frac 1{1+\\alpha}}\\int_0^1 \\big(1 + \\frac{\\red{x_2}}{a}\\big)^{-1-\\alpha}\\, \\red{dx_2}.\n\t\\end{align*}\n\t\\red{For $a=|x_1 - y_1|<1$ we have $x_2\/a > x_2$, so the latter integral is bounded from above by $C\\in (0,1)$.} Thus the whole expression is approximately \\red{equal to a positive} constant which proves our claim.\n\t\n\tThe shape of $\\Omega$ grants that for every $\\theta\\in(0,1]$ we have\n\t\\begin{align*}\n\t&\\int_{\\Omega}\\int_{B(\\red{x,\\theta\\delta(x)})} \\frac{(f_n(x) - f_n(y))^2}{|x-y|^{2+\\alpha}} \\red{\\, dy \\, dx}\\\\ \\leq &\\int_{\\mR}\\int_{B(\\red{x_1},1)} (f_n(x_1,\\xi) - f_n(y_1,\\xi))^2 \\kappa(x_1,y_1) \\, \\red{dy_1 \\, dx_1}. \n\t\\end{align*}\n\tTo simplify the notation we will write $f_n(x_1) = f_n(x_1,\\xi)$ for some fixed $\\xi\\in (0,1)$, $x\\in\\mR$. Since $f_n$ is Lipschitz \\red{with constant $1\/n$}, we have\n\t\\begin{align*}\n\t&\\int_{\\mR}\\int_{B(\\red{x_1},1)} (f_n(x_1) - f_n(y_1))^2\\kappa(x_1,y_1) \\, \\red{dy_1 \\, dx_1}\\\\ \\approx &\\int_{\\mR} \\int_{B(\\red{x_1},1)} (f_n(x_1) - f_n(y_1))^2 |x_1-y_1|^{-1-\\alpha} \\red{\\, dy_1 \\, dx_1}\\\\\n\t=&\\int_{-n-1}^{n+1}\\int_{B(\\red{x_1},1)} (f_n(x_1) - f_n(y_1))^2|x_1-y_1|^{-1-\\alpha}\\, \\red{dy_1 \\, dx_1}\\\\ \\lesssim &\\frac 1{n^2} \\int_{-n-1}^{n+1}\\int_{B(\\red{x_1},1)} |x_1-y_1|^{1-\\alpha}\\, \\red{dy_1 \\, dx_1} \\approx \\frac 1n.\n\t\\end{align*}\n\tThanks to the fact that $\\alpha < 1$, the full seminorm is significantly greater as $n\\to\\infty$:\n\t\\begin{align*}\n\t&\\int_{\\mR}\\int_{\\mR} (f_n(x_1) - f_n(y_1))^2 \\kappa(x_1,y_1) \\, \\red{dy_1 \\, dx_1} \\gtrsim \\int_{-\\frac n2}^{0}\\int_{-\\infty}^{-n} |x_1-y_1|^{-2-\\alpha} \\red{\\, dy_1 \\, dx_1}\\\\\n\t= &\\int_{-\\frac n2}^{0} \\frac 1{1+\\alpha} \\frac 1{(\\red{x_1}+n)^{1+\\alpha}}\\, d\\red{x_1}\n\t\\geq \\frac 1{1+\\alpha}\\frac{n\/2}{\\red{n}^{1+\\alpha}} \\approx \\frac 1{n^\\alpha}.\n\t\\end{align*}\n\\end{example}\t\n\\begin{lemma}\\label{lem:tech}\n\tLet $\\Omega = \\mR\\times (0,1)$. If $f\\colon \\mR^2 \\to [0,\\infty)$ is radial, then \\red{$\\int_\\Omega (1\\vee |x|)f(x) \\, dx \\approx \\int_{\\mR^2} f(x) \\, dx < \\infty$ with a constant independent of $f$.}\n\\end{lemma}\n\\begin{proof}\n\tNote that for $n \\in \\mathbb{N}$ the area of $\\Omega\\cap(B_n\\setminus B_{n-1})$ is comparable to the $1\/n$-th of the area of the annulus $B_n\\setminus B_{n-1}$. Therefore by the rotational symmetry of $f$ we get\n\t\\begin{align*}\n\t\\int_\\Omega \\red{(1\\vee |x|)}f(x) \\, dx \\approx \\sum\\limits_{n\\in \\mathbb{N}} \\int_{\\Omega\\cap(B_n\\setminus B_{n-1})}nf(x) \\, dx &\\approx \\sum\\limits_{n\\in \\mathbb{N}}\\int_{B_n \\setminus B_{n-1}} f(x) \\, dx\\\\ &= \\int_{\\mR^2} f(x) \\, dx.\\qedhere\n\t\\end{align*}\n\\end{proof}\nThe case of $\\alpha \\in (1,2)$ is included in the following result.\n\\begin{theorem}\\label{th:strip}\n\tLet $\\Omega = \\mR \\times (0,1)$. Assume that $K$ satisfies \\AJ,\\ \\AD,\\ \\AT\\ and $\\sum_{n\\geq 1} \\int_{B(0,n)^c} K(0,x) \\, dx < \\infty$. Then the seminorms \\eqref{eq:fullsemi} and \\eqref{eq:truncated} are comparable.\n\\end{theorem}\n\\begin{proof}\n\tWe split the domain $\\Omega$ into open unit cubes $Q_n$ centered in $(n,1\/2)$, $n\\in \\mathbb{Z}$, so that we have $\\Omega \\subseteq \\bigcup\\limits_{n\\in\\mathbb{Z}} \\overline{Q_n}$. If we let $L_n = \\mathrm{Int}[\\overline{Q_{n-1}\\cup Q_n\\cup Q_{n+1}}]$, then $L_n$ is a uniform domain, hence by Theorem \\ref{th:main}\n\t\\begin{align*}\n\t\\int_{L_n}\\int_{L_n} (f(x) - f(y))^2 K(x,y) \\red{\\, dy \\, dx}\\approx \\int_{L_n}\\int_{B(x,\\theta\\delta(x))} (f(x) - f(y))^2 K(x,y) \\red{\\, dy \\, dx}\n\t\\end{align*}\n\twith the constant independent of $n$.\n\tTherefore for every $0<\\theta\\leq 1$\n\t\\begin{align}\n\t&\\int_{\\Omega}\\int_{B(x,\\theta\\delta(x))} (f(x) - f(y))^2 K(x,y) \\red{\\, dy \\, dx}\\nonumber\\\\ \\approx  &\\sum_{n\\in\\mathbb{Z}}\\int_{L_n}\\int_{L_n} (f(x) - f(y))^2 K(x,y)\\, \\red{dy \\, dx}\\nonumber\\\\\n\t\\approx &\\sum_{n\\in\\mathbb{Z}}\\int_{Q_n}\\int_{L_n} (f(x) - f(y))^2 K(x,y)\\, \\red{dy \\, dx},\\label{eq:kl}\n\t\\end{align}\n\tso it suffices to show that the latter expression is comparable with the integral over $\\Omega\\times \\Omega$.\n\tWe have\n\t\\begin{align*}\n\t&\\int_{\\Omega}\\int_{\\Omega} (f(x) - f(y))^2 K(x,y) \\red{\\, dy \\, dx}\\\\ = &\\sum_{i,j\\in\\mathbb{Z}} \\int_{Q_i}\\int_{Q_j} (f(x) - f(y))^2 K(x,y) \\red{\\, dy \\, dx}\\\\\n\t\\approx &\\sum_{i \\in \\mathbb{Z}}\\sum_{j + 1 < i} \\int_{Q_i}\\int_{Q_j}(f(x) - f(y))^2 K(x,y)\\,\\red{dy \\, dx}\\\\ &+ \\sum_{i \\in \\mathbb{Z}} \\int_{Q_i}\\int_{L_i}(f(x) - f(y))^2 K(x,y)\\, \\red{dy \\, dx}.\n\t\\end{align*}\n\tClearly it suffices to estimate the first summand. Since the cubes are far apart, we have $|x-y| \\approx |i-j|$ for $x\\in Q_{i}$, $y\\in Q_{j}$. Hence\n\t\\begin{align}\n\t&\\sum_{i \\in \\mathbb{Z}}\\sum_{j + 1 < i} \\int_{Q_i}\\int_{Q_j}(f(x) - f(y))^2 K(x,y)\\, \\red{dy \\, dx}\\nonumber\\\\\n\t\\lesssim\\, &\\red{\\sum_{i \\in \\mathbb{Z}}\\sum_{j + 1 < i} \\int_{Q_{i}}\\int_{Q_{j}} (f(x) - f_{Q_i})^2 K(x,y) \\, dy \\, dx}\\label{eq:alfa}\\\\\n\t+&\\red{\\sum_{i \\in \\mathbb{Z}}\\sum_{j + 1 < i} \\int_{Q_{i}}\\int_{Q_{j}} (f(y) - f_{Q_j})^2 K(x,y) \\, dy \\, dx}\t\\nonumber\\\\\n\t+&\\sum_{i \\in \\mathbb{Z}}\\sum_{j + 1 < i} \\sum_{j\\leq n < i}\\int_{Q_i}\\int_{Q_j} (f_{Q_{n+1}} - f_{Q_n})^2 |x-y|K(x,y) \\, \\red{dy \\, dx}.\\nonumber\n\t\\end{align} \n\tIn this inequality we have used $(a_1 + \\ldots + a_m)^2 \\leq m(a_1^2 + \\ldots +a_m^2)$ and $|Q_i| = |Q_j| = 1$.\n\tFor the first term we use Jensen's inequality and the fact that the sum over $j$ is \\red{uniformly bounded with respect to $i$ and $x\\in Q_i$:}\n\t\\begin{align*}\n\t&\\sum_{i \\in \\mathbb{Z}} \\int_{Q_i} (f(\\red{x}) - f_{Q_i})^2\\sum_{j + 1 < i}\\int_{Q_j} K(x,y) \\red{\\, dy \\, dx}\\\\ \\lesssim &\\sum_{i\\in\\mathbb{Z}}\\int_{Q_i}\\int_{Q_i} (f(y) - f(x))^2 \\, \\red{dy \\, dx}.\n\t\\end{align*}\n\tThe latter expression does not exceed \\eqref{eq:kl}. \\red{The second term can be estimated in a similar way after changing the order of summation}.\n\t\n\tBy Lemma \\ref{lem:tech} the additional assumption on $K$ is equivalent to $$\\sum_{n\\geq 1}\\int_{B(0,n)^c\\cap \\Omega} |x| K(0,x) \\, dx < \\infty.$$ We change the order of summation and use that fact to estimate the last term on the right hand side of \\eqref{eq:alfa}:\n\t\\begin{align*}\n\t&\\sum_{i \\in \\mathbb{Z}}\\sum_{j + 1 < i} \\sum_{j\\leq n < i} (f_{Q_{n+1}} - f_{Q_n})^2\\int_{Q_i}\\int_{Q_j} |x-y|K(x,y) \\red{\\, dy \\, dx}\\\\\n\t= &\\sum_{n\\in\\mathbb{Z}}(f_{Q_{n+1}} - f_{Q_n})^2\\sum\\limits_{i> n}\\sum_{\\substack{j+1<i\\\\j\\leq n}}\\int_{Q_i}\\int_{Q_j} |x-y|K(x,y) \\red{\\, dy \\, dx}\\\\\n\t\\lesssim &\\sum_{n\\in\\mathbb{Z}}(f_{Q_{n+1}} - f_{Q_n})^2 \\leq \\sum_{n\\in\\mathbb{Z}}\\int_{Q_n}\\int_{Q_{n+1}}(f(x) - f(y))^2 \\red{\\, dy \\, dx}\\\\\n\t\\lesssim &\\sum_{n\\in\\mathbb{Z}}\\int_{Q_n}\\int_{Q_{n+1}}(f(x) - f(y))^2 \\red{K(x,y)}\\red{\\, dy \\, dx}.\n\t\\end{align*}\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{th:stripmd}]\n\tThe idea is similar as above. We split $\\Omega$ into a family of unit cubes $(Q_i)_{i\\in \\mathbb{Z}^k}$ and we let $L_i = \\mathrm{Int}\\big[\\bigcup \\{\\overline{Q_j}\\colon B(x_{Q_i},\\sqrt{d}) \\cap Q_j \\ne \\emptyset\\}\\big]$. By Theorem \\ref{th:main}, for $0<\\theta\\leq 1$ we have\n\t\\begin{align*}\n\t&\\int_\\Omega\\int_{B(x,\\theta\\delta(x))} (f(x) - f(y))^2 |x-y|^{-d-\\alpha} \\, dy \\, dx\\\\ \\approx &\\sum_{i\\in \\mathbb{Z}^k}\\int_{Q_i}\\int_{L_i} (f(x) - f(y))^2 |x-y|^{-d-\\alpha} \\, dy \\, dx.\n\t\\end{align*}\n\tFor $i=(i_1,\\ldots,i_k),\\ j = (j_1,\\ldots, j_k)$ and $m\\in \\mathbb{N}_0$, we say that $j>i+m$ if $j_1 > i_1+m,\\ldots,\\, j_k>i_k + m$. \\red{By $j>m$ we mean $j>0+m$ and $j\\geq i + m$ is defined by replacing \\textit{all} the inequalities by weak ones.} By the radial symmetry of $|x-y|^{-d-\\alpha}$ it suffices to show that under our assumptions on $l$ and $\\alpha$ we have\n\t\\begin{align*}\n\t&\\sum_{i\\in \\mathbb{Z}^k}\\int_{Q_i}\\int_{L_i} (f(x) - f(y))^2 |x-y|^{-d-\\alpha} \\, dy \\, dx\\\\ \\gtrsim &\\sum_{i\\in \\mathbb{Z}^k}\\int_{Q_i}\\sum_{j>i+1}\\int_{Q_j} (f(x) - f(y))^2 |x-y|^{-d-\\alpha} \\, dy \\, dx.\n\t\\end{align*}\n\tIn order to perform a decomposition similar to \\eqref{eq:alfa} we fix a method of communication from $Q_i$ to $Q_j$, $j>i$: first we move \\red{on the} coordinate $i_1$ until we reach $j_1$, and then we do the same with the next coordinates. The set of indexes of the cubes connecting $Q_i$ and $Q_j$ \\red{in the way presented above, with $Q_i$ included and $Q_j$ excluded,} will be called $i\\to j$. Note that $|i\\to j| \\approx |i-j|$. Let $\\mathcal{N}(Q)$ be the successor of $Q$ on the way from $Q_i$ to $Q_j$. As before, we have $|i-j| \\approx |x-y|$ for $x\\in \\red{Q_i}$, $y\\in \\red{Q_j}$, therefore\n\t\\begin{align*}\n\t&\\sum_{i\\in \\mathbb{Z}^k}\\int_{Q_i}\\sum_{j>i+1}\\int_{Q_j} (f(x) - f(y))^2 |x-y|^{-d-\\alpha} \\, dy \\, dx\\\\ \\lesssim &\\sum_{i\\in \\mathbb{Z}^k}\\int_{Q_i}\\sum_{j>i+1}\\int_{Q_j} (f(x) - f_{Q_i})^2 |x-y|^{-d-\\alpha} \\, dy \\, dx\\\\\n\t+ &\\sum_{i\\in \\mathbb{Z}^k}\\int_{Q_i}\\sum_{j>i+1}\\int_{Q_j} (f(y) - f_{Q_j})^2 |x-y|^{-d-\\alpha} \\, dy \\, dx\\\\\n\t+ &\\sum_{i\\in \\mathbb{Z}^k}\\int_{Q_i}\\sum_{j>i+1}\\int_{Q_j}\\sum_{n\\in i\\to j} (f_{Q_n} - f_{\\mathcal{N}(Q_n)})^2 |x-y|^{-d-\\alpha+1} \\, dy \\, dx.\n\t\\end{align*}\n\tThe first \\red{two terms} can be handled as in the previous theorem. In the \\red{latter} we change the order of summation and we get \\red{that up to a constant it does not exceed}\n\t\\begin{align*}\n\t\\sum_{n\\in\\mathbb{Z}^k} \\red{\\bigg(\\int_{L_n} |f_{Q_n} - f(\\xi)|\\, d\\xi\\bigg)^2} \\sum_{j \\geq n}\\sum_{\\substack{i\\leq n\\\\i+1<j}}\\int_{Q_{\\red{i}}}\\int_{Q_{\\red{j}}} |x-y|^{-d-\\alpha+1} \\red{\\, dy \\, dx}.\n\t\\end{align*}\n\tTo finish the proof we note that the double sum over $i,j$ does not depend on $n$, hence we take $n=\\red{(1,\\ldots,1)}$ (for short, $n=1$) and \\red{we estimate as follows:}\n\t\\begin{align*}\n\t&\\sum_{j\\geq \\red{1}}\\sum_{\\substack{i\\leq \\red{1}\\\\i+1<j}}\\int_{Q_{\\red{i}}}\\int_{Q_{\\red{j}}} |x-y|^{-d-\\alpha+1} \\, \\red{dy \\, dx}\\\\ \\approx &\\sum_{j\\geq \\red{1}} \\int_{Q_{j}} \\int_{B(y,|j|)^c\\cap\\Omega} |x-y|^{-d-\\alpha+1} \\, dx \\, dy\\\\\n\t= &\\sum_{j\\geq \\red{1}} \\int_{Q_j} \\sum_{m=0}^\\infty \\int_{(B(0,2^{m+1}|j|)\\setminus B(0,2^m|j|))\\cap\\Omega} |x|^{-d-\\alpha+1} \\, dx \\, dy\\\\ \\approx &\\sum_{j\\geq \\red{1}} \\int_{Q_{j}} \\sum_{m=0}^\\infty (2^m|j|)^k(2^m|j|)^{-d-\\alpha+1} \\, dy\\\\\n\t\\approx & \\sum_{j\\geq \\red{1}} |j|^{\\red{k-d-\\alpha + 1}} = \\sum_{j\\geq \\red{1}} |j|^{\\red{-l-\\alpha+1}} \\approx \\red{\\sum_{j\\in \\mathbb{Z}^k\\setminus \\{0\\}} |j|^{-l-\\alpha+1},}\n\t\\end{align*}\n\twhich is finite provided that \\red{$k-l - \\alpha < -1$}.\n\\end{proof}\n\\section{Application: a new class of Markov processes}\\label{sec:proces}\nIn this section we present how our comparability results can be applied to prove the existence of Markov stochastic processes corresponding to the truncated seminorms \\eqref{eq:truncated}. Hereafter we work with Sobolev spaces, i.e. $p=q=2$.\n\n\\red{In this Section we will discuss several cases which depend on various results concerning Sobolev spaces and censored\/reflected Markov processes, each with its own assumptions. Therefore we refrain from formulating any theorems here, as they would be unnecessarily complicated. Interested readers may gather the assumptions from the references provided to each case.}\n\n\\red{We will gradually introduce some notions concerning the Dirichlet forms in \\textit{italics}, for details we refer to Fukushima, Oshima, and Takeda \\cite[Chapter 1.1]{MR2778606}. Let $\\mathcal E$ be a symmetric bilinear form with domain $D[\\mathcal E] \\subseteq L^2(\\Omega)$ for some $\\Omega\\subseteq \\mR^d$. Let $\\mathcal E_1(u,u) = \\mathcal E(u,u) + \\|u\\|_{L^2(\\Omega)}^2$. We say that $(\\mathcal{E},D[\\mathcal E])$ (this pair will also be called form below) is \\textit{closed} if, with respect to $\\mathcal E_1$, every Cauchy sequence has a limit in $D[\\mathcal E]$. We say that the form is \\textit{closable} if it has a closed extension. In what follows we write\n\t$$\\mathcal{E}^{\\rm cen}(u,u) = \\int_\\Omega\\int_\\Omega (u(x) - u(y))^2 K(x,y) \\, \\red{dy \\, dx},$$\n\tand for $\\theta\\in (0,1]$\n\t$$\\mathcal{E}^{\\rm tr} (u,u) = \\int_\\Omega \\int_{B(x,\\theta\\delta(x))} (u(x) - u(y))^2 K(x,y) \\, dy \\, dx.$$\n\tThe symbol $\\mathcal E^{\\rm cen}$ refers to the censored stable processes introduced by Bogdan, Burdzy, and Chen \\cite{MR2006232}. There, the kernel was the one known from the fractional Sobolev spaces: $K(x,y) = c|x-y|^{-d-\\alpha}$. Censored processes for more general $K$ corresponding to a class of subordinated Brownian motions were studied by Wagner \\cite{Vanja}.}\n\n\\red{We will consider the above forms in two contexts. In the first one, we start with the space of smooth functions, compactly supported in $\\Omega$: $C_c^\\infty(\\Omega)$. Using the arguments which follow equation (2.4) on \\cite[page 93]{MR2006232} it can be shown that $(\\mathcal E^{\\rm cen},C_c^\\infty(\\Omega))$ is closable and \\textit{Markovian} for arbitrary L\\'evy kernel $K$ (in particular, any which satisfies \\AJ) and set $\\Omega$. If we let\n\t$$\\mathcal{F} := \\textrm{'Completion of }C^\\infty_c(\\Omega)\\textrm{ with respect to }\\mathcal{E}^{\\rm cen}_1\\,',$$\n\tthen by \\cite[Theorem 3.1.1]{MR2778606} $(\\mathcal E^{\\rm cen},\\mathcal F)$ is closed and Markovian, that is, a \\textit{Dirichlet} form. Furthermore, by construction, it is obvious that $C_c^\\infty(D)$ is a \\textit{core} for $(\\mathcal E^{\\rm cen},\\mathcal F)$, hence the form is \\textit{regular} and by \\cite[Theorem 7.2.1]{MR2778606} to every regular Dirichlet form corresponds a Hunt process. Thus, when $\\mathcal E^{\\rm cen}$ is comparable to $\\mathcal E^{\\rm tr}$ we obtain the existence of a Hunt process with the Dirichlet form $(\\mathcal E^{\\rm tr},\\mathcal F)$. We note that the arguments from \\cite[page 93]{MR2006232} may be used directly with $\\mathcal E^{\\rm tr}$ because it has a similar structure. Then, independently of comparability results, we obtain a regular Dirichlet form $(\\mathcal E^{\\rm tr}, \\mathcal F^{\\rm tr})$, where\n\t$$\\mathcal F^{\\rm tr}  := \\textrm{'Completion of }C^\\infty_c(\\Omega)\\textrm{ with respect to }\\mathcal{E}^{\\rm tr}_1\\,'.$$}\n\\red{\\indent The second approach is by considering the domain corresponding to the active reflected form $\\mathcal F^{\\rm ref} := F_{2,2}(\\Omega)$. Here the situation becomes more tedious, since in general $C_c^\\infty(\\Omega)$ (or even $C_c(\\Omega)$) need not be dense in $\\mathcal F^{\\rm ref}$. However for some $K$ and $\\Omega$ the density holds true, see e.g., \\cite[Corollary~2.6]{MR2006232} and \\cite[Corollary~2.9]{Vanja}. In that case we get that $\\mathcal F = F_{2,2}(\\Omega)$ and when the comparability holds, we in fact have $\\mathcal F^{\\rm tr} = F_{2,2} (\\Omega)$. Then, the form $(\\mathcal E^{\\rm tr}, F_{2,2}(\\Omega))$ is a regular Dirichlet form and there exists an associated Hunt process. If the density does not hold, the technical remedy is to change the reference set to $\\overline \\Omega$, cf. \\cite[Remark 2.1]{MR2006232}. If $K$ and $\\Omega$ are sufficiently regular, then there exist extension (and trace) operators between $F_{2,2}(\\Omega)$ and $F_{2,2}(\\mR^d)$, see e.g. Jonsson and Wallin \\cite[Chapter V]{MR820626} or Rutkowski \\cite[Section 6]{2018AR}. Thanks to them we may show that $C_c^\\infty(\\overline \\Omega)$ is dense with respect to $\\mathcal E^{\\rm cen}_1$ in $F_{2,2}(\\Omega)$ by using the results for the functions on the whole space $\\mR^d$, available for very general L\\'evy kernels, see, e.g., Bogdan, Grzywny, Pietruska-Pa\\l{}uba, and Rutkowski \\cite[Lemma A.5]{MR4088505} or Fiscella, Servadei, and Valdinoci \\cite{MR3310082}. Then we obtain the existence of a process on $\\overline \\Omega$ corresponding to the regular Dirichlet form $(\\mathcal E^{\\rm cen}, F_{2,2}(\\Omega))$ and the comparability yields the existence of the process corresponding to $(\\mathcal E^{\\rm tr}, F_{2,2}(\\Omega))$.}\n\n\\red{The latter case seems more interesting in terms of applying the comparability results as we build regular Dirichlet forms from the truncated form $\\mathcal E^{\\rm tr}$ on the well-established Sobolev\/Triebel--Lizorkin space $F_{2,2}(\\Omega)$, which is then its natural domain.}\n\\bibliographystyle{abbrv}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec:intro}\n\nThe characterization of gravitational radiation escaping (or entering) asymptotically flat spacetimes was firmly established in the 1950-60's \\cite{Trautman58,Pirani57,Bel1962,Bondi1962,Sachs1962,Penrose62}, see \\cite{Zakharov} and references therein for a comprehensive review of 1973. The covariant approach uses Penrose's conformal completions \\cite{Penrose65,Penrose1966,Frauendiener2004,Kroon} and the basic ingredient is the {\\em News} tensor field \\cite{Bondi1962,Sachs1962}, a tensor that lives at infinity and which, when non-zero, determines univocally the existence of gravitational radiation escaping (or entering) the spacetime. \n\nUnfortunately, results based on the News tensor apply only to the case with vanishing cosmological constant $\\Lambda =0$. Since the beginning of this century we know that the Universe is in accelerated expansion, that proves the existence of a {\\em positive} cosmological constant $\\Lambda >0$. This constant might be an effective one, or a true new universal constant, but either way it destroys the asymptotically-flat picture, independently of the value of $\\Lambda$. Even if $\\Lambda$ is minuscule the problem remains. The difficulties were pointed out in \\cite{Penrose2011} and largely explained in \\cite{Ashtekar2014,Ashtekar2017} where the various problems arising were clearly exposed.\n\nThis situation prompted many scientists to attack the problem which resulted in a plethora of results, new techniques, new definitions, and various attempts to recover the neat and nice picture we had when $\\Lambda=0$. Nowadays, there is a vast literature on the subject and a better understanding of the predicament when $\\Lambda>0$, which can be categorized in the following points\n\\begin{itemize}\n\\item Linearized approximations \\cite{Ashtekar2015-b}, including  a version of the quadrupole formula in the linear regime \\cite{Ashtekar2015,Hoque2019}, the power radiated by a binary system in a de Sitter background \\cite{Bonga2017}, or intended definitions of energy \\cite{Bishop2016,Kolanowski2020}.\n\\item Studies using techniques of exact solutions, analyzing the asymptotic behaviour of the Weyl tensor \\cite{Krtous2004}, or the radiation generated by accelerating balck holes \\cite{Podolsky2009,Griffiths-Podolsky2009}\n\\item Definitions of mass-energy, by using spinorial techniques \\cite{Szabados2013,Szabados2015}, or Newman-Penrose expansions in preferred coordinate systems \\cite{Saw2018}, or on null hypersurfaces \\cite{Chrusciel2016}, or for weak gravitational waves \\cite{Chrusciel2021,Chrusciel2021b}, or using Hamiltonian techniques \\cite{Chrusciel2013}, or --for the case of a black hole-- assuming the existence of a timelike Killing vector \\cite{Dolan2019}. For a review, see \\cite{Szabados2019}.\n\\item Searching for mass-loss formulas by means of Newman-Penrose formalism using Bondi-type coordinate expansions \\cite{Saw2016,Saw2017ziu,Saw2017,Saw2018i,He2016}\n\\item Using holographic methods, gauge fixing and foliations on $\\mathscr{J}$ , in particular to study asypmtotic symmetries  \\cite{Compere2019,Compere2020}  or in combination with Bondi-like coordinate expansions \\cite{Poole2019} \n\\item Looking for charges and conservation laws \\cite{Chrusciel2013,Virmani2019,Kolanowski2021,Poole2021} and references therein.\n\\item Relation between the radiation and the properties of the sources \\cite{He2018}\n\\end{itemize}\n\nDespite all these advances, a basic problem remained: how to characterize, unambiguously, the presence of gravitational radiation at $\\mathscr{J}$. To solve this funsamental problem, we explored alternative, but physically equivalent, descriptions of the existence of radiation at infinity when $\\Lambda =0$.\nThe main aim in this quest was to find alternatives that could perform equally well in the presence of a positive cosmological constant too. We found an appropriate characterization of gravitational radiation at $\\mathscr{J}$ {\\em fully equivalent} to the standard one based on the News tensor \\cite{Fernandez-Alvarez_Senovilla20}.\nOur proposal was based on a re-scaled version of the {\\em Bel-Robinson tensor} \\cite{Bel1958,Senovilla97,Bel1962,Senovilla2000} at $\\mathscr{J}$, which describes the tidal energy-momentum of the gravitational field. The News tensor encodes information about {\\em quasi-local} energy-momentum radiated away by an isolated system, while the Bel-Robinson tensor describes energy-momentum properties of the {\\em tidal} gravitational field ---for historical reasons, one uses the name `super-energy' for this, see Appendix \\ref{App:1}. There is a relation between superenergy and quasi-local energy-momentum quantities on closed surfaces \\cite{Horowitz82,Senovilla2000,Szabados2004} that can be exploited. Furthermore, actual measurements of gravitational waves are basically of tidal nature.\nHence, it seemed like a good idea to explore the re-scaled Bel-Robinson tensor as a viable object detecting the existence of gravitational radiation. \n\nOnce we had the novel, but equivalent, characterization of radiation we were able to simply use their appropriate version when $\\Lambda >0$ and check whether or not it was able to do the job. It certainly is \\cite{Fernandez-Alvarez_Senovilla20b}, and we found the fundamental object that can be used for that purposes: the {\\em asymptotic (radiant) super-momentum}. This is introduced in section \\ref{sec:Q}, where I present our radiation criteria for general $\\Lambda\\geq 0$. The next section is devoted to clarify the equivalence with the News prescription when $\\Lambda =0$, and then section \\ref{sec:dS} is devoted to the case with positive $\\Lambda$. The problem of the existence of news-like objects in this case, and the question of in- and out-going radiation are discussed in section \\ref{sec:news} and the existence of asymptotic symmetries is studied in section \\ref{sec:sym}. I end the paper with a list of examples presented in \\cite{Fernandez-Alvarez_Senovilla-afs,Fernandez-Alvarez_Senovilla-dS} and some closing comments.\n\nBefore that, let us present the set up.\n\n\\subsection{Weakly asymptotically simple spacetimes}\\label{subsec:prelim}\n\nThroughout, I will assume that the spacetime $(\\hat M,\\hat g)$ is weakly asymptotically simple admitting a conformal compactification \\`a la Penrose \\cite{Penrose65,Kroon,Frauendiener2004,Stewart1991}, so that there exists a (unphysical) spacetime $(M,g)$ and a conformal embedding $\\Phi : \\hat M \\hookrightarrow M$ such that \n$$\\Phi^* (\\Omega^{-2} g) \\stackrel{\\hat M}{=} \\hat g, \\hspace{1cm} \\Omega\\in C^\\infty (M), \\hspace{1cm} \\Omega |_{\\Phi(\\hat M)} >0$$\nwhere $\\Phi^*$ is the pullback of $\\Phi$, and that the boundary of the image of $\\hat M$ in $M$, denoted by {\\color{blue} $\\mathscr{J} :=\\partial [\\Phi(\\hat M)]$}, is a smooth hypersurface where $\\Omega$ vanishes:\n$$\n\\Omega \\ \\stackrel{\\scri}{=}\\  0, \\hspace{1cm}  \\bm{n} := d\\Omega \\ \\stackrel{\\scri}{\\neq}\\  0 .\n$$\n\n$\\mathscr{J}$ is called ``null infinity''. When $\\Lambda \\geq 0$ it consists of two (not necessarily connected) subsets: future ($\\mathscr{J}^+$) and past ($\\mathscr{J}^-$) null infinity, distinguished by the absence of endpoints of past or future causal curves contained in $(M,g)$, respectively.\nUnder appropriate decaying conditions for the physical Ricci tensor $\\hat R_{\\mu\\nu}$ one has \\cite{Penrose65,Kroon}\n\\begin{equation}\\label{causalcharacter}\nn_\\mu n^\\mu \\ \\stackrel{\\scri}{=}\\  -\\frac{\\Lambda}{3} \\hspace{4mm} \\Longrightarrow \\mathscr{J} \\, \\, \\mbox{is} \n\\left\\{\n\\begin{array}{lcr}\n\\mbox{timelike} & \\mbox{if} & \\Lambda <0\\\\\n\\mbox{null} & \\mbox{if} & \\Lambda =0\\\\\n\\mbox{{\\color{blue} spacelike}} & \\mbox{if} & \\Lambda >0\n\\end{array}\n\\right.\n\\end{equation}\nIn the cases with $\\Lambda \\geq 0$, $n_\\mu$ is taken to be future pointing.\n\nThere is a {\\em gauge freedom} by changing the conformal factor by an arbitrary positive factor\n\\begin{equation}\\label{gauge}\n\\Omega \\rightarrow \\Omega\\omega, \\hspace{1cm} 0< \\omega\\in C^\\infty (M).\n\\end{equation}\nThough this is not necessary, in order to concorde with references \\cite{Fernandez-Alvarez_Senovilla20,Fernandez-Alvarez_Senovilla20b,Fernandez-Alvarez_Senovilla-afs,Fernandez-Alvarez_Senovilla-dS} I am going to {\\em partly} fix this gauge freedom by choosing $\\Omega$ such that $\\nabla_\\mu \\nabla^\\mu \\Omega \\ \\stackrel{\\scri}{=}\\  0$, which in turn implies\n\\begin{equation}\\label{nablan}\n\\nabla_\\mu n_\\nu =\\nabla_\\mu \\nabla_\\nu \\Omega \\ \\stackrel{\\scri}{=}\\  0.\n\\end{equation}\nThe remaining gauge freedom is given by functions $\\omega>0$ restricted to \n$$\n\\pounds_n \\omega =n^\\mu\\nabla_\\mu \\omega\\ \\stackrel{\\scri}{=}\\  0.\n$$\n\n$\\mathscr{J}$ being a hypersurface, it inherits a metric from $(M,g)$, its first fundamental form: \n$$\nh(X,Y) := g(X,Y),  \\hspace{1cm} \\forall X,Y \\in \\mathfrak{X}(\\mathscr{J}).\n$$\nGiven any basis $\\{\\vec{e}{}_a\\}$ ($a,b,\\dots =1,2,3$) of vector fields in $\\mathfrak{X}(\\mathscr{J})$ the corresponding components are denoted by\n$$\nh_{ab} = g(\\vec{e}_a,\\vec{e_b}) .\n$$\nDue to (\\ref{causalcharacter}) the metric $h_{ab}$ is Riemannian (positive definite) if $\\Lambda >0$, Lorentzian if $\\Lambda <0$ and degenerate if $\\Lambda =0$. In the latter case, $n^\\mu$ is tangent to $\\mathscr{J}$ so that $n^\\mu =n^a e^\\mu_a$, and then $n^a$ is the degeneration direction\n\\begin{equation}\nh_{ab} n^a =0, \\hspace{1cm} (\\Lambda =0).\n\\end{equation}\n\nFor general $\\Lambda$, and according to (\\ref{nablan}), $\\mathscr{J}$ is a {\\em totally geodesic} hypersurface, its second fundamental form vanishing\\footnote{In the general case where the partial gauge fixing (\\ref{nablan}) is not enforced $\\mathscr{J}$ is a {\\em totally umbilic} hypersurface, the second fundamental form being proportional to the first fundamental form.}:\n$$\nK(X,Y) =0 \\hspace{1cm} \\forall X,Y \\in \\mathfrak{X}(\\mathscr{J}) .\n$$\nThis leads to the existence of a canonical torsion-free connection $\\overline\\nabla$ on $\\mathscr{J}$, inherited from $(M,g)$, {\\em independently of the sign of $\\Lambda$}:\n$$\n\\overline\\nabla_X Y := \\nabla_X Y  \\hspace{1cm} \\forall X,Y \\in \\mathfrak{X}(\\mathscr{J}).\n$$\nThis connection is, of course, the Levi-Civita connection of $(\\mathscr{J},h_{ab})$ whenever $\\Lambda \\neq 0$. Actually, one has\n\\begin{equation}\\label{nablah}\n\\overline\\nabla_c h_{ab} =0\n\\end{equation}\nfor all values of $\\Lambda$.\n\nOne can also define a volume 3-form $\\epsilon_{abc}$ by\n$$\n- n_\\alpha \\epsilon_{abc} :\\ \\stackrel{\\scri}{=}\\  V \\eta_{\\alpha\\mu\\nu\\rho} e^\\mu{}_a e^\\nu{}_b e^\\rho{}_c .\n$$\nwhere $\\eta_{\\alpha\\mu\\nu\\rho}$ is the canonical volume 4-form in $(M,g)$ and the constant\n$$\nV=\\left\\{\\begin{array}{ccc}\n(|\\Lambda|\/3)^{1\/2} & \\mbox{if} & \\Lambda \\neq 0\\\\\n1 & \\mbox{if} & \\Lambda =0 .\n\\end{array}\n\\right.\n$$\nAgain $\\overline{\\nabla}_d \\epsilon_{abc}=0$ in all cases.\n\nHenceforth, we will say that $S\\subset \\mathscr{J}$ is a {\\em cut} on $\\mathscr{J}$ if it is a 2-dimensional spacelike submanifold immersed in $\\mathscr{J}$. When $\\Lambda >0$ the `spacelike' character is ensured and all possible 2-dimensional submanifolds are cuts. For $\\Lambda =0$, cuts are cross sections of the null $\\mathscr{J}$ transversal to the null generators everywhere. In many cases cuts will have $\\mathbb{S}^2$ topology, and these always exist in the regular (or asymptotically Minskowskian) case when $\\Lambda =0$ as the topology of $\\mathscr{J}$ is $\\mathbb{R}\\times \\mathbb{S}^2$ \\cite{Geroch1977}. However, this will not be necessarily the case when $\\Lambda >0$ and, furthermore, even in the case with  $\\mathscr{J} \\simeq \\mathbb{R}\\times \\mathbb{S}^2$ one might be interested in preferred cuts with non-$\\mathbb{S}^2$ topology. Examples are given in \\cite{Fernandez-Alvarez_Senovilla-dS}.\n\n\n\\section{Asymptotic (radiant) super-momentum: the radiation criterion}\\label{sec:Q}\nThe fact is that a real gravitational field is described by the curvature of the spacetime. In particular, gravitational radiation is the propagation of curvature, the propagation of changing geometrical properties, in space and time. Hence, the existence of gravitational radiation carrying energy-momentum --lost by isolated systems in their dynamical evolution-- should be amenable to a description that considers the strength of the curvature, that is, the strength of the tidal gravitational effects, as the fundamental variable. This is the basic idea to be developed in what follows which was put forward and developed in detail in \\cite{Fernandez-Alvarez_Senovilla20,Fernandez-Alvarez_Senovilla20b,Fernandez-Alvarez_Senovilla-afs,Fernandez-Alvarez_Senovilla-dS}.\n\nThe strength\\footnote{This could be called the `energy' of the Weyl curvature, but I prefer to use the word `strength' to avoid misunderstandings, as the physical units are not those of energy \\cite{Senovilla97,Senovilla2000}. Actually the name `super-energy' has been traditionally used for these quantities quadratic in the curvature, but this may also lead to confusion. A better name could be the {\\em tidal energy}, but we will have to wait to see if this will eventually catch up.} of the tidal gravitational forces can be appropriately described by the {\\em Bel-Robinson tensor} (see Appendix \\ref{App:1}), defined by\n$$\n{\\cal T}_{\\alpha\\beta\\lambda\\mu}=C_{\\alpha\\rho\\lambda}{}^{\\sigma}\nC_{\\mu\\sigma\\beta}{}^{\\rho}+\\stackrel{*}{C}_{\\alpha\\rho\\lambda}{}^{\\sigma}\n\\stackrel{*}{C}_{\\mu\\sigma\\beta}{}^{\\rho} .\n$$\n${\\cal T}_{\\alpha\\beta\\lambda\\mu}$ is conformally invariant, fully symmetric and traceless\n$${\\cal T}_{\\alpha\\beta\\lambda\\mu}={\\cal T}_{(\\alpha\\beta\\lambda\\mu)},\\hspace{1cm} {\\cal T}^{\\rho}{}_{\\rho\\lambda\\mu}=0\n$$\nand satisfies the dominant property\n\\begin{equation}\\label{DP}\n{\\cal T}_{\\alpha\\beta\\lambda\\mu}u^{\\alpha}v^{\\beta}w^{\\lambda}z^{\\mu}\\geq 0\n\\end{equation}\nfor arbitrary future-pointing vectors $u^{\\alpha}$, $v^{\\beta}$, $w^{\\lambda}$, and $z^{\\mu}$ (inequality is strict if all of them are timelike).\nThe Bel-Robinson tensor is also covariantly conserved \n$$\\nabla^{\\alpha}{\\cal T}_{\\alpha\\beta\\lambda\\mu} =0$$\nif the $\\Lambda$-vacuum Einstein's field equations $R_{\\beta\\mu}=\\Lambda g_{\\beta\\mu}$ hold.\nThis provides conserved quantities if there are (conformal) Killing vector fields \\cite{Senovilla2000,Lazkoz2003}.\nNevertheless, ${\\cal T}_{\\alpha\\beta\\lambda\\mu}$ is not a good tensor to describe radiation arriving at {\\em infinity}. The reason is that one can prove under very general circumstances that the Weyl tensor vanishes at $\\mathscr{J}$ \\cite{Penrose65,Kroon,Geroch1977}: $$C_{\\alpha\\beta\\mu}{}^\\nu \\ \\stackrel{\\scri}{=}\\  0.$$ Therefore, the Bel-Robinson tensor vanishes there too.\n\nHowever, the vanishing of the Weyl tensor at $\\mathscr{J}$ allows us to introduce the re-scaled Weyl tensor \n\\begin{equation}\\label{def:d}\nd_{\\alpha\\beta\\mu}{}^\\nu :=\\frac{1}{\\Omega} C_{\\alpha\\beta\\mu}{}^\\nu \n\\end{equation}\nwhich is well defined, and generically non-vanishing,  at $\\mathscr{J}$.\nThis is a conformally invariant traceless tensor field defined on $M$ with the same symmetry and trace properties as the Weyl tensor: it is a Weyl-tensor candidate ---see Appendix \\ref{App:1}. In the physical spacetime one has\n$$\\nabla_{\\nu} d_{\\alpha\\beta\\mu}{}^\\nu \\stackrel{\\hat M}{=} \\Omega^{-1} \\hat\\nabla_\\nu \\hat C_{\\alpha\\beta\\mu}{}^\\nu$$\nso that $d_{\\alpha\\beta\\mu}{}^\\nu$ is divergence-free on $\\hat M$ and also at $\\mathscr{J}$ in $\\Lambda$-vacuum\\footnote{Actually, at $\\mathscr{J}$ it is enough that the physical Cotton tensor decays quickly enough.}.\nThe gauge behaviour of the re-scaled Weyl tensor under the remaining gauge freedom (\\ref{gauge}) is simply\n$$\\hspace{1cm} d_{\\alpha\\beta\\mu}{}^\\nu \\rightarrow \\frac{1}{\\omega} d_{\\alpha\\beta\\mu}{}^\\nu .$$\nThe Bianchi identities imply that\n\\begin{equation}\\label{dn}\nd_{\\alpha\\beta\\mu}{}^\\nu n_\\nu +2 \\nabla_{[\\alpha} S_{\\beta]\\mu} \\ \\stackrel{\\scri}{=}\\  0\n\\end{equation}\nwhere $S_{\\beta\\mu} := \\frac{1}{2}(R_{\\beta\\mu} -\\frac{1}{6} g_{\\beta\\mu})$ is the Schouten tensor on $(M,g)$.\n\n\nGiven that $d_{\\alpha\\beta\\mu}{}^\\nu$ is a Weyl-tensor candidate, we can build its super-energy tensor $T\\{d\\}$ as shown in Appendix \\ref{App:1} \n$$\n\t\t\t\tT\\{d\\}_{\\alpha\\beta\\gamma\\delta} := D_{\\alpha\\beta\\gamma\\delta} := \\Omega^{-2}\\mathcal{T}_{\\alpha\\beta\\gamma\\delta}=  d_{\\alpha\\mu\\gamma}{}^\\nu d_{\\delta\\nu\\beta}{}^\\mu +\\stackrel{*}{d}_{\\alpha\\mu\\gamma}{}^\\nu \\stackrel{*}{d}_{\\delta\\nu\\beta}{}^\\mu \n$$\nwhich can also be considered as a \\emph{re-scaled Bel-Robinson tensor}.\n$\\mathcal{D}_{\\alpha\\beta\\gamma\\delta}$ is regular at $\\mathscr{J}$, non-vanishing in general.\n$\\mathcal{D}_{\\alpha\\beta\\gamma\\delta}$ has all the properties of the Bel-Robinson tensor, in particular is fully symmetric and traceless. It is also divergence-free at $\\mathscr{J}$ under the decaying conditions for the physical energy-momentum tensor that imply $\\nabla_\\nu d_{\\alpha\\mu\\gamma}{}^\\nu\\ \\stackrel{\\scri}{=}\\  0$.\n Its gauge behaviour under (\\ref{gauge}) is \n\t\t\t\t$$\\mathcal{D}_{\\alpha\\beta\\gamma\\delta}\\rightarrow \\frac{1}{\\omega^2}\\mathcal{D}_{\\alpha\\beta\\gamma\\delta}.$$\n\t\t\t\t\nFrom now on we will concentrate in the physical relevant case with non-negative $\\Lambda\\geq 0$.\nThe fundamental object on which the entire approach is based is the following one-form\t\t\t\n\\begin{equation}\\label{Pi}\n\\fbox{$\\Pi_\\alpha := -n^\\mu n^\\nu n^\\rho \\mathcal{D}_{\\alpha\\mu\\nu\\rho} =-\\nabla^\\mu \\Omega \\nabla^\\nu \\Omega \\nabla^\\rho \\Omega \\mathcal{D}_{\\alpha\\mu\\nu\\rho}  $}\n\\end{equation}\nwhich is geometrically well and uniquely defined at $\\mathscr{J}$. We will mainly use the properties of $\\Pi_\\alpha$ at $\\mathscr{J}$. From the general dominant property of super-energy tensors (Appendix \\ref{App:1}) one knows that $\\Pi_\\alpha|_\\mathscr{J}$ is causal and future pointing ---this is also true on a neighbourhood of $\\mathscr{J}$ when $\\Lambda >0$, and can always be achieved on such a neighbourhood when $\\Lambda =0$ by appropriate choices of $\\Omega$. In general, we call $\\Pi_\\alpha|_\\mathscr{J}$ the {\\em asymptotic super-momentum}. Actually, in the $\\Lambda =0$ situation, $\\Pi_\\alpha|_\\mathscr{J}$ is null, and to stress this fact we add the adjective ``radiant'' and then a specific notation is used:\n\\begin{eqnarray*}\n&\\Lambda =0:& \\hspace{7mm} \\Pi_\\mu |_\\mathscr{J} := \\mathcal{Q}_\\mu, \\hspace{7mm} \\mathcal{Q}_\\mu \\mathcal{Q}^\\mu =0 \\hspace{2mm} \\mbox{(Asymptotic radiant super-momentum)}\\\\\n&\\Lambda >0:& \\hspace{7mm} \\Pi_\\mu |_\\mathscr{J} := p_\\mu, \\hspace{7mm} p_\\mu p^\\mu \\leq 0 \\hspace{2mm} \\mbox{(Asymptotic super-momentum)}\n\\end{eqnarray*}\n\nThe gauge behaviour under (\\ref{gauge}) is the same for both $\\mathcal{Q}_\\mu$ and $p_\\mu$, namely we have in general\n$$\\Pi_\\alpha |_\\mathscr{J} \\rightarrow \\omega^{-5} \\Pi_\\alpha |_\\mathscr{J} .\n$$\nFurthermore we have the following important property\n\\begin{equation} \\label{divPi}\n\t\t\t\t\t\\nabla_\\mu\\Pi^\\mu\\ \\stackrel{\\scri}{=}\\  0  \n\\end{equation}\nwhich holds in full generality when $\\Lambda =0$ \\cite{Fernandez-Alvarez_Senovilla-afs}, but needs to assume \nthat the energy-momentum tensor of the physical space-time $(\\hat M,\\hat g_{\\mu\\nu})$ behaves approaching $\\mathscr{J}$ as $ \\hat{T}_{\\alpha\\beta}|_\\mathscr{J} \\sim \\mathcal{O}(\\Omega^3)$ \\cite{Fernandez-Alvarez_Senovilla-dS} (this includes the vacuum case $ \\hat{T}_{\\alpha\\beta}=0$).\t\n\nThe existence of gravitational radiation cannot be detected at a given point, due to the non-local nature of the gravitational field. Thus, the maximum one can aspire for is to detect the radiation by tidal deformations of cuts \\cite{Penrose1986}. Consider thus any cut $S\\subset \\mathscr{J}$ and let $\\ell^\\mu$ be a null normal to $S$ such that $\\bm{\\ell}\\wedge \\bm{n} \\neq 0$. \nThe criteria that we found to detect the existence or absence of gravitational radiation arriving at $\\mathscr{J}^+$ (or departing from $\\mathscr{J}^-$) are as follows \\cite{Fernandez-Alvarez_Senovilla20,Fernandez-Alvarez_Senovilla20b,Fernandez-Alvarez_Senovilla-afs,Fernandez-Alvarez_Senovilla-dS}\n\\begin{crit}[Absence of radiation on a cut]\\label{crit1}\nWhen $\\Lambda \\geq 0$, there is no gravitational radiation on a cut $S\\subset \\mathscr{J}$ with spherical topology if and only if $\\Pi_\\alpha|_S$ is orthogonal to $S$ pointing along the direction $\\ell_\\alpha +\\mbox{{\\rm sgn}} (\\Lambda) \\left(n_\\alpha -\\ell_\\alpha\\right)$.\n\\end{crit}\t\t\nObserve that this criterion states that $p_\\mu$ points along $n_\\mu$ if $\\Lambda >0$, and that if $\\Lambda =0$, $\\mathcal{Q}_\\mu$ points along $\\ell_\\mu$ (which in this case is uniquely determined as the null direction orthogonal to $S$ other than $n_\\mu$).\n\n\nThe restriction on the topology of the cut will be justified later when we discuss the equivalence with the standard characterization of a vanishing news tensor if $\\Lambda =0$. However, such a restriction can be somewhat relaxed if one considers open portions of $\\mathscr{J}$. Thus, \nlet now $\\Delta\\subset \\mathscr{J}$ denote an open portion of $\\mathscr{J}$ with the same topology of $\\mathscr{J}$.\n\\begin{crit}[Absence of radiation on $\\Delta \\subset \\mathscr{J}$]\\label{crit2}\nWhen $\\Lambda \\geq 0$, there is no gravitational radiation on an open portion $\\Delta\\subset \\mathscr{J}$ that admits a cut with $\\mathbb{S}^2$-topology if and only if $\\Pi_\\alpha|_\\Delta$ is transversal to $\\mathscr{J}$ and orthogonal to $\\Delta$. This is the same as saying that $\\Pi_\\alpha|_\\Delta$ is orthogonal to every cut within $\\Delta$.\n\nEquivalently, there is no gravitational radiation on such open portion $\\Delta\\subset \\mathscr{J}$ if and only if $n_\\alpha|_\\Delta$ is a principal direction of the re-scaled Weyl tensor $d_{\\alpha\\beta\\lambda\\mu}$ there.\n\\end{crit}\t\t\nObserve that these criteria are identical for both cases with positive or zero $\\Lambda$, and that they are purely geometrical and fully determined by the algebraic properties of $d_{\\alpha\\beta\\lambda\\mu}$. Here, the principal directions of the Weyl-tensor candidate $d_{\\alpha\\beta\\lambda\\mu}$ are considered in the classical sense \\cite{Pirani57,Bel1962}, that is, those lying in the intersection of the principal planes, or in other words, the common directions of the eigen-2-forms of $d_{\\alpha\\beta\\lambda\\mu}$ when seen as an endomorphism on 2-forms. Recall that, considering only the {\\em causal} principal vectors, for Petrov type I there is one principal {\\em timelike} vector and no null one, for Petrov type D there is an entire 2-plane of causal principal directions --which contains the two null multiple null ones-- and finally for Petrov types II, III, or N there is just one null principal vector and no timelike one.\n\n\nLet me then make some brief considerations about the implications of these criteria from the viewpoint of the algebraic properties of the re-scaled Weyl tensor. In the case with $\\Lambda =0$, stating that $\\mathcal{Q}_\\alpha$ is orthogonal to $\\Delta\\subset \\mathscr{J}$ {\\em and} transversal to $\\mathscr{J}$ can only happen if $\\mathcal{Q}_\\alpha$ actually vanishes there $\\mathcal{Q}_\\alpha|_\\Delta =0$. But this is known to imply \\cite{Bergqvist98,Senovilla2011} that the null $n^\\mu$ is actually a multiple principal null direction of $d_{\\alpha\\beta\\lambda\\mu}|_\\Delta$, that is to say, the re-scaled Weyl tensor is algebraically special and, at least, of Petrov type II there, which is in accordance with the discussion in \\cite{Krtous2004}. Hence, if $d_{\\alpha\\beta\\lambda\\mu}$ is type I and $\\Lambda =0$, the existence of radiation is ensured. In the case with $\\Lambda >0$, $p_\\mu$ is orthogonal to $\\Delta\\subset \\mathscr{J}$ (and then automatically transversal too) if $p_\\mu$ points along the normal $n_\\mu$, so that $\\bm{p}\\wedge \\bm{n}=0$. This states that the `asymptotic' super-Poynting (see later section \\ref{subsec:superp}) relative to the frame defined by $n^\\mu$ vanishes, that is\n$$\n\\left(\\delta^\\mu_\\nu -\\frac{3}{\\Lambda}  n^\\mu n_\\nu\\right)p^\\nu \\stackrel{\\Delta}{=} 0 \n$$\nwhich implies that $n^\\mu$ is a principal vector of $d_{\\alpha\\beta\\lambda\\mu}$ \\cite{Bel1962,Ferrando1997}. As $n_\\mu$ is timelike in this situation, absence of radiation in this case requires that $d_{\\alpha\\beta\\lambda\\mu}|_\\Delta$ is of Petrov type I or D. The converse does not hold; for instance, the $C$-metric is Petrov type D and contains gravitational radiation, see section \\ref{sec:fin} and \\cite{Fernandez-Alvarez_Senovilla-dS}.\n\nThere should be no confusion between the Petrov type of the physical Weyl tensor $\\hat{C}_{\\alpha\\beta\\lambda}{}^\\mu$ and that of $d_{\\alpha\\beta\\lambda}{}^{\\mu}$. Of course, there is a relation between them, as the Petrov type of the latter can only be equally, or more, degenerate than that of the former in the asymptotic region. This follows because the Weyl tensor is conformally invariant so that $\\hat{C}_{\\alpha\\beta\\lambda}{}^\\mu\\stackrel{\\hat M}{=} C_{\\alpha\\beta\\lambda}{}^\\mu$ and therefore, using (\\ref{def:d}) the Petrov type of $d_{\\alpha\\beta\\lambda}{}^{\\mu}$ is the same as that of $\\hat{C}_{\\alpha\\beta\\lambda}{}^\\mu$ on a neighbourhood of $\\mathscr{J}$. By using any invariant characterization of the Petrov types, for instance with curvature invariants or the number of principal null directions, one easily deduces that the Petrov type of $d_{\\alpha\\beta\\lambda}{}^\\mu$ at $\\mathscr{J}$ is as degenerate, or more, than that of the physical Weyl tensor near $\\mathscr{J}$. The reasoning is that, if one of the invariants used in the classification \\cite{Stephani2003} vanishes in the neighbourhood of $\\mathscr{J}$ then it will also vanish at $\\mathscr{J}$, while if it does not vanish on the neighbourhood, it may vanish or not at $\\mathscr{J}$. Therefore, the possible Petrov types of $d_{\\alpha\\beta\\lambda}{}^{\\mu}$ are restricted as follows\n\\begin{itemize}\n\\item If the Petrov type of $\\hat{C}_{\\alpha\\beta\\lambda}{}^{\\mu}$ in the asymptotic region is I, then $d_{\\alpha\\beta\\lambda}{}^{\\mu}$ can have {\\em any} Petrov type at $\\mathscr{J}$\n\\item If the Petrov type of $\\hat{C}_{\\alpha\\beta\\lambda}{}^{\\mu}$ in the asymptotic region is II, then $d_{\\alpha\\beta\\lambda}{}^{\\mu}$ can have any Petrov type at $\\mathscr{J}$ except I\n\\item If the Petrov type of $\\hat{C}_{\\alpha\\beta\\lambda}{}^{\\mu}$ in the asymptotic region is III, then $d_{\\alpha\\beta\\lambda}{}^{\\mu}$ can have Petrov types III, N and 0 at $\\mathscr{J}$\n\\item If the Petrov type of $\\hat{C}_{\\alpha\\beta\\lambda}{}^{\\mu}$ in the asymptotic region is N, then $d_{\\alpha\\beta\\lambda}{}^{\\mu}$ is either Petrov type N or 0 at $\\mathscr{J}$\n\\item If the Petrov type of $\\hat{C}_{\\alpha\\beta\\lambda}{}^{\\mu}$ in the asymptotic region is D, then $d_{\\alpha\\beta\\lambda}{}^{\\mu}$ is either Petrov type D or 0 at $\\mathscr{J}$\n\\item If $\\hat{C}_{\\alpha\\beta\\lambda}{}^{\\mu}=0$ on an open asymptotic region, then $d_{\\alpha\\beta\\lambda}{}^{\\mu}\\ \\stackrel{\\scri}{=}\\  0$\n\\end{itemize}\nHence, all Petrov types on the asymptotic region of the physical spacetime --except 0-- are compatible with the existence, and with the absence, of gravitational radiation crossing $\\mathscr{J}$.\n\nIn what follows, first I will show that criterion \\ref{crit2} coincides with the traditional one when  $\\Lambda =0$, and then I will discuss the implications that it has when $\\Lambda >0$.\n\n\n\\section{The case with $\\Lambda =0$: equivalence with the news criterion}\\label{sec:Lambda=0}\nAs we saw in subsection \\ref{subsec:prelim}, if $\\Lambda =0$ $n_\\mu $ is null, $h_{ab}$ is degenerate, $n^\\mu \\ \\stackrel{\\scri}{=}\\  n^a e^\\mu{}_a$ and $n^a$ is the degeneration vector field at $\\mathscr{J}$, ergo tangent to its null generators: $h_{ab}n^a =0$. Using the canonical connection and (\\ref{nablan}), $n^a$ is parallel on $\\mathscr{J}$: \n\\begin{equation}\\label{nablan1}\n\\overline\\nabla_b n^a =0.\n\\end{equation}\n\nThe topology of $\\mathscr{J}$ is usually taken to be $\\mathbb{R}\\times \\mathbb{S}^2$, though there are cases where this does not hold if there are singularities or incompleteness of $\\mathscr{J}$. In the standard case with $\\mathscr{J}\\simeq \\mathbb{R}\\times \\mathbb{S}^2$, the cuts $\\mathcal{S}$ can be chosen to be topologically $\\mathbb{S}^2$, see figure \\ref{fig:cut}.\nFor any cut $\\mathcal{S}$ there is a {\\em unique} lightlike vector field $\\ell^\\mu$ orthogonal to $\\mathcal{S}$ and such that $n_\\mu \\ell^\\mu =-2$ ---this is the vector field $\\ell^\\mu$ used in criterion \\ref{crit1}. I will denote by $\\{\\vec{E}_A\\}$ any basis of $\\mathfrak{X}(\\mathcal{S})$ ($A,B,\\dots =2,3$). These can be extended to vector fields on $\\mathscr{J}$ by choosing them on any cut and then propagating them such that $\\pounds_n E^a_A = M_A n^a$ (for some $M_A$ which will be irrelevant in what follows), where $\\pounds_v$ is the Lie derivative with respect to $v^a$ on $\\mathscr{J}$. Then $\\{\\vec{e}_a\\}=\\{\\vec n,\\vec{E}_A\\}$ are a basis of vector fields on $\\mathscr{J}$. Let $h^{ab}$ represent any tensor field satisfying \n$$h^{ab} h_{ac} h_{bd} = h_{cd}.$$\nSuch $h^{ab}$ suffers from an indeterminacy as $h^{ab} +n^a s^b + n^b s^a$ also satisfies the condition, for arbitrary $s^b$. Nevertheless, $h^{ab}$  allows us to raise indices and take traces {\\em unambiguously} when acting on covariant tensors fully orthogonal to $n^a$.\n\n\\begin{figure}[h]\n\\includegraphics[height=10cm]{FlatScri.pdf}\n\\caption{This is a schematic representation of $\\mathscr{J}^+$ when $\\Lambda =0$, where $\\vec n$ is the null degeneration vector field, $\\mathcal{S}$ is a cut, $\\vec\\ell$ is the unique null vector orthogonal to $\\mathcal{S}$ and transversal to $\\mathscr{J}$, and $\\vec E_A$ are vector fields tangent to the cut. Cuts are 2-dimensional surfaces, usually with $\\mathbb{S}^2$ topology. In the picture, one dimension is suppressed, so that here this topology of the cut is represented as a circumference. \\label{fig:cut}}\n\\end{figure}\n\nThe connection $\\overline\\nabla$, which is inherited from the spacetime, has a curvature tensor $\\overline{R}_{abc}{}^d$ and the corresponding (symmetric) Ricci tensor $\\overline{R}_{ac}:=\\overline{R}_{adc}{}^d$. It happens that\n$$\n\\overline{R}_{ab} n^b =0\n$$\nand therefore \n\\begin{equation}\\label{Rbar}\n\\overline{R} := h^{ab} \\overline{R}_{ab} \n\\end{equation}\nis well defined.\n\nDue to (\\ref{nablah}) and to the vanishing of the second fundamental form on $\\mathscr{J}$, which induces (\\ref{nablan1}), in this case we also have on $\\mathscr{J}$\n$$\\pounds_n h_{ab}=0.$$\nHence, {\\em all possible cuts are isometric}, with a first fundamental form \n$$\nq_{AB}:=h_{ab} E^a_A E^b_B, \\hspace{15mm} \\pounds_n q_{AB} =n^c\\overline \\nabla_c q_{AB} =0\n$$\nwhich is basically the non-degenerate part of $h_{ab}$. Its covariant derivative will be denoted by $D_A$. The scalar curvature (or twice the Gaussian curvature) of the cuts is precisely (\\ref{Rbar}) ---and $\\pounds_n \\overline R =0$. Of course, only the conformal class is fixed because of the gauge freedom (\\ref{gauge}): \n\\begin{equation}\\label{gaugeh}\nh_{ab} \\rightarrow \\tilde{h}_{ab}\\ \\stackrel{\\scri}{=}\\  \\omega^2 h_{ab},\\hspace{1cm}   \\tilde{q}_{AB} \\ \\stackrel{\\mathcal{S}}{=}\\  \\omega^2 q_{AB}.\n\\end{equation}\n\nThe structure $(h_{ab},n^a)$ on $\\mathscr{J}$ is universal. Nevertheless, observe that it does not contain any {\\em dynamical} behaviour. The dynamics, and therefore the possible existence of gravitational radiation, is not encoded in this universal structure: it comes from structure {\\em inherited} from the physical spacetime. In this $\\Lambda =0$ situation the time dependence along $\\mathscr{J}$ is actually encoded in the connection $\\overline\\nabla$ and its curvature. This is crucial. Notice that\n$$\n\\pounds_n \\overline\\nabla \\neq 0, \\hspace{1cm} [\\pounds_n,\\overline\\nabla]\\neq 0\n$$\nIn particular, for any one-form $\\bm{t}$ \n\\begin{equation}\n[\\pounds_n,\\overline\\nabla_b]t_a = -n^c t_c \\, \\left(\\overline{S}_{ab}-\\frac{1}{2}h^{ef}\\overline{S}_{ef} h_{ab}\\right)\n\\end{equation}\nwhere $\\overline{S}_{ab}$ is the pull-back of the Schouten tensor to $\\mathscr{J}$:\n$$\\overline{S}_{ab}:\\ \\stackrel{\\scri}{=}\\  S_{\\mu\\nu}e^\\mu{}_a e^\\nu{}_b , \\hspace{1cm} n^a \\overline{S}_{ab} =0$$\nalso given by\n$$\\overline{S}_{ab}-\\frac{1}{2}h^{ef}\\overline{S}_{ef} h_{ab}= \\overline{R}_{ab} -\\frac{1}{2}\\overline R h_{ab}.$$\n\nIn plain words, $\\overline{S}_{ab}$ encodes the time variations within $\\mathscr{J}$, hence it contains the information about any gravitational radiation crossing $\\mathscr{J}$.\nHowever, $\\overline{S}_{ab}$ has non-trivial gauge behaviour:\n\\begin{equation}\\label{gaugeS}\n\\overline{S}_{ab} \\rightarrow \\overline{S}_{ab} - \\frac{1}{\\omega}\\overline\\nabla_a \\overline\\nabla_b \\omega + \\frac{2}{\\omega^2}\\overline\\nabla_a\\omega\\overline\\nabla_b\\omega-\\frac{1}{2\\omega^2} h_{ab}\\, \\omega^c\\overline\\nabla_c \\omega\n\\end{equation}\n(here $g^{\\mu\\nu}\\nabla_\\nu\\omega : \\ \\stackrel{\\scri}{=}\\  \\omega^c e^\\mu{}_c$).\nOne needs to extract the relevant gauge-invariant part of $\\overline{S}_{ab}$: this is the News tensor field.\n\n\nThere are many ways to define the News tensor field, such as by using expansions in Bondi coordinates \\cite{Bondi1960,Bondi1962,Sachs1962}, or defining the asymptotic outgoing shear \\cite{Ashtekar81,Kroon,Penrose65,Stewart1991}, or by computing the limit at $\\mathscr{J}$ of $\\Omega^{-1} \\nabla_\\mu n_\\nu$ in certain gauges \\cite{Winicour}.\nTo our purposes, the best suited definition is just the dynamical (time-dependent) and gauge invariant part of $\\overline{S}_{ab}$, in accordance with \\cite{Geroch1977}. This is a geometrically neat and physically clarifying definition. \n\nTo find the explicit expression, I start by noticing that $\\overline{S}_{ab}$ is orthogonal to $n^a$, so that only the components $S_{AB} =\\overline{S}_{ab}E^a_A E^b_B$ are non-zero. Nevertheless, these components change from cut to cut, due to the dynamical dependence of $\\overline{S}_{ab}$ itself. By projecting \\eqref{dn} to $\\mathscr{J}$ one has\n\\begin{equation}\\label{bianchi}\n2\\overline \\nabla_{[a}\\overline{S}_{b]c}=-e^\\alpha_a e^\\beta_b e^\\lambda_c d_{\\alpha\\beta\\lambda}{}^\\mu n_\\mu\n\\end{equation}\nfrom where it easily follows\n$$\n\\pounds_n \\overline{S}_{bc} = n^c\\overline \\nabla_c \\overline{S}_{bc} = -n^\\alpha e^\\beta_b e^\\lambda_c d_{\\alpha\\beta\\lambda}{}^\\mu n_\\mu\\neq 0\n$$\nwhich is non-vanishing in general. In particular\n$$\n\\pounds_n S_{AB} = n^c\\overline \\nabla_c S_{AB} \\neq 0\n$$\nso that $S_{AB}$ depend on the cut. Such a time-dependent part is what interests us. Consequently, we need tu subtract, from $\\overline{S}_{ab}$, a tensor field that is symmetric, orthogonal to $n^a$, time-independent and with a gauge behaviour that compensates \\eqref{gaugeS}. Explicitly, we need a tensor field $\\rho_{ab}$ such that\n\\begin{equation}\\label{rhoprop}\n\\rho_{ab}=\\rho_{ba}, \\hspace{1cm} n^a\\rho_{ab}=0, \\hspace{8mm} \\overline\\nabla_{[c}\\rho_{a]b} =0,\n\\end{equation}\nand with the following gauge behaviour under \\eqref{gaugeh}\n$$\n\\tilde{\\rho}_{ab}= \\rho_{ab} - \\frac{1}{\\omega}\\overline\\nabla_a \\overline\\nabla_b \\omega + \\frac{2}{\\omega^2}\\overline\\nabla_a\\omega\\overline\\nabla_b\\omega-\\frac{1}{2\\omega^2} h_{ab}\\, \\omega^c\\overline\\nabla_c \\omega .\n$$\nNote that $n^c\\overline \\nabla_c \\rho_{ab}=0$ follows from the above, so that $\\rho_{ab}$ is actually a true 2-dimensional tensor field, with only $\\rho_{AB}$ non-zero components and these are time-independent $n^c\\overline \\nabla_c \\rho_{AB}=0$. Therefore, it is enough to have this tensor field on any cut. {\\em But this is the tensor $\\rho_{AB}$ studied in Appendix \\ref{App:rho}}. Observe that then we have, in addition, $h^{ab} \\rho_{ab} =\\overline{R}\/2$.\n\nThe News tensor field is defined by \\cite{Geroch1977}\n\\begin{equation}\\label{news}\n\\fbox{$N_{ab}:= \\overline{S}_{ab} -\\rho_{ab}$}\n\\end{equation}\nand has the following properties\n$$\nN_{ab}=N_{ba}, \\hspace{1cm} n^a N_{ab} =0, \\hspace{1cm} h^{ab} N_{ab} =0\n$$\nand, more importantly, $N_{ab}$ {\\em is gauge invariant} under \\eqref{gaugeh}\n$$\nN_{ab}=\\tilde{N}_{ab}.\n$$\nFrom \\eqref{bianchi}, \\eqref{news} and \\eqref{rhoprop} we derive\n\\begin{equation}\\label{bianchiN}\n2\\overline \\nabla_{[a}N_{b]c}=-e^\\alpha_a e^\\beta_b e^\\lambda_c d_{\\alpha\\beta\\lambda}{}^\\mu n_\\mu\n\\end{equation}\nfrom where, as before,\n$$\n\\pounds_n N_{ab} \\neq 0\n$$\nin general, so that the News tensor generically changes from one cut to another.\nThe pullback of $N_{ab}$ to any cut $\\mathcal{S}$ is denoted by \n$$\nN_{AB}(\\mathcal{S})\\ \\stackrel{\\mathcal{S}}{=}\\  N_{ab} E^a{}_A E^b{}_B.\n$$\nI will also use the notation\n$$\n\\dot N_{AB}(\\mathcal{S}) :\\ \\stackrel{\\mathcal{S}}{=}\\  E^a{}_A E^b{}_B \\pounds_n N_{ab}\n$$\nThe classical characterization of gravitational radiation in the case $\\Lambda =0$ is given as follows\n\\begin{Definition}[Classical radiation characterization]\\label{def:news}\nThere is no gravitational radiation on a given cut $\\mathcal{S}\\subset \\mathscr{J}$ if and only if the News tensor vanishes there:\n$$\nN_{AB}(\\mathcal{S})=0 \\Longleftrightarrow N_{ab}\\ \\stackrel{\\mathcal{S}}{=}\\  0 \\Longleftrightarrow \\mbox{no gravitational radiation on $\\mathcal{S}$}\n$$\n\\end{Definition}\n\\begin{Remark}\nObserve that $N_{ab}$ is a tensor field, and its vanishing at any point is an invariant statement. Nevertheless, one cannot aspire to localize gravitational radiation at a point, and thus the vanishing of $N_{ab}$ at a given point has no meaning in principle --see e.g. the discussion in \\cite{Penrose1986}. On the other hand, the vanishing of $N_{ab}$ on an entire cut does have a meaning, as this is a quasilocal statement. In this sense, $N_{ab}$ is related to the quasi-local energy-momentum properties of the gravitational field at $\\mathscr{J}$.\n\\end{Remark}\n\nTo justify the previous definition, a description of the gravitational energy-momentum properties at infinity is needed, which in turn requires the knowledge of the asymptotic symmetries, that is, the symmetries of $\\mathscr{J}$: the BMS group \\cite{Geroch1977,Bondi1962,Winicour,Penrose1966,Sachs62}. A convenient characterization of the infinitesimal isometries of $\\mathscr{J}$ that is independent of the gauge choice is given by the vector fields $\\vec Y\\in \\mathfrak{X}(\\mathscr{J})$ satisfying\n$$\n\\pounds_Y (n^a n^b h_{cd})=0.\n$$\nThis can be shown to be equivalent to ($\\phi\\in C^\\infty(\\mathscr{J})$)\n$$\n\\pounds_Y n^b =-\\phi n^b, \\hspace{1cm} \\pounds_Y h_{ab} = 2\\phi h_{ab}\n$$\nand the set of such vector fields is a Lie algebra.\nAny vector field of the form $Y^a=\\alpha n^a$, with $\\pounds_n \\alpha =0$ (and gauge behaviour $\\tilde\\alpha =\\omega\\alpha$), satisfies these relations. These are called {\\em infinitesimal super-translations}, and constitute an infinite-dimensional Abelian ideal. The rest of the BMS algebra is given by the conformal Killing vectors of $(\\mathcal{S},q_{AB})$ (i.e., the Lorentz group for round spheres).\nThere exists, however, a 4-dimensional Abelian sub-ideal constituted by the solutions of the linear equation ($\\Delta$ is the Laplacian on $(\\mathcal{S},q_{AB})$, see Appendix \\ref{App:rho})\n$$\n\\overline\\nabla_a \\overline\\nabla_b \\alpha +\\alpha \\rho_{ab}-\\frac{1}{2}h_{ab} \\left(\\Delta \\alpha + \\frac{\\overline R}{2} \\right)=0 \n$$\nwhose elements are called {\\em infinitesimal translations}. This equation is fully orthogonal to $n^a$ and time independent (its Lie derivative with respect to $n^a$ vanishes), and thus it is actually fully equivalent to the equation on any given cut\n$$\nD_A D_B \\alpha -\\frac{1}{2}q_{AB} \\Delta \\alpha +\\alpha \\left(\\rho_{AB} -\\frac{\\overline R}{4}q_{AB} \\right)=0 .\n$$\n{\\em This is precisely equation \\eqref{eq:Hess-gen}} whose four independent solutions are denoted by $\\pi_{(\\mu)}$.\nUsing these solutions $Y^b_{(\\mu)} :=\\pi_{(\\mu)} n^b$, the corresponding {\\em Bondi-Trautman 4-momentum} on any given cut $\\mathcal{S}$ can be expressed as \\cite{Geroch1977}\n$$\nB_{(\\mu)} (\\mathcal{S}) :=-\\frac{1}{32\\pi}  \\int_\\mathcal{S} \\pi_{(\\mu)} \\left( d_{\\beta\\mu\\nu}{}^\\rho n_\\rho\\ell^\\beta n^\\mu\\ell^\\nu +2\\sigma^{AB}N_{AB}\\right)\n$$\nwhere $\\sigma_{AB}$ is the shear tensor of $\\mathcal{S}$ along $\\ell^\\mu$, that is to say, the trace-free part of $E^\\mu{}_A E^\\nu{}_B \\nabla_\\mu\\ell_\\nu$ on $\\mathcal{S}$.\n\\begin{figure}\n\\includegraphics[height=10cm]{FlatScriPortion.pdf}\n\\caption{Schematic representation of a portion $\\Delta$ of $\\mathscr{J}^+$ delimited by two cuts $\\mathcal{S}_1$ and $\\mathcal{S}_2$ when $\\Lambda =0$, one dimension suppressed. The cut $\\mathcal{S}_2$ is to the future of $\\mathcal{S}_1$. The portion $\\Delta$ has the same topology as $\\mathscr{J}^+$ and is depicted by the shadowed part.\\label{fig:Delta}}\n\\end{figure}\nLet now $\\Delta\\subset\\mathscr{J}^+$ be a connected open portion of $\\mathscr{J}^+$, with the same topology as $\\mathscr{J}^+$, and limited by two cuts $\\mathcal{S}_1$ and $\\mathcal{S}_2$, with $\\mathcal{S}_2$ entirely to the future of $\\mathcal{S}_1$, as shown in figure \\ref{fig:Delta}. \nOne can compute the Bondi-Trautman 4-momentum for both cuts, and check what is the difference. \nThe result is, removing any matter content around $\\mathscr{J}^+$ for simplicity and to make things clearer (for the general case see e.g. \\cite{Geroch1977,Fernandez-Alvarez_Senovilla-afs}), \n$$\nB_{(\\mu)}(\\mathcal{S}_2)-B_{(\\mu)}(\\mathcal{S}_1) = -\\frac{1}{32\\pi}\\int_{\\Delta} \\pi_{(\\mu)}  h^{ab} h^{cd} N_{ac} N_{bd} \n$$\nwhich is a null vector in the auxiliary Minkowski metric of Appendix \\ref{App:rho} where $\\eta^{\\mu\\nu} \\pi_{(\\mu)}\\pi_{(\\mu)}=0$ and in particular has a strictly negative 0-component. This leads to the interpretation of News of definition \\ref{def:news}.\n\nWe can finally prove the equivalence of definition \\ref{def:news} with Criteria \\ref{crit1} and \\ref{crit2}. \nOn a given cut $ \\mathcal{S} $, one can split the radiant super-momentum into its null transverse (along $\\ell^\\alpha$) and tangent parts to $ \\mathscr{J} $,\n$$\n\t\t\t\\mathcal{Q}^\\alpha\\ \\stackrel{\\mathcal{S}}{=}\\ \n\t\t\t\\frac{1}{2}\\mathcal{W} \\ell^\\alpha + \\overline{\\mathcal{Q}}^a e^\\alpha{}_a,$$\n\t\twhere\n$\\mathcal{W} := -n^\\mu \\mathcal{Q}_\\mu \\geq 0$ and \n$$\n\t\t\t\\overline{\\mathcal{Q}}^a :=\\frac{1}{2} \\mathcal{Z} n^a + \\overline{\\mathcal{Q}}^A E^a{}_A\\quad \\text{with}\n\t\t\t\\hspace{3mm} \\mathcal{Z} := -\\ell_\\mu \\mathcal{Q}^\\mu \\geq 0 .\n$$\nThese quantities are observer-independent: $\\mathcal{Z}$ and $\\overline{\\mathcal{Q}}^A$ depend only on the cut, while $\\mathcal{W}$ is fully intrinsic to $\\mathscr{J}$. \n\nThe theorem that proves equivalence with criterion \\ref{crit1} is:\n\\begin{Theorem}[Radiation condition]\\label{th}\n\t\t\t\t There is no gravitational radiation on a given cut $ \\mathcal{S} \\subset \\mathscr{J} $ with $\\mathbb{S}^2$ topology if and only $\\mathcal{Q}^\\mu$ points along $\\ell^\\mu$ on that cut:\n$$\n\t\t\t\t   N_{AB}(\\mathcal{S})= 0 \\quad\\Longleftrightarrow \\quad \\overline{\\mathcal{Q}}^a\\ \\stackrel{\\mathcal{S}}{=}\\  0 \\quad (\\Longleftrightarrow \\quad \\mathcal{Z} = 0).\n$$\n\t\t\t\t\n\\end{Theorem}\n\\begin{proof}\nProjecting \\eqref{bianchiN} to $\\mathcal{S}$, a somehow long calculation leads to\n\\begin{align}\n\t\t\t\\mathcal{W}&\\ \\stackrel{\\mathcal{S}}{=}\\  2\\dot {N}^{RT}\\dot {N}_{RT} \\geq 0 ,\\label{nQ}\\\\\n\t\t\t\\mathcal{Z}&\\ \\stackrel{\\mathcal{S}}{=}\\  8 D^{[A} N^{B]C} D_{[A} N_{B]C}=4 D_C N^C{}_A D_B N^{BA}  \\geq 0 ,\\label{lQ}\\\\\n\t\t\t\\overline{\\mathcal{Q}}^A &\\ \\stackrel{\\mathcal{S}}{=}\\  \t 8\\dot N_{BC} D^{[B} N^{A]C} = -4 \\dot N^{BA} D_C N^C{}_B .\\label{FQ}\t\n\t\t\t\\end{align}\nEq. \\eqref{lQ} implies that $\\mathcal{Z}= 0 \\iff D_{[A} N_{B]C} =  0 $. Using \\eqref{FQ}, this happens if and only if $ \\overline\\mathcal{Q}^a= 0 $, that is, if and only if $2\\mathcal{Q}^\\mu \\ \\stackrel{\\mathcal{S}}{=}\\  \\mathcal{W} \\ell^\\mu$.\nBut $D_{[A} N_{B]C} =  0$ ---or equivalently $D_A N^A{}_B =0$--- informs us that $N_{AB}$ is a traceless symmetric Codazzi (and divergence-free) tensor on the compact $\\mathcal{S}$, which implies \\cite{Liu1998} that $N_{AB}=0$. Hence $  N_{AB}= 0  \\iff \\overline{\\mathcal{Q}}^a= 0 $ on $\\mathcal{S}$ .\n\t\t\t\\end{proof}\n\t\t\t\n\\begin{Remark}\nAs the radiant super-momentum $\\mathcal{Q}^\\mu$ is always null, this theorem can be equivalently stated as: there is no gravitational radiation on a given cut $ \\mathcal{S} \\subset \\mathscr{J} $ if and only if the radiant super-momentum is orthogonal to $\\mathcal{S}$ everywhere and not co-linear with $n^\\alpha$. \nNotice that, given a cut, this statement is totally unambiguous.\t\t\t\n\\end{Remark}\n\nSimilarly, the theorem that proves equivalence with criterion \\ref{crit2} is:\n\\begin{Theorem}[No radiation on $ \\Delta\\subset\\mathscr{J}$] \nThere is no gravitational radiation on an open portion $ \\Delta \\subset \\mathscr{J} $ which contains a cut with topology $\\mathbb{S}^2$ if and only if the radiant super-momentum $ \\mathcal{Q}{^\\alpha} $ vanishes on $\\Delta$:\n$$\n\t\t\t\t   N_{ab}\\stackrel{\\Delta}{=}  0 \\quad\\Longleftrightarrow \\quad \\mathcal{Q}{^\\alpha}\\stackrel{\\Delta}{=}  0 .\n$$\n\\end{Theorem}\t\n\\begin{proof}\nIf one can find cuts with $\\mathbb{S}^2$ topology in $\\Delta$, then according to the previous remark and theorem \\ref{th}, absence of radiation on $\\Delta$ requires that $2\\mathcal{Q}^\\alpha\\ \\stackrel{\\mathcal{S}}{=}\\  \\mathcal{W} \\ell^\\alpha$ on {\\em every possible such cut} $\\mathcal{S}$ included in $\\Delta$. But this is only possible if $\\mathcal{Q}^\\alpha\\stackrel{\\Delta}{=} 0$. \nMore generally, observe first that $N_{ab}\\stackrel{\\Delta}{=}  0$ trivially implies $\\mathcal{Q}^\\alpha\\stackrel{\\Delta}{=}  0$ due to \\eqref{nQ}-\\eqref{FQ} independently of the topologies. Conversely, if $\\mathcal{Q}^\\alpha \\stackrel{\\Delta}{=}  0$, then from \\eqref{nQ} $\\dot N_{AB}\\stackrel{\\Delta}{=}  0$, so that $N_{ab}$ is time independent and $N_{AB}$ is the same for all possible cuts (as they are all locally isometric). From \\eqref{lQ} we also have $D_{[A} N_{B]C}=0$ on every cut. Thus, if a compact cut has a positive Gaussian curvature --so that its topology is necessarily $\\mathbb{S}^2$, then a known theorem \\cite{Liu1998} implies that $N_{AB}=0$.\n\\end{proof}\n\n\\begin{Remark}\nIf there is gravitational radiation at $\\mathscr{J}$, there can arise situations where actually $2\\mathcal{Q}^\\mu =\\mathcal{W}\\ell^\\mu\\neq 0$ for a given foliation of cuts, with $\\mathcal{Z}= 0$ on them. Of course, this is only possible if the cuts have a non-$\\mathbb{S}^2$ topology. In this case, on those cuts $D_{[A} N_{B]C}=0$ (and $D_B N^{BA}=0$). In particular, for instance if $\\overline R =0$ one further has $D_C N_{AB}=0$, so that $N_{AB}$ is constant on those cuts. Hence, $N_{ab}=N_{ab}(v)$ are functions of a single coordinate $v$ such that the foliation is defined by $v=$const.\\, and necessarily $n^a\\overline \\nabla_a v \\neq 0$. For any other cut not in this special foliation $\\mathcal{Z}\\neq 0$. In any case, the non-vanishing of $\\mathcal{Q}^\\mu$ detects the radiation in this case correctly. Some examples of this situation exist in the C-metric and the Robinson-Trautman solutions.\n\\end{Remark}\n\n\n\\section{The case with $\\Lambda >0$}\\label{sec:dS}\nThe case of asymptotically de Sitter spacetimes is much harder and of a different nature. The main differences and the basic complications arise due to the fact that $\\bm n$ is now timelike, and thus $\\mathscr{J}$ is a spacelike hypersurface: there is no notion of `evolution'. The topology of $\\mathscr{J}$ is not determined, and it has no `universal' structure. The existence of infinitesimal symmetries is not guaranteed. There is a big issue concerning in- and out-going gravitational radiation. The very notion of energy is unclear because there cannot be any globally defined timelike Killing vector ---actually all possible Killing vectors on $(\\hat M,\\hat g)$ become tangent to $\\mathscr{J}$ at $\\mathscr{J}$.\nAnd there are other issues, see e.g., \\cite{Penrose2011,Ashtekar2017, Ashtekar2014,Szabados2019}. Still, criteria \\ref{crit1} and \\ref{crit2} appropriately identify the cases without radiation, even though there remain some subtleties to be understood concerning the mixture (or possible anihilation) of in- and out-going radiation.\n\nLet us start by noticing that, contrary to the asymptotically flat case where generally one deals with a nice topology $\\mathbb{R}\\times \\mathbb{S}^2$, in the case with $\\Lambda>0$ the topology of any connected component of $\\mathscr{J}$ is not determined, figure \\ref{fig:scri}. Its topology can be (see e.g. \\cite{Mars2017} with examples)\n\\begin{figure}\n\\includegraphics[width=12cm]{ScridS0.pdf}\n\\caption{This is a schematic representation of $\\mathscr{J}^+$ when $\\Lambda >0$, where $n^\\mu$ is timelike and normal to $\\mathscr{J}^+$, and $\\mathcal{S}$ represents a cut with spherical topology. As usual, one dimension is suppressed. The topology of $\\mathscr{J}$ is not fixed, the manifold can be $\\mathbb{R}^3$, $\\mathbb{R}\\times\\mathbb{S}^2$, $\\mathbb{S}^3$, or even $\\mathbb{S}^3\\setminus \\{p_1,\\dots, p_n\\}$ with $n>2$, see the main text. If, for instance, the topology is $\\mathbb{S}^3$, the shown schematic representation should be understood as a stereographic projection onto Euclidean space. Thus, the best way to always imagine $\\mathscr{J}^+$ when $\\Lambda >0$ is as $\\mathbb{S}^3$, possibly with a number of points removed.\\label{fig:scri}}\n\\end{figure}\n\\begin{enumerate}\n\\item $\\mathbb{S}^3$. This is the case for de Sitter or Taub-NUT-de Sitter spacetimes.\n\\item $\\mathbb{R}\\times \\mathbb{S}^2$. This happens in Kerr-de Sitter spacetime, including Kottler with spherical symmetry.\n\\item $\\mathbb{R}^3$, such as in Kottler spacetimes with non-positively curved group orbits.\n\\item Others, $\\mathbb{S}^3\\setminus \\{p_1,\\dots, p_n\\}$ with $n>2$.\n\\end{enumerate}\nThe {\\em conformal} geometry of $(\\mathscr{J},h_{ab})$ is given by the completion of the physical spacetime. In particular \n\\begin{itemize}\n\\item its intrinsic Schouten tensor, which actually coincides with the pull-back of the Schouten tensor on $(M,g)$:\n$$\\overline{S}_{ab}:=\\overline{R}_{ab}-\\frac{\\overline{R}}{4} h_{ab} \\ \\stackrel{\\scri}{=}\\  S_{\\mu\\nu}e^\\mu{}_a e^\\nu{}_b$$\n\\item and the corresponding Cotton-York tensor $C_{ab}$, which coincides with the {\\em magnetic} part of the re-scaled Weyl tensor \\cite{Kroon,Ashtekar2014,Friedrich1986a}\n\\begin{equation}\\label{eq1}\n\\left(\\frac{\\Lambda}{3}\\right)^{1\/2} C_{ab}:= \\epsilon_a{}^{cd} \\overline\\nabla_c \\overline{S}_{db} \\ \\stackrel{\\scri}{=}\\  \\stackrel{*}{d}_{\\mu\\nu\\rho}{}^\\sigma \\bar{n}_\\sigma e^\\mu{}_a \\bar{n}^\\nu e^\\rho{}_b\n\\end{equation}\nwhere $\\bar{n}^\\mu $ is the normalized version of $n^\\mu$.\n\\end{itemize}\n\n\nOnly the trace-free part of $\\overline{S}_{ab}$ enters into the previous equation.\nGiven the foliation by spacelike hypersurfaces $\\Omega =$  const.\\ around $\\mathscr{J}$ determined by $\\bm{n}=d\\Omega$, the time derivative of its shear $\\sigma_{\\mu\\nu}$ coincides, on $\\mathscr{J}$, with the mentioned trace-free part:\n$$\n\\dot \\sigma_{ab}:\\ \\stackrel{\\scri}{=}\\   e^\\mu{}_a e^\\nu{}_b \\pounds_{\\bar{n}} \\sigma_{\\mu\\nu}=\\overline{S}_{ab}-\\frac{1}{12} \\overline{R} h_{ab}  .\n$$\nThe completion of the physical spacetime also provides the {\\em electric} part of the re-scaled Weyl tensor\\footnote{The standard notation for this electric part is $D_{ab}$  \\cite{Fernandez-Alvarez_Senovilla20b,Fernandez-Alvarez_Senovilla-dS,Friedrich1986a,Friedrich1986b,Friedrich2002}, but I will use $\\mathcal F_{ab}$ herein to avoid notational conflicts.}\n$$\n\\mathcal F_{ab} :\\ \\stackrel{\\scri}{=}\\  d_{\\mu\\nu\\rho}{}^\\sigma \\bar{n}_\\sigma e^\\mu{}_a \\bar{n}^\\nu e^\\rho{}_b\n$$\nbut this is not intrinsic to $(\\mathscr{J},h_{ab})$. $\\mathcal F_{ab}$ can be seen to coincide with the second time-derivative of the shear: \n$$\n\\ddot \\sigma_{ab} \\ \\stackrel{\\scri}{=}\\  2\\left(\\frac{\\Lambda}{3}\\right)^{1\/2} \\mathcal F_{ab}.\n$$\nIn general, $C_{ab}$ and $\\mathcal F_{ab}$ are {\\em trace-free} tensors with gauge behaviour under \\eqref{gauge}\n$$\n\\{C_{ab},\\mathcal F_{ab}\\} \\rightarrow \\omega^{-1} \\{C_{ab}, \\mathcal F_{ab} \\}.\n$$\nFrom the Bianchi identities, $C_{ab}$ is also divergence free, that is to say, it is a TT-tensor. For appropriate decaying condition of the physical energy-momentum tensor, $\\mathcal F_{ab}$ is also a TT-tensor. Under these decaying conditions the Bianchi identities reduce to\n\\begin{equation}\n\\overline \\nabla_a C^{ab}=0, \\hspace{4mm} \\overline \\nabla_a \\mathcal F^{ab}=0, \\hspace{4mm}\n\\overline \\nabla_{[c} C_{a]b} =\\frac{1}{2} \\epsilon_{cad}\\dot{\\mathcal F}^d{}_b, \\hspace{4mm}\n\\overline \\nabla_{[c} \\mathcal F_{a]b} =\\frac{1}{2} \\epsilon_{cad}\\dot{C}^d{}_b .\\label{Bianchi}\n\\end{equation}\nNote that the first two are consequences of the second pair by using the traceless property of $\\mathcal F_{ab}$ and $C_{ab}$. In the above, the dot means derivative along the unit normal to $\\mathscr{J}$\n\nThere are several fundamental results demonstrating that the geometry of the physical spacetime is fully encoded, as initial conditions of a well-posed initial value problem, on $(\\mathscr{J},h_{ab})$ {\\em together} with a symmetric and trace-free tensor field ($\\mathcal F_{ab}$). This can be seen as an initial or final value problem. Specifically, I refer to\n\\begin{itemize}\n\\item A classical result by Starobinsky \\cite{Starobinsky1983}. An expansion in powers of $e^{-\\left(\\frac{\\Lambda}{3}\\right)^{1\/2} t}$ as $t\\rightarrow \\infty$ shows that the first term is a spatial 3-dimensional metric $h_{ab}$, then the next two terms are determined by the curvature of $h_{ab}$ and a traceless symmetric tensor $\\mathcal F_{ab}$ whose divergence depends on the matter contents --and is divergence free in vacuum--, and these three terms determine the whole expansion.\n\\item A more mathematical (and more general) similar result due to Fefferman and Graham \\cite{Fefferman2002,Fefferman2005} showing that given any conformal geometry $(\\Sigma,h_{ab})$ the addition of a TT-tensor $\\mathcal F_{ab}$ provides, via a well determined expansion, a 4-dimensional spacetime whose conformal completion has $(\\mathscr{J} ,h_{ab})=(\\Sigma,h_{ab})$.\n\\item The results by Friedrich \\cite{Friedrich1986a,Friedrich1986b,Friedrich2002,Kroon} proving that the $\\Lambda$-vacuum Einstein field equations are equivalent to a set of symmetric hyperbolic partial differential equations on the unphysical spacetime and the solutions are fully determined by initial\/final data consisting of a 3-dimensional Riemannian manifold with the metric conformal class plus a TT-tensor. The Riemannian manifold turns out to be (a representative of the conformal class of) $(\\mathscr{J},h_{ab})$ while the TT-tensor coincides with the electric part $\\mathcal F_{ab}$ of the re-scaled Weyl tensor.\n\\end{itemize}\n In summary, we now know that {\\em any property of the physical spacetime is fully encoded in the triplet $(\\mathscr{J},h_{ab},\\mathcal F_{ab})$}. Consequently, the existence, or absence, of gravitational radiation {\\em is also fully encoded} in $(\\mathscr{J},h_{ab},\\mathcal F_{ab})$. Our criteria fulfil this completely, because the\n asymptotic super-momentum can be split into the parts tangent and normal to $\\mathscr{J}$\n$$\np^\\alpha := -\\mathcal{D}^\\alpha{}_{\\beta\\mu\\nu} n^\\beta n^\\mu n^\\nu \\ \\stackrel{\\scri}{=}\\  \\overline W \\bar{n}^\\alpha +\\bar p^a e^\\alpha{}_a\n$$ \nand  \\eqref{divPi}, that now requieres appropriate matter decaying conditions, gives\n\\begin{equation}\\label{continuity}\n\\nabla_\\mu p^\\mu \\ \\stackrel{\\scri}{=}\\  0 \\hspace{2mm} \\Longrightarrow \\hspace{2mm} \\dot{\\overline W} +\\overline\\nabla_a \\bar p^a=0.\n\\end{equation}\n$\\bar p^a$ is called the {\\em asymptotic super-Poynting} vector.\nObserve that criterion \\ref{crit1} (respectively criterion \\ref{crit2}) states that there is no gravitational radiation crossing a cut $\\mathcal{S}\\subset \\mathscr{J}$ (resp.\\ $\\Delta$) if $\\bar p^a$ vanishes on $\\mathcal{S}$ (resp.\\ $\\Delta$). From well-known old results \\cite{Bel1962,Maartens1998,Alfonso2008}\n\\begin{equation}\n\\bar p_a=2\\left(\\frac{\\Lambda}{3}\\right)^{(3\/2)} \\epsilon_{abc} C^{bd} \\mathcal F^c{}_d \\label{superp}\n\\end{equation}\nso that there is no gravitational radiation crossing $\\mathscr{J}$ if and only if $C^a{}_b$ and $\\mathcal F^a{}_b$ conmute:\n$$\n \\bar p_a =0 \\hspace{4mm} \\Longleftrightarrow \\hspace{4mm}  \\epsilon_{abc} C^{bd} \\mathcal F^c{}_d =0.\n$$\nThis condition is truly encoded on $(\\mathscr{J},h_{ab},\\mathcal F_{ab})$ and it takes all its elements into account, as required.\n\\begin{Remark}[Radiation encoded at $\\mathscr{J}$]\nFrom the perspective of the initial, or final, value problem, given a particular conformal geometry representing $(\\mathscr{J},h_{ab})$, one only needs to add a TT tensor $\\mathcal F_{ab}$ such that it does (not) conmute with the Cotton-York tensor $C_{ab}$ if the spacetime is going to (not) be free of gravitational radiation. Observe that there is a special possibility when $(\\mathscr{J},h_{ab})$ is conformally flat, so that $C_{ab}=0$, in which case no matter which TT-tensor field $\\mathcal F_{ab}$ one adds the resulting spacetime will not contain gravitational radiation.\n\\end{Remark}\n\nLet now $\\Delta\\subset \\mathscr{J}$ be an open region of $\\mathscr{J}$ bounded by two disjoint cuts $\\mathcal{S}_1$ and $\\mathcal{S}_2$, as shown in figure \\ref{fig:2cuts}. From \\eqref{continuity} one easily gets \n\\begin{equation}\\label{balance}\n\\int_\\Delta \\dot{\\overline{W}}\\bm{\\epsilon} = \\int_{\\mathcal{S}_1} m^a_1 \\bar p_a \\bm{\\epsilon}_2 -\\int_{\\mathcal{S}_2} m^a_2 \\bar p_a \\bm{\\epsilon}_2\n\\end{equation}\nwhere $m^a_1$ and $m^a_2$ are the unit normals to $\\mathcal{S}_1$ and $\\mathcal{S}_2$ within $\\mathscr{J}$, respectively. We will later see that $\\bar{p}_a m^a$ has a sign in relevant cases.\n\\begin{figure}[!ht]\n\\includegraphics[width=12cm]{ScridS1.pdf}\n\\caption{Schematic representation of a region $\\Delta$ in $\\mathscr{J}^+$ when $\\Lambda >0$ bounded by two disjoint cuts $\\mathcal{S}_1$ and $\\mathcal{S}_2$. The vector fields $m_1^a$ and $m_2^a$ are the unit normal vectors to the cuts $\\mathcal{S}_1$ and $\\mathcal{S}_2$ within $\\mathscr{J}^+$, respectively. \\label{fig:2cuts}}\n\\end{figure}\n\n\n\n\n\\subsection{Geometry of cuts on $\\mathscr{J}$}\nOur criteria for absence of radiation are primarily associated to cuts, and thus it is convenient to develop some formalism for the geometry of these cross-sections of $\\mathscr{J}$ in relation with the physical quantities relevant for the criteria. Let $\\mathcal{S}$ be any cut on $\\mathscr{J}$ and let $m^b$ denote the unit vector field normal to ${\\cal S}$ within $\\mathscr{J}$ and, as before, $\\{E_A^a\\}$ a basis of tangent vector fields on ${\\cal S}$. The first fundamental form of the cut is denoted by\n$$\nq_{AB}= h_{ab} E^a_A E^b_B\n$$\nand \\eqref{gaugeh} still holds now. Define for every symmetric tensor field $\\bar{t}^{ab}$ on $\\mathscr{J}$ its corresponding parts in an orthogonal decomposition relative to $\\mathcal{S}$ and thereby introduce the notation for all such tensor decompositions:\n$$\n\\bar{t}^{ab} = t^{AB} E^a_A E^b_B +t^A E^a_A m^b + t^B E^b_B m^a + t m^a m^b\n$$ \nand then raise and lower indices of the objects on ${\\cal S}$ with the inherited metric $q_{AB}$. The Levi-Civita connection of $({\\cal S},q_{AB})$ is denoted by $\\gamma^A_{BC}$ and one then has\n$$\nE^a_A \\overline \\nabla_a E^b_B = \\gamma^C_{AB} E^b_C -\\varkappa_{AB} m^b, \n$$\nwhere $\\varkappa_{AB}$ is the 2nd fundamental form of ${\\cal S}$ in $\\mathscr{J}$ ---and also the unique non-zero 2nd fundamental form of ${\\cal S}$ in the unphysical spacetime. One can decompose this object as usual\n$$\n\\varkappa_{AB}:=\\Sigma_{AB} +\\frac{1}{2} \\varkappa q_{AB}, \\hspace{7mm} \\varkappa := q^{AB}\\varkappa_{AB} , \\hspace{7mm} q^{AB}\\Sigma_{AB}=0 \n$$\nwhere $\\Sigma_{AB}$ is the shear of ${\\cal S}$ in $\\mathscr{J}$ ---or the unique non-zero shear of ${\\cal S}$ in the unphysical spacetime. \nFurthermore, for any symmetric $t_{ab}$\n$$\nE^a_A E^b_B E^c_C  \\overline \\nabla_c \\bar{t}_{ab} = D_C t_{AB} +t_A \\varkappa_{BC} +t_B \\varkappa_{AC} .\n$$\n\nUnder the allowed gauge transformations \\eqref{gaugeh} the above objects and those relative to $\\overline{S}_{ab}$ transform as follows ($\\omega_A := D_A\\omega$, $\\omega_m := m^b \\overline \\nabla_b \\omega$)):\n\\begin{eqnarray} \n\\tilde{m}_a &=&\\omega m_a,\\\\\n\\tilde{\\gamma}^C_{AB} &=&\\gamma^C_{AB} +\\frac{1}{\\omega}\\left(\\delta^C_A\\omega_B +\\delta^C_B \\omega_A-\\omega^C q_{AB} \\right),\\label{gammagauge}\\\\\n\\tilde{\\varkappa}_{AB}&=& \\omega \\varkappa_{AB} + \\omega_m q_{AB},\\label{kappagauge}\\\\\n\\tilde{\\Sigma}_{AB} &=& \\omega \\Sigma_{AB},\\label{Sigmagauge}\\\\\n\\tilde{\\varkappa} &=&\\frac{1}{\\omega} \\varkappa + \\frac{2}{\\omega^2} \\omega_m,\\label{trkappagauge}\\\\\n\\tilde{S}_{AB}&=& S_{AB}-\\frac{1}{\\omega}D_A\\omega_B +\\frac{2}{\\omega^2} \\omega_A\\omega_B -\\frac{1}{2\\omega^2} \\omega^D\\omega_D q_{AB}-\\frac{\\omega_m}{\\omega} \\left(\\varkappa_{AB}+\\frac{1}{2\\omega} \\omega_m q_{AB}\\right),\\label{gaugeS'}\\\\\n\\tilde{S}_A &=& \\frac{1}{\\omega} \\left(S_A -\\frac{1}{\\omega} D_A\\omega_m+\\frac{1}{\\omega}\\varkappa_{AB}\\omega^B +\\frac{2}{\\omega^2} \\omega_m \\omega_A \\right),\\\\\n\\tilde{S} &=& \\frac{1}{\\omega^2} \\left(S- \\frac{1}{\\omega} m^a m^b \\overline \\nabla_a \\overline \\nabla_b \\omega +\\frac{2}{\\omega^2} \\omega_m^2 -\\frac{1}{2\\omega^2} \\overline \\nabla_c\\omega \\overline \\nabla^c\\omega\\right).\n\\end{eqnarray}\n\nThe projections of the gauge-invariant equation (\\ref{eq1}) onto to cut ${\\cal S}$ lead to the following relations\n\\begin{eqnarray}\nD_{[C}S_{A]B}+\\varkappa_{B[C} S_{A]} =\\frac{1}{2}\\left(\\frac{\\Lambda}{3}\\right)^{1\/2} \\epsilon_{CA} C_B, \\label{eq11}\\\\\nE^a_A E^b_B m^c  \\overline \\nabla_c S_{ab} -D_A S_B+\\varkappa_A^D S_{BD}-S \\varkappa_{AB}= \\left(\\frac{\\Lambda}{3}\\right)^{1\/2}\\epsilon_A{}^D C_{DB}  \\label{eq12}\n\\end{eqnarray}\nwhere $\\epsilon_{AB}$ is the canonical volume element 2-form on $({\\cal S},q_{AB})$. Relation (\\ref{eq11}) is gauge invariant, while (\\ref{eq12}) is gauge homogeneous with a factor $1\/\\omega$. As the righthand side of (\\ref{eq11}) is easily seen to be gauge invariant (because $\\tilde{C}_{ab}=(1\/\\omega) C_{ab}$), it follows that $D_{[C}S_{A]B}+\\varkappa_{B[C} S_{A]} $ is also gauge invariant. The skew-symmetric part of (\\ref{eq12}) reads\n$$\nD_{[C} S_{A]} -\\varkappa^D_{[C}S_{A]D}=\\frac{1}{2} \\left(\\frac{\\Lambda}{3}\\right)^{1\/2} \\epsilon_{CA} C\n$$\n(notice that $C:=C_{ab}m^am^b =-C^E_E$, as follows from $C^b_b=0$),  while the symmetric part reads\n$$\nE^a_A E^b_B m^c  \\overline \\nabla_c S_{ab} -D_{(A} S_{B)}+\\varkappa_{(A}^D S_{B)D}-S \\varkappa_{AB}= \\left(\\frac{\\Lambda}{3}\\right)^{1\/2}\\epsilon_{(A}{}^D C_{B)D}=\\left(\\frac{\\Lambda}{3}\\right)^{1\/2}\\epsilon_{A}{}^D \\hat{C}_{BD}\n$$\nwhere we use a hat over the matrices to denote its trace-free part:\n\\begin{equation}\n\\hat{C}_{AB} := C_{AB} -\\frac{1}{2} q_{AB} C^E{}_E, \\hspace{1cm} \\epsilon_{DA}\\hat{C}_{B}{}^D=\\epsilon_{DB}\\hat{C}_{A}{}^D = \\epsilon_{D(A} C_{B)}{}^D \n\\end{equation}\nand similarly for $\\hat{\\mathcal F}_{AB}$. Using the 2-dimensional identity\n$$\n\\varkappa_{(A}^DS_{B)D} -\\frac{1}{2} \\varkappa S_{AB} -\\frac{1}{2} S^D_D \\varkappa_{AB}+\\frac{1}{2}\\left(\\varkappa S^D_D-\\varkappa^{CD} S_{CD} \\right)q_{AB}=0\n$$\nthe previous symmetric part can be recast into the form\n\\begin{eqnarray}\nE^a_A E^b_B m^c  \\overline \\nabla_c S_{ab} -D_{(A} S_{B)}+\\frac{1}{2} \\varkappa S_{AB}+\\left(\\frac{1}{2} S^D_D-S\\right) \\varkappa_{AB}\\nonumber \\\\\n-\\frac{1}{2}\\left(\\varkappa S^D_D-\\varkappa^{CD} S_{CD} \\right)q_{AB}= \\left(\\frac{\\Lambda}{3}\\right)^{1\/2}\\epsilon_{A}{}^D \\hat{C}_{BD} .\n\\end{eqnarray}\n An equivalent form of (\\ref{eq11}) is\n$$\nD_B S^B_A -D_A S^D_D +\\varkappa S_A -\\varkappa_A^B S_B =\\left(\\frac{\\Lambda}{3}\\right)^{1\/2} C^D \\epsilon_{DA}.\\label{eq11'}\n$$\n\n\n\nOne can rewrite (\\ref{eq11}) in a form without $S_A$. This can be achieved by using the Gauss and Codazzi relations for ${\\cal S}$, which can be checked to read\n\\begin{eqnarray}\nS_{A[C}q_{D]B}+q_{A[C}S_{D]B}&=&Kq_{A[C}q_{D]B}-\\varkappa_{A[C}\\varkappa_{D]B}, \\label{gauss0}\\\\\nD_{[C}\\varkappa_{A]B}&=&q_{B[C}S_{A]} \\label{cod}\n\\end{eqnarray}\nRelation (\\ref{cod}) is equivalent to its trace\n\\begin{equation}\nS_A=D_E\\varkappa^E_A-D_A\\varkappa . \\label{cod'}\n\\end{equation}\nThe Gauss equation (\\ref{gauss0}) is also fully equivalent to its trace and also to its double trace\n\\begin{eqnarray}\nS^D_D q_{AB} &=&K q_{AB} +\\varkappa_A^D\\varkappa_{DB} -\\varkappa \\varkappa_{AB},\\label{gauss}\\\\\nS^D_D &=&K +\\frac{1}{2}\\left(\\varkappa^{AB}\\varkappa_{AB}- \\varkappa^2 \\right)=K-\\det(\\varkappa^E_F) \\label{gauss'},\n\\end{eqnarray}\nwhich can be easily checked by using a typical 2-dimensional identity, and for the last part also using the Caley-Hamilton theorem \n$$\n\\varkappa_A^D\\varkappa_{DB} -\\varkappa \\varkappa_{AB}+q_{AB} \\det(\\varkappa^E_F)=0.\n$$\nAnother simpler version of this relation is simply\n\\begin{equation}\n\\Sigma_A{}^D\\Sigma_{DB} =\\frac{1}{2} \\Sigma_{DE} \\Sigma^{DE} q_{AB} . \\label{Sigmasquare}\n\\end{equation}\nNotice that\n$$\n\\varkappa_{AB} \\varkappa^{AB} = \\Sigma_{AB} \\Sigma^{AB} +\\frac{1}{2} \\varkappa^2 .\n$$\nUsing \\eqref{cod'}, equation (\\ref{eq11}) can be rewritten as\n\\begin{equation}\nD_{[C}S_{A]B}+\\varkappa_{B[C}\\left(D^E\\varkappa_{A]E} -D_{A]}\\varkappa \\right)=\\frac{1}{2}\\left(\\frac{\\Lambda}{3}\\right)^{1\/2} \\epsilon_{CA} C_B,\n\\end{equation}\nwhose lefthand side is (must be!) gauge invariant, in accordance with (\\ref{DSgauge}). This is still equivalent, after some calculation, to\n\\begin{eqnarray}\nD_C \\left(S^C{}_A -\\frac{1}{2} \\Sigma^{CE}\\Sigma_{EA}+ \\frac{\\varkappa}{2}\\Sigma^C{}_A +\\frac{\\varkappa^2}{8} \\delta^C_A -K\\delta^C_A \\right)=\\nonumber \\\\\n\\frac{3}{2}D_B(\\Sigma^{BE}\\Sigma_{EA})-\\Sigma^{CE}D_E \\Sigma_{CA} +\\left(\\frac{\\Lambda}{3}\\right)^{1\/2} \\epsilon_{EA}C^E\\label{eq11''}.\n\\end{eqnarray}\nObserve that the righthand side in this expression is gauge homogeneous with a factor $1\/\\omega^2$.\n\nProjecting the Bianchi equations (\\ref{Bianchi}) to the cut ${\\cal S}$ as before one derives\n\\begin{eqnarray}\nD_{[C}C_{A]B} +\\varkappa_{B[C} C_{A]} =\\frac{1}{2} \\epsilon_{CA} \\dot\\mathcal F_B,\\\\\nE^a_A E^b_B m^c\\overline \\nabla_c C_{ab} -D_A C_B +\\varkappa_A{}^D C_{BD} +C^E{}_E \\varkappa_{AB} =\\epsilon_{AD} \\dot\\mathcal F^D{}_B,\\\\\nD_{[C}\\mathcal F_{A]B} +\\varkappa_{B[C} \\mathcal F_{A]} =\\frac{1}{2} \\epsilon_{CA} \\dot{C}_B,\\\\\nE^a_A E^b_B m^c\\overline \\nabla_c \\mathcal F_{ab} -D_A \\mathcal F_B +\\varkappa_A{}^D \\mathcal F_{BD} +\\mathcal F^E{}_E \\varkappa_{AB} =\\epsilon_{AD} \\dot{C}^D{}_B\n\\end{eqnarray}\n\nAnalogously to Lemma \\ref{lem:dt2} one can prove the following result for cuts on $\\mathscr{J}$ when $\\Lambda >0$\n\\begin{Lemma}\\label{lem:dt1}\nLet $p_{AB}=p_{(AB)}$ be any symmetric tensor field on $({\\cal S},q_{AB})$ whose gauge behaviour under residual gauge transformations (\\ref{gaugeh}) is\n$$\n\\tilde{p}_{AB}= p_{AB}-\\frac{1}{\\omega}D_A\\omega_B +\\frac{2}{\\omega^2} \\omega_A\\omega_B -\\frac{1}{2\\omega^2} \\omega^D\\omega_D q_{AB}-\\frac{\\omega_m}{\\omega} \\left(\\varkappa_{AB}+\\frac{1}{2\\omega} \\omega_m q_{AB}\\right)\n$$\nThen, \n\\begin{eqnarray*}\n\\tilde{D}_{[C}\\tilde{p}_{A]B}+\\tilde{\\varkappa}_{B[C}\\left(\\tilde{D}^E\\tilde{\\varkappa}_{A]E}-\\tilde{D}_{A]}\\tilde{\\varkappa} \\right)&=&D_{[C}p_{A]B}+\\varkappa_{B[C}\\left(D^E\\varkappa_{A]E}-D_{A]}\\varkappa \\right) \\\\\n&+&\\frac{1}{\\omega} \\left(p_{B[C}-S_{B[C}\\right)\\omega_{A]}+\\frac{1}{\\omega}q_{B[C} \\left(p^D_{A]}-S^D_{A]} \\right)\\omega_D \n\\end{eqnarray*}\n\\end{Lemma}\nThe proof is again by direct calculation. As a corollary we immediately have\n\\begin{equation}\n\\tilde{D}_{[C}\\tilde{S}_{A]B}+\\tilde{\\varkappa}_{B[C}\\left(\\tilde{D}^E\\tilde{\\varkappa}_{A]E}-\\tilde{D}_{A]}\\tilde{\\varkappa} \\right)=D_{[C}S_{A]B}+\\varkappa_{B[C}\\left(D^E\\varkappa_{A]E}-D_{A]}\\varkappa \\right) \\label{DSgauge}\n\\end{equation}\n\n\n\n\\subsubsection{The super-Poynting vector and asymptotic radiant super-momenta on cuts of $\\mathscr{J}$}\\label{subsec:superp}\nLet me denote by\n$$\n\\vec k_\\pm := \\vec \\bar{n} \\pm \\vec m, \\hspace{1cm} k^\\mu_+ k_{-\\mu}=-2\n$$\nthe two future null normals to the cut $\\mathcal{S}$ (see figure \\ref{fig:kpm}) and, given that $\\Sigma_{AB}$ is the only non-zero shear of $\\mathcal{S}$ in $\\mathscr{J}$, the corresponding two null shears are simply $\\pm\\Sigma_{AB}$.\n\\begin{figure}[!ht]\n\\includegraphics[width=12cm]{ScridS.pdf}\n\\caption{Schematic representation of the two null normals $k^\\mu_\\pm =\\bar{n}^\\mu \\pm m^\\mu$ to the cut $\\mathcal{S}$ at a given point of the cut. \\label{fig:kpm}}\n\\end{figure}\nWe introduce, for each cut $\\mathcal{S}$, the two {\\em asymptotic radiant super-momenta} as\n\\begin{equation}\\label{Qpm}\nQ_\\pm^\\alpha := -D^\\alpha{}_{\\mu\\nu\\rho} k_\\pm^\\mu k_\\pm^\\nu k_\\pm^\\rho, \n\\end{equation}\nand they are always, by construction, null and future. It is convenient to have formulae for $\\bar p_a$ and also for $Q_\\pm^\\alpha$ in terms of $C_{ab}$ and $\\mathcal F_{ab}$. To that end, we write the asymptotic  radiant super-momenta in the given bases\n\\begin{equation}\\label{Qs}\nQ_\\pm^\\alpha = \\frac{1}{2} \\mathcal{W}_\\pm k_\\mp^\\alpha +\\frac{1}{2} \\mathcal{Z}_\\pm k^\\alpha_\\pm +Q^A_\\pm E^\\alpha_A\n\\end{equation}\nor equivalently\n\\begin{equation}\\label{Qs1}\nQ_\\pm^\\alpha = \\frac{1}{2}(\\mathcal{W}_\\pm +\\mathcal{Z}_\\pm) \\bar{n}^\\alpha \\pm \\frac{1}{2} (\\mathcal{Z}_\\pm -\\mathcal{W}_\\pm) m^\\alpha \n+Q^A_\\pm E^\\alpha_A\n\\end{equation}\nwhere by direct (long) calculation one finds\n\\begin{eqnarray}\n\\mathcal{W}_\\pm &:=& -k_\\alpha^\\pm Q_\\pm^\\alpha =8(\\hat{\\mathcal F}_{AB}\\mp \\epsilon_{DA}\\hat{C}_{B}{}^D)\n(\\hat{\\mathcal F}^{AB}\\mp \\epsilon^{CA}\\hat{C}^{B}{}_C)\\geq 0, \\label{Ws}\\\\\n\\mathcal{Z}_\\pm &:= & -k_\\alpha^\\mp Q_\\pm^\\alpha = 4(\\mathcal F_A\\pm\\epsilon_{AB}C^B)(\\mathcal F^A\\pm\\epsilon^{AD}C_D)\\geq 0,\n\\label{Zs}\\\\\nQ_\\pm^A &:=& W^A_\\alpha Q_\\pm^\\alpha =\\pm 8 (\\hat{\\mathcal F}_{AB}\\mp \\epsilon_{D(A}\\hat{C}_{B)}{}^D)(\\mathcal F^B\\pm\\epsilon^{BE}C_E)\\label{QAs}.\n\\end{eqnarray}\nSome useful formulas are\n\\begin{eqnarray}\n\\mathcal{Z}_+ -\\mathcal{Z}_- &=& 16\\epsilon_{AB}\\mathcal F^AC^B,\\hspace{1cm} \\mathcal{Z}_+ +\\mathcal{Z}_- = 8(\\mathcal F_A\\mathcal F^A+C_A C^A),\\label{Zs1}\\\\\n\\mathcal{W}_+ -\\mathcal{W}_- &=&\n32\\epsilon_{AB}\\hat{\\mathcal F}^{AD}\\hat{C}^B{}_D, \\hspace{3mm} \\mathcal{W}_+ +\\mathcal{W}_- =16(\\hat{\\mathcal F}_{AB}\\hat{\\mathcal F}^{AB}+\\hat{C}_{AB}\\hat{C}^{AB}),\\label{Ws1}\\\\\nQ_+^A -Q_-^A &=& 16 (\\hat{C}^{AB} C_B + \\hat{\\mathcal F}^{AB}\\mathcal F_B), \\, \\, \nQ_+^A +Q_-^A = 16\\epsilon_{AB} (\\hat{C}^{BD} \\mathcal F_D - \\hat{\\mathcal F}^{BD}C_D).\n\\end{eqnarray}\nThen, the expressions of the components of $\\bar p_a$ can be easily found. Orthogonally decomposing the super-Poynting on $\\mathcal{S}$ as\n$$\n\\left(\\frac{3}{\\Lambda}\\right)^{3\/2}\\bar p^a = p_m m^a + p^A E_A^a \n$$\nanother straightforward calculation leads to\n\\begin{equation}\\label{pm}\np_m =\\frac{1}{16} \\left(\\mathcal{Z}_+-\\mathcal{Z}_--\\mathcal{W}_++\\mathcal{W}_-\\right)+3\\epsilon_{AB}C^A\\mathcal F^B=\\frac{1}{16} \\left(\\mathcal{W}_--\\mathcal{W}_++2\\mathcal{Z}_--2\\mathcal{Z}_+ \\right)\n\\end{equation}\n(where the first in (\\ref{Zs1}) has been used) and to\n\\begin{eqnarray}\np_A &=& 2\\epsilon_{AB} \\left(C^{BD}\\mathcal F_D -\\mathcal F^{BD} C_D +C^E{}_E \\mathcal F^B-\\mathcal F^E{}_E C^B \\right)\\nonumber\\\\\n&=& 2\\epsilon_{AB} \\left(\\hat{C}^{BD}\\mathcal F_D -\\hat{\\mathcal F}^{BD} C_D +\\frac{3}{2}C^E{}_E \\mathcal F^B-\\frac{3}{2}\\mathcal F^E{}_E C^B \\right) \\label{pA}\\\\\n&=& \\frac{1}{8} \\left(Q^+_A +Q^-_A\\right)+3\\epsilon_{AB}\\left(C^E{}_E \\mathcal F^B-\\mathcal F^E{}_E C^B\\right).\\nonumber\n\\end{eqnarray}\nFor completeness, we note in passing that\n\\begin{equation}\\label{sumQs}\nQ_+^\\alpha+Q_-^\\alpha= \\frac{1}{2} (\\mathcal{W}_++\\mathcal{W}_-+\\mathcal{Z}_+ +\\mathcal{Z}_-) n^\\alpha +\\frac{1}{2}(\\mathcal{Z}_+ -\\mathcal{Z}_- -\\mathcal{W}_++\\mathcal{W}_-)m^\\alpha+\n\\left(Q_+^A+Q_-^A \\right)E^\\alpha_A .\n\\end{equation}\n\n\n\n\n\\section{Are there any News for cuts (and for $\\mathscr{J}$)?}\\label{sec:news}\nThere are some objects in the literature that are called ``news'' tensor in the case with $\\Lambda >0$ based on analogies with the asymptotically flat case. None of them seem to have led to properties similar to that of the News tensor when $\\Lambda=0$, and one can raise some doubts about the existence of news in the general case with $\\Lambda >0$. Nevertheless, in this section I describe a general method to search for such `News', and also a tensor field is uncovered that will certainly be part of any news tensor, if this exists. \n\nRecall first of all that, when $\\Lambda =0$, $N_{ab}$ is the pull-backed Schouten tensor gauge corrected, and that one can unambiguously define the news tensor associated to any cut $\\mathcal{S}$ by projecting into the cut. An interesting idea, given the previous considerations, is to try to assign to any possible cut ${\\cal S}\\subset \\mathscr{J}$ ---and especially when the cut is topologically $\\mathbb{S}^2$--- a gauge invariant tensor field contained {\\em partly} in the pullback to ${\\cal S}$ of $\\overline{S}_{ab}$. \n\nWhy partly? Well, there are crucial differences now with respect to the case with $\\Lambda =0$, as now the Schouten tensor $\\overline{S}_{ab}$ is fully intrinsic to $(\\mathscr{J},h_{ab})$, in contrast with the asymptotically flat case where it arises as the curvature of the connection, and this is inherited from the ambient manifold but not intrinsic to the null $(\\mathscr{J},h_{ab})$. In this sense, note that \\eqref{eq1} is fully intrinsic to the spacelike $(\\mathscr{J},h_{ab})$ showing in particular that $\\overline{S}_{ab}$ is {\\em determined exclusively by $C_{ab}$} and thus {\\em it cannot contain by itself any gauge-invariant part that describes the existence of radiation, which as explained before, must be encoded in the triplet $(\\mathscr{J},h_{ab},\\mathcal F_{ab})$.}  A key equation now is the identity\n$$\n\\frac{1}{2} \\frac{3}{\\Lambda} \\bar p_c = \\overline\\nabla_c (\\mathcal F^{ab} \\overline{S}_{ab} ) -\\overline\\nabla_a (\\mathcal F^{ab}\\overline{S}_{bc}) -\\overline{S}_{ab} \\overline\\nabla_c \\mathcal F^{ab}\n$$\nwhich graphically shows that the asymptotic super-Poynting depends on the interplay between $\\overline{S}_{ab}$ and $\\mathcal F_{ab}$. In this formula, every term on the righthand side has a complicated gauge behaviour yet their combination equals $\\bar p_c$, whose gauge behaviour is simply $\\bar p_c \\rightarrow \\omega^{-5} \\bar p_c$. Given that the vanishing of $\\bar p_c$ characterizes the absence of radiation, the existence of any `source' of type News for $\\bar p_c$ requires a splitting of the righthand terms in gauge well-behaved parts plus a remainder that must be uniquely determined. Such a ``News tensor'' should then satisfy appropriate differential equations.\n\nDespite these difficulties, $\\overline{S}_{ab}$ will probably entail the part of the news (if this exists) not related to the TT-tensor $\\mathcal F_{ab}$. This is the part that we were able to identify \\cite{Fernandez-Alvarez_Senovilla-dS}, as I discuss in the following.\n\n\nLet us generalize Corollary \\ref{coroRho} by finding the general form of the tensor fields defined by Corollary \\ref{coroDt} but with a general, non-vanishing, $D_{[C}t_{A]B}$.\n\\begin{Proposition}\nLet $\\mathcal{S}\\subset \\mathscr{J}$ by a cut on $\\mathscr{J}$. If the equation \n\\begin{equation}\\label{DU}\nD_{[C}W_{A]B}= X_{CAB} \n\\end{equation}\nfor a given \\underline{{\\em gauge invariant}} tensor field $X_{CAB}=X_{[CA]B}$ has a solution for $W_{AB}=W_{(AB)}$ whose gauge behaviour is (\\ref{normalgauge}) with $a=1$, then this solution is given by\n\\begin{equation}\\label{U}\nW_{AB} = S_{AB} -\\frac{1}{2} \\Sigma_{A}{}^D \\Sigma_{BD} +\\frac{\\varkappa}{2}\\Sigma_{AB} +\\frac{\\varkappa^2}{8} q_{AB}+M_{AB}\n\\end{equation}\nwhere $M_{AB}$ is a trace-free, gauge invariant and symmetric tensor field solution of\n\\begin{equation}\nD_{[C}M_{A]B}= X_{CAB} -\\frac{1}{2}\\left(\\frac{\\Lambda}{3}\\right)^{1\/2} \\epsilon_{CA} C_B +D_{[C} \\left(\\Sigma_{A]E}\\Sigma_{B}{}^E \\right)-\\frac{1}{2} D_B \\Sigma_{[C}{}^E \\Sigma_{A]E}.\\label{DM}\n\\end{equation}\n\\end{Proposition}\n{\\bf Remark}: The righthand side of (\\ref{DM}) is gauge invariant. If the cut has $\\mathbb{S}^2$ topology the solution is unique. More generally, $M_{AB}$ (and a fortiori $W_{AB}$) is unique whenever $(\\mathcal{S},q_{AB})$ has a conformal Killing vector with a fixed point \\cite{Fernandez-Alvarez_Senovilla-dS}.\n\\begin{proof}\nBy using (\\ref{gammagauge}), (\\ref{Sigmagauge}), (\\ref{trkappagauge}) and (\\ref{gaugeS'}) it is a matter of checking that the tensor \\eqref{U} has the gauge behaviour (\\ref{normalgauge}) with $a=1$, provided $M_{AB}$ is gauge invariant. Its trace, on using (\\ref{gauss'}) and (\\ref{Sigmasquare}) is \n\\begin{equation}\\label{trW}\nW^E{}_E =K.\n\\end{equation}\n Therefore, Corollary \\ref{coroDt} applies and $D_{[C}W_{A]B}$ is gauge invariant.\nFor the second part, using (\\ref{eq11''}) and manipulating a little one arrives at\n$$\nD_{[C}W_{A]B}= \\frac{1}{2}\\left(\\frac{\\Lambda}{3}\\right)^{1\/2} \\epsilon_{CA} C_B -D_{[C} \\left(\\Sigma_{A]E}\\Sigma_{B}{}^E \\right)+\\frac{1}{2} D_B \\Sigma_{[C}{}^E \\Sigma_{A]E}+D_{[C}M_{A]B}\n$$\nfrom where (\\ref{DM}) immediately follows. Due to the second part in Corollary \\ref{coroDt} $D_{[C}M_{A]B}$ is gauge invariant.\n\\end{proof}\n\nNow, notice that the tensor field $W_{AB}-M_{AB}$, that is,\n$$\nU_{AB}:=S_{AB} -\\frac{1}{2} \\Sigma_{A}{}^D \\Sigma_{BD} +\\frac{\\varkappa}{2}\\Sigma_{AB} +\\frac{\\varkappa^2}{8} q_{AB}\n$$\nhas the following trace\n\\begin{equation}\\label{trU}\nU^E_E = K\n\\end{equation}\nand that equation (\\ref{eq11''}) can be rewritten, in terms of $U_{AB}$ as\n\\begin{equation}\\label{paso}\nD_C(U^C{}_A -U^E{}_E \\delta^C_A)=\\frac{3}{2}D_B(\\Sigma^{BE}\\Sigma_{EA})-\\Sigma^{CE}D_E \\Sigma_{CA} +\\left(\\frac{\\Lambda}{3}\\right)^{1\/2} \\epsilon_{EA}C^E .\n\\end{equation}\nContracting this equation with any conformal Killing vector field $\\xi^A$ and integrating its lefthand side on $\\mathcal{S}$ \n\\begin{eqnarray*}\n\\int_\\mathcal{S} \\xi^A [D_C(U^C{}_A -U^E{}_E \\delta^C_A)] = \\int_\\mathcal{S} D_C [\\xi^A(U^C{}_A -U^E{}_E \\delta^C_A)] - \\int_\\mathcal{S} (U^{CA} -U^E{}_E q^{CA})D_C\\xi_A\\\\\n=  \\int_\\mathcal{S} D_C [\\xi^A(U^C{}_A -U^E{}_E \\delta^C_A)] - \\frac{1}{2} \\int_\\mathcal{S} (U^{CA} -U^E{}_E q^{CA})q_{CA} D_B\\xi^B\\\\\n= \\int_\\mathcal{S} D_C [\\xi^A(U^C{}_A -U^E{}_E \\delta^C_A)] +\\frac{1}{2} \\int_\\mathcal{S} K D_B\\xi^B\n\\end{eqnarray*}\nwhere in the last equality I have used \\eqref{trU}. If $\\mathcal{S}$ is compact the first summand here vanishes. Concerning the second, a non-trivial result proved in Appendix \\ref{App:rho}, namely (\\ref{intLieK}), shows that this term also vanishes if $\\mathcal{S}$ is compact. Therefore, whenever the cut $\\mathcal{S}$ is compact we arrive at\n\\begin{equation}\\label{CKV}\n\\int_{\\cal S} \\xi^A\\left(\\frac{3}{2}D_B(\\Sigma^{BE}\\Sigma_{EA})-\\Sigma^{CE}D_E \\Sigma_{CA} +\\left(\\frac{\\Lambda}{3}\\right)^{1\/2} \\epsilon_{EA}C^E \\right) =0\n\\end{equation}\nfor every conformal Killing vector fields $\\xi^A$ if $\\mathcal{S}$ is compact.\n\nDefine the {\\em first piece of news} on $\\mathcal{S}$ as the tensor field\n\\begin{equation}\\label{V}\nV_{AB} := U_{AB} -\\rho_{AB}\n\\end{equation}\nwhere $\\rho_{AB}$ is the tensor field of Corollary \\ref{coroRho}. Explicitly, the first piece of news is given by\n$$\nV_{AB}=S_{AB} -\\frac{1}{2} \\Sigma_{A}{}^D \\Sigma_{BD} +\\frac{\\varkappa}{2}\\Sigma_{AB} +\\frac{\\varkappa^2}{8} q_{AB}-\\rho_{AB} .\n$$\nBy construction, $V_{AB}$ is gauge invariant and trace free, so that \n$$D_{[C}V_{A]B}=D_{[C}U_{A]B}$$\n is also  gauge invariant. However, $V_{AB}$\ndepends {\\em only} on the intrinsic geometry of $(\\mathscr{J},h_{ab})$ and the cut, and therefore it simply cannot contain the desired News tensor, which must involve, as explained, ${\\cal F}_{ab}$. It follows that the part described by $M_{AB}$ must be related to ${\\cal F}_{ab}$, thereby bringing the information encoded in $\\mathcal F_{ab}$  into the total tensor (\\ref{U}). Hence, it follows that the `source' $X_{CAB}$ in the equation (\\ref{DU}) has to also entail somehow ${\\cal F}_{ab}$. The definition of $V_{ab}$ induces \n\\begin{equation}\nW_{AB}= U_{AB}+M_{AB} = \\rho_{AB} + V_{AB} +M_{AB} , \\label{U=r+N} \n\\end{equation}\nso that $M_{AB}$ is the {\\em second piece of news} and the total News tensor field of the cut $\\mathcal{S}$ is\n\\begin{equation}\\label{N=V+M}\nN_{AB} =V_{AB} +M_{AB} .\n\\end{equation}\n$N_{AB}$ is symmetric, traceless, gauge invariant and satisfies the gauge invariant equation\n\\begin{equation}\\label{DN}\nD_{[C}N_{A]B}= X_{CAB} .\n\\end{equation}\nNotice that $N_{AB}$ is partly known, as the first piece $V_{AB}$ is explicitly known for any cut $\\mathcal{S}$. To find the complete news tensor one needs to identify the appropriate tensor field $X_{CAB}=X_{[CA]B}$ the provides, via \\eqref{DM}, the second piece $M_{AB}$.\nThus, the problem of the existence of $N_{AB}$ reduces to the existence of a tensor field $X_{CAB}$, or equivalently of the one-form $X_A:=X^C{}_{AC}$ with\n$$\nX_{CAB}= 2 q_{B[C} X_{A]} , \n$$\nsuch that the equation (\\ref{DM}) has a solution for $M_{AB}$ and the vanishing of $X_A$ be equivalent, on the entire cut ${\\cal S}$, to the vanishing of $N_{AB}$.\n\nTo ascertain under which circumstances such choices allow for the existence of the tensor $M_{AB}$, let us consider the trace of (\\ref{DM}) which is actually equivalent to (\\ref{DM}) itself:\n\\begin{equation}\\label{divM}\n\\frac{1}{2} D_C M^C{}_A =X_A+\\frac{1}{2}\\left(\\frac{\\Lambda}{3}\\right)^{1\/2} \\epsilon_{AB}C^B-\\frac{3}{8} D_A(\\Sigma_{DE}\\Sigma^{DE}) +\\frac{1}{2} \\Sigma^{CE}D_C\\Sigma_{EA}.\n\\end{equation}\nWe know that this provides the tensor field $M_{AB}$ if and only if the righthand side is $L^2$-orthogonal to every conformal Killing vector field on ${\\cal S}$ (there is a 6-parameter family of these in the sphere, Appendix \\ref{App:rho}). Therefore, by using here the relations (\\ref{CKV}) for every conformal Killing $\\xi^A$, the existence of $N_{AB}$ requires that\n\\begin{equation}\n\\int_{\\cal S} \\xi^A X_A   =0 \\label{cond}\n\\end{equation}\nfor every conformal Killing vector $\\xi^A$. An analysis of this condition is performed in Appendix \\ref{App:Hodge}. Observe that, given that $X_{CAB}$ is gauge invariant, the gauge behaviour of $X_A$ is simply\n\\begin{equation}\n\\tilde{X}_A = \\omega^{-2} X_A \\label{Xgauge} \n\\end{equation}\nand therefore the statement (\\ref{cond}) is gauge independent (because $\\xi^A X_A \\epsilon_{BC}$ is gauge invariant). \nUsing here Lemma \\ref{useful}, a plausible solution for $X_A$ is any one-form of the form\n\\begin{equation}\\label{possibility}\nX_A = \\Delta f D_A f\n\\end{equation}\nfor a choosable function $f$ on ${\\cal S}$. Observe that, due to\n$$\n\\tilde{\\Delta} f =\\frac{1}{\\omega^2} \\Delta f, \\hspace{1cm} \\forall f\\in C^2({\\cal S})\n$$\nany such one-form has the correct gauge behaviour (\\ref{Xgauge}) for $f$ gauge invariant. Moreover, the physical units of $X_A$ are $L^{-2}$, and thus $f$ carries no physical units. Notice finally that $X_A=0$ if and only if $f$ is constant in the sphere topology.\n\nIn principle, if one wishes that $X_A$ be related to the existence or not of radiation, so that the vanishing of a would-be news tensor field $N_{AB}$ implies the vanishing of $X_A$ and, hopefully, viceversa, the function $f$ in (\\ref{possibility}) should be related to the triplet $(\\mathscr{J},h_{ab},\\mathcal F_{ab})$ including explicitly $\\mathcal F_{ab}$. One possibility is that $f$ be a (known) function of the potentials $H_C,h_C$ and $H_\\mathcal F, h_\\mathcal F$ that $\\hat{C}_{AB} $ and $\\hat{\\mathcal F}_{AB}$ possess according to formula (\\ref{Hodge2}). Observe that these potentials have the right physical dimensions (a-dimensional), they do not have a simple gauge behaviour though.\n\n\n\n\\subsection{The problem of incoming and outgoing radiation: The case with $Q_-^\\alpha =0$}\\label{Q-=0}\nAs mentioned at the beginning of section \\ref{sec:dS}, one of the big differences of the $\\Lambda>0$-case with respecto to the $\\Lambda=0$-case is the existence of possible in-coming radiation that arrives at $\\mathscr{J}^+$ mingling with the outgoing flux of radiation. This is a complicated matter, and there is no easy way to try to identify in- or out-going components of the radiation. It should be remarked that our criteria \\ref{crit1} and \\ref{crit2}, based on the vanishing of the asymptotic super-Poynting $\\bar{p}^a$ in the case with $\\Lambda >0$, does not discriminate between those types of radiations. The absence\/presence of radiation on a cut may in general be due to a balance between several possible components, and this varies from one cut to another. This was somehow recognized time ago as a dependence of the radiative part of the field on the direction of approach to $\\mathscr{J}$ if $\\mathscr{J}$ is not a null hypersurface \\cite{Penrose65,Krtous2004,Fernandez-Alvarez_Senovilla2022}.This issue is of special importance when considering isolated sources of the radiation, or sources that are confined to a compact region of the spacetime, emitting gravitational radiation. \n\nIn the asymptotically flat scenario the lightlike character of $\\mathscr{J}^+$ implies that any radiation escaping from the space-time through infinity necessarily travels along lightlike directions {\\em transversal} to $\\mathscr{J}^+$. The generators of $\\mathscr{J}^+$ are the only exceptions and they provide an evolution direction which can be seen as `incoming direction' and thus, radiation from the physical spacetime is exclusively outgoing. In contrast, when $\\Lambda>0$ every radiation component, without exception, crosses $\\mathscr{J}^+$ and escapes from the space-time. In this case one needs to find physically reasonable conditions ruling out undesired radiative components, just leaving the radiation emitted by the isolated system of sources. In \\cite{Ashtekar2019} a proposal to solve this problem was presented, but this relies on information from the physical spacetime. In our opinion, and according to the entire philosophy of this paper, everything happening at the portion of the physical spacetime given by the past domain of dependence of $\\mathscr{J}^+$ is determined by the information encoded in the triplet $(\\mathscr{J}^+,h_{ab},\\mathcal F_{ab})$ ---plus the conformal re-scalings--- so that any `incoming radiation' or any undesired radiation components are {\\em encoded in that triplet too}. I wish to stress that this is independent of the existence of multiple isolated sources emitting the radiation, or of the possibility of scattering of the radiation by other components or matter, etcetera, because {\\em everything} that happens in the (domain of dependence of $\\mathscr{J}$ in the) physical spacetime is encoded in the initial\/final data $(\\mathscr{J},h_{ab},\\mathcal F_{ab})$. \n\n\\begin{figure}[!ht]\n\\includegraphics[width=8cm]{FlatScriQ=0.pdf}\n\\hspace{-1.8cm}\n\\includegraphics[width=10cm]{ScridSQ=0.pdf}\n\\caption{Comparison of $\\mathscr{J}^+$ and null directions orthogonal to a cut $\\mathcal{S}$ for the case with $\\Lambda =0$ (left) and the case with $\\Lambda >0$ (right). On the left the physical spacetime is the region below the cone representing $\\mathscr{J}^+$ and on the right the region below the plane that represents $\\mathscr{J}^+$. In both cases two points $p$ and $q$ belonging to the cut are shown, as well as the two null normals to the cut $\\mathcal{S}$ at those points. On the left, they are given by $n^\\mu$ itself, and $\\ell^\\mu$; on the right by $k^\\mu_\\pm = \\bar{n}^\\mu\\pm m^\\mu$, where $\\vec m$ is the unit normal to $\\mathcal{S}$ within $\\mathscr{J}^+$, so that $m^\\mu =m^ae^\\mu_a$. We know that, on the left, the vanishing of the asymptotic radiant super-momentum $\\mathcal{Q}^\\mu =0$ is equivalent to the vanishing of the news tensor and thus to the absence of radiation crossing $\\mathscr{J}^+$. If one modifies the cut passing through, say, $p$ the picture would be similar, but with a different $\\vec\\ell$. {\\em All possible} such null $\\vec \\ell$, for all possible cuts through $p$, span the little cone shown above $p$, and similarly for $q$. Hence, vanishing of $\\mathcal{Q}^\\mu$ implies that there is no radiation on any of all those transversal directions spanning the little cone with the exception, of course, of $\\vec n$, which is not transversal but tangent to $\\mathscr{J}^+$ and actually defines an evolution direction to the future. Notice that $\\mathcal{Q}^\\mu=0$ states that $n^\\mu$ is a multiple principal null direction of the re-scaled Weyl tensor $d_{\\alpha\\beta\\lambda}{}^\\mu$. Inspiration from these properties on the left is used on the right picture to try to isolate a unique component of radiation arriving at the cut $\\mathcal{S}$: when $\\Lambda >0$ (right picture) set $\\mathcal{Q}^\\mu_-=0$, and assume that this implies absence of radiation arriving along the directions spanned by the little cones shown above $p$ or $q$ except along $k^\\mu_-$, in analogy with the left situation. This would mean that the radiation is arriving basically along the null direction $k^\\mu_-$, which again is a multiple null direction of the re-scaled Weyl tensor, so that this makes sense. If this interpretation is accepted, the vector $m^a$ on the right defines, in analogy with $n^a$ on the left, an evolution direction towards the ``future'' within the spacelike $\\mathscr{J}^+$. In a way, one can think that the radiation is crossing $\\mathcal{S}$ towards its exterior (the projection of $k^\\mu_-$). \\label{fig:doble}}\n\\end{figure}\n\nMoreover, one can try to get some inspiration from the asymptitcally flat situation. The vanishing of the radiant super-momentum when $\\Lambda=0$ entails the absence of radiation transversal to $\\mathscr{J}^+$, and thus we may suspect that absence of radiation propagating {\\em transversally} to some null direction is also encoded in the analogous radiant super-momenta. More specifically, in our setup the vanishing of one of the radiant supermomenta \\eqref{Qpm} may mean absence of radiation components travelling along the corresponding transversal directions on that particular cut $\\mathcal{S}$. This is graphically explained in figure \\ref{fig:doble}.\n\n\nConsider for instance the case with $\\mathcal{Q}^\\mu_- =0$ on a cut $\\mathcal{S}$. By the previous discussion, this may indicate that there are no radiation components along directions transversal to $k^\\mu_-$, see figure \\ref{fig:doble}, in particular along the second null normal to $\\mathcal{S}$, $k^\\mu_+$. Observe that $\\mathcal{Q}^\\mu_- =0$ signifies that $k^\\mu_-$ is a repeated principal null direction of the re-scaled Weyl tensor, and in this sense it may be thought of as the direction of propagation of asymptotic radiation. In turn, this signifies that $m^a$ is, on the given cut $\\mathcal{S}$, an `incoming' direction that provides the direction of `evolution' of radiation at $\\mathcal{S}$ within $\\mathscr{J}^+$ --in analogy with the null $n^a$ in the asymptotically flat case, figure \\ref{fig:doble}. More importantly, as I am going to prove next, the condition $\\mathcal{Q}^\\mu_- =0$ can be expressed, in explicit manner, in terms of the triplet $(\\mathscr{J}^+,h_{ab},\\mathcal F_{ab})$.\nAssuming $Q_-^\\alpha =0$ on ${\\cal S}$ is equivalent, due to (\\ref{Ws}), (\\ref{Zs}) and (\\ref{QAs}) for the minus sign, to \n\\begin{equation}\n\\mathcal F_A =\\epsilon_{AB} C^B  \\hspace{3mm} \\mbox {and} \\hspace{3mm} \\hat{\\mathcal F}_{AB}= \\epsilon_{AD}\\hat{C}_{B}{}^D .\n\\label{FC}\n\\end{equation}\nThese conditions are actually stating that, on the cut $\\mathcal{S}$\n\\begin{equation}\\label{DisC}\n\\fbox{$\\mathcal F_{ab}-\\frac{1}{2} \\mathcal F_{cd}m^c m^d (3m_a m_b -h_{ab}) \\ \\stackrel{\\mathcal{S}}{=}\\  m^d\\epsilon_{ed(a} \\left(C_{b)}{}^e +m_{b)}m^f C_f{}^e \\right).$}\n\\end{equation}\nThis is our fundamental relation for cuts with only one radiation component. Note that this condition states that $\\mathcal F_{ab}$ is determined by $C_{ab}$ (which is intrinsic to $(\\mathscr{J}, h_{ab})$) except for the one single component $\\mathcal F_{cd}m^c m^d$, which is the only extra degree of freedom not given by the conformal geometry of $(\\mathscr{J}, h_{ab})$. This free degree of freedom concerns the Coulombian part of the gravitational field, proving that \\eqref{DisC} certainly affects the radiative degrees of freedom. \n\nUsing \\eqref{DisC} one can readily compute the asymptotic super-Poynting vector on $\\mathcal{S}$\n$$\n\\left(\\frac{3}{\\Lambda}\\right)^{3\/2} \\bar{p}^a\\ \\stackrel{\\mathcal{S}}{=}\\  -2m^a \\left(C_{bc} C^{bc} + m^b C_{be} m_c C^{ce}\\right) +4C^{ab} C_{bc} m^c +C_{bc} m^b m^c C^{ae} m_e -3(\\mathcal F_{bc}m^b m^c)\\epsilon^{ade}m_dC_{ef}m^f\n$$\nor equivalently (these can also be obtained from (\\ref{pm}) and (\\ref{pA}))\n\\begin{eqnarray}\np_m =-2\\left(\\hat{\\mathcal F}_{AB} \\hat{\\mathcal F}^{AB} +\\mathcal F_A \\mathcal F^A\\right)=-2\\left(\\hat{C}_{AB} \\hat{C}^{AB} +C_A C^A\\right)\\leq 0,\\label{pm2} \\\\\np_A = \\left[4\\hat{\\mathcal F}_{AB}+3\\left(C^E{}_E \\epsilon_{AB} - \\mathcal F^E{}_E q_{AB} \\right) \\right] \\mathcal F^B=\n\\left[4\\hat{C}_{AB}+3\\left(C^E{}_E q_{AB} - \\mathcal F^E{}_E \\epsilon_{AB} \\right) \\right] C^B . \\label{pA2}\n\\end{eqnarray}\n\n\nConcerning the asymptotic super-momentum $\\mathcal{Q}^\\alpha_+$, using again (\\ref{Ws}), (\\ref{Zs}) and (\\ref{QAs}), now for the $+$ sign, one derives\n$$\n\\mathcal{W}_+ =32 \\hat{\\mathcal F}_{AB} \\hat{\\mathcal F}^{AB}, \\hspace{8mm} \\mathcal{Z}_+ =16 \\mathcal F_A \\mathcal F^A ,  \\hspace{8mm} Q^+_A = 32 \\hat{\\mathcal F}_{AB}\\mathcal F^B.\n$$\nor equivalently\n\\begin{eqnarray}\nQ_+^\\alpha =8 \\left( 2\\hat{\\mathcal F}_{AB} \\hat{\\mathcal F}^{AB} k_-^\\alpha + \\mathcal F_A \\mathcal F^A k^\\alpha_+ +4 \\hat{\\mathcal F}^{AB}\\mathcal F_B E^\\alpha_A\\right)\\nonumber \\\\\n=8 \\left( 2\\hat{C}_{AB} \\hat{C}^{AB} k_-^\\alpha + C_A C^A k^\\alpha_+ +4 \\hat{C}^{AB}C_B E^\\alpha_A\\right)\\label{Q+}.\n\\end{eqnarray}\n{\\bf Remark}: It is remarkable that, with the restrictions put on $\\mathcal F_{ab}$ in this case, $Q^\\alpha_+$ is fully determined by the intrinsic geometry of $(\\mathscr{J},h_{ab})$ and the cut $\\mathcal{S}$ as follows from (\\ref{Q+}). This is also true for $p_m$, see (\\ref{pm2}). The only remaining `extrinsic' quantity identified above, $\\mathcal F^E{}_E=\\-\\mathcal F_{ab}m^am^b$, only affects the components $p_A$ tangential to the cut. Another important point to remark is that $p_m=\\bar{p}_a m^a \\leq 0$ is non-positive, in accordance with our intuition that radiation in this situation travels towards the exterior of the cut $\\mathcal{S}$ (figure \\ref{fig:doble}), and provides an interesting interpretation for the balance law \\eqref{balance}. Furthermore, $p_m=0$ implies that the entire $\\bar{p}_a=0$ vanishes, and this statement again depends only on the intrinsic geometry of $(\\mathscr{J},h_{ab})$ and the cut now.\n\nIf the discussed interpretation of the condition $\\mathcal{Q}^\\mu_- \\ \\stackrel{\\mathcal{S}}{=}\\  0$ is to be accepted, then the absence of radiation determined by $\\bar p_a$ should equivalently eliminate the unique radiative component that was left on the cut $\\mathcal{S}$. This is proven in the following proposition.\n\\begin{Proposition}\\label{prop:typeD}\nThe following conditions are all equivalent at any point of ${\\cal S}$:\n\\begin{enumerate}\n\\item $Q_-^\\mu =Q_+^\\mu =0$.\n\\item $Q_-^\\mu =0$ and $p_m=0$.\n\\item $Q_-^\\mu =0$ and $\\bar p_a=0$.\n\\item $\\hat{\\mathcal F}_{AB}=\\hat{C}_{AB}=0$ and $\\mathcal F_A =C_A=0$.\n\\item In the basis $\\{\\vec m,\\vec E_A\\}$\n\\begin{equation}\n(\\mathcal F_{ab})= \\mathcal F^E{}_E \\left(\n\\begin{array}{ccc}\n-1 & 0 & 0 \\\\\n0 & 1\/2 & 0 \\\\\n0 & 0 & 1\/2\n\\end{array}\n \\right) ,\n \\hspace{8mm}\n (C_{ab})= C^E{}_E \\left(\n\\begin{array}{ccc}\n-1 & 0 & 0 \\\\\n0 & 1\/2 & 0 \\\\\n0 & 0 & 1\/2\n\\end{array}\n \\right) \\label{typeD}\n\\end{equation}\n\\end{enumerate}\n\\end{Proposition}\n\\begin{proof} \nI provide a circular proof $1\\Rightarrow 2 \\Rightarrow 3\\Rightarrow 4 \\Rightarrow 5 \\Rightarrow 1$:\n\\begin{itemize}\n\\item If $Q_-^\\mu =Q_+^\\mu =0$ then from (\\ref{Q+}) $\\hat{C}_{AB}=0=C_A$ so that (\\ref{pm2}) gives $p_m=0$.\n\\item If $Q_-^\\mu =0$ and $p_m=0$, (\\ref{pm2}) implies $\\hat{C}_{AB}=0=C_A$ and together with (\\ref{pA2}) gives that the full $\\bar p_a$ vanishes.\n\\item If $Q_-^\\mu =0$ and $\\bar p_a=0$, (\\ref{pm2}) implies $\\hat{C}_{AB}=0=C_A$ and then (\\ref{FC}) that also $\\hat{\\mathcal F}_{AB}=0=\\mathcal F_A$. \n\\item $\\hat{\\mathcal F}_{AB}=\\hat{C}_{AB}=0$ and $\\mathcal F_A =C_A=0$ is just saying that, in the mentioned basis, the matrices of $\\mathcal F_{ab}$ and $C_{ab}$ take the form displayed in (\\ref{typeD}).\n\\item If (\\ref{typeD}) holds in the given basis, then $\\hat{\\mathcal F}_{AB}=\\hat{C}_{AB}=0$ and $\\mathcal F_A =C_A=0$ so that (\\ref{Ws}--\\ref{QAs}) imply $\\mathcal{W}_\\pm =\\mathcal{Z}_\\pm =0=Q_\\pm^A$ and thus $Q_\\pm^\\mu =0$. $\\Box$\n\\end{itemize}\n\\end{proof}\n{\\bf Remark}: This case corresponds to the situation where the rescaled Weyl tensor has Petrov type D at $\\mathscr{J}$ and is aligned at the cut $\\mathcal{S}$, that is, the two multiple principal null directions are $\\vec k_\\pm$ (unless when also $\\mathcal F^E{}_E=C^E{}_E=0$, but this corresponds to the de Sitter spacetime if $\\mathscr{J} \\sim \\mathbb{S}^3$).\n\nSimilar formulas and results are valid if one assumes $\\mathcal{Q}^\\mu_+ =0$ instead of $\\mathcal{Q}^\\mu_-=0$.\n\nAccording to the nomenclature introduced in \\cite{Fernandez-Alvarez_Senovilla-dS}, if on $\\Delta\\subset \\mathscr{J}$ there exists a foliation by cuts, all of them satisfying the property $\\mathcal{Q}^\\mu_-=0$, then we say that $\\Delta$ is {\\em strictly equipped and strongly oriented}, the vector field $m^a$ orthogonal to the cuts providing the orientation and equipment. If in addition the cuts are umbilical ($\\Sigma_{AB}=0$), $\\Delta$ is both {\\em strongly equipped and oriented} by $m^a$. The existence of news under such circumstances, as well as other possibilities, were explored at large in \\cite{Fernandez-Alvarez_Senovilla-dS}. In particular we proved that the first component of news provides a good total News tensor field in the case of strongly equipped and oriented $\\mathscr{J}$.\n\n\\subsection{A conserved charge in vacuum}\nAs yet another justification for criterion \\ref{crit2} let me present a conserved charge, built from the re-scaled Bel-Robinson tensor, that identifies the existence of radiation in asymptotic vacuum (this could be generalized to the case with matter) when the spacetime possesses conformal Killing vector fields. If the energy-momentum tensor of the physical spacetime vanishes in a neighbourhood ${\\cal U}$ of $\\mathscr{J}^+$, then on that neighbourhood \n$$\n\\nabla_\\rho {\\cal D}^\\rho{}_{\\mu\\nu\\tau} \\stackrel{{\\cal U}}{=} 0.\n$$\nIf $\\xi^\\mu_{{i}}$ are {\\em any} three conformal Killing vectors on $(M,g)$ (they can be repeated) then the currents \n$$\n{\\cal B}^\\rho (i,j,k):= \\xi^\\mu_{(i)} \\xi^\\nu_{(j)} \\xi^\\tau_{(k)}{\\cal D}^\\rho{}_{\\mu\\nu\\tau}\n$$\nare divergence-free \\cite{Senovilla2000,Lazkoz2003} on ${\\cal U}$\n$$\n\\nabla_\\rho {\\cal B}^\\rho (i,j,k)\\stackrel{{\\cal U}}{=} 0.\n$$\nThis implies that the `charges' defined on any spacelike hypersurface $\\Sigma$ without edge within ${\\cal U}$ by\n$$\n{\\cal B}_\\Sigma (i,j,k) := \\int_\\Sigma {\\cal B}^\\rho(i,j,k) t_\\rho\n$$\n(where $t_\\rho$ is the unit normal to $\\Sigma$) are conserved, in the sense that they are independent of the choice of $\\Sigma$. In particular, they are equal to ${\\cal B}_{\\mathscr{J}^+}(i,j,k)$.\n\nIf the $\\xi^\\mu_{(i)}=\\xi^a_{(i)} e^\\mu_a$ happen to be tangent to $\\mathscr{J}^+$, by using the explicit formulae in \\cite{Alfonso2008} one can find (for instance, and for simplicity, for three copies of the same $\\xi^\\mu_{(1)}:=\\xi^\\mu$)\n$$\n{\\cal B}_{\\mathscr{J}^+}(1,1,1) =\\int_{\\mathscr{J}^+}  \\left( \\left(\\frac{3}{\\Lambda}\\right)^{(3\/2)} \\bar{p}_a\\xi^a-\\xi_a\\epsilon^{abc} \\xi^d C_{bd} \\xi^e \\mathcal F_{ce} \\right).\n$$\nThis charge is generically non-zero. Nevertheless, if \\eqref{DisC} holds and $\\bar p_a=0$ then it vanishes. This is precisely the case of proposition \\ref{prop:typeD}. This seems to hint in the direction that (non-zero) values of ${\\cal B}_{\\mathscr{J}^+}(1,1,1)$ arise when there is gravitational radiation arriving at $\\mathscr{J}^+$.\n\n\\section{Symmetries with $\\Lambda >0$}\\label{sec:sym}\nOne of the missing elements to complete the picture in the $\\Lambda >0$ scenario are the asymptotic symmetries. There is nothing like the BMS algebra\/group and, the lack of a universal structure on $\\mathscr{J}$ is an impediment to provide a general notion of symmetries and, thereby, to look for appropriate conservation and balance laws. Still, one can try to find such missing symmetries in restricted situations, such as the one described in the previous section \\ref{Q-=0} with strictly equipped and strongly oriented $\\mathscr{J}$, that is, if \\eqref{DisC} holds on $\\mathscr{J}$.\n\nTo start with, let me argue that the `natural' definition for (infinitesimal) symmetries is any vector field $\\vec Y\\in\\mathfrak{X}(\\mathscr{J})$ leaving invariant the the tensor field \n$$\nX_{abcdef}:= h_{ab} \\mathcal F_{cd} \\mathcal F_{ef}.$$\n$X_{abcdef}$ is {\\em gauge invariant} and contains the elements needed to determine any property of the physical spacetime, the triplet $(\\mathscr{J},h_{ab},\\mathcal F_{ab})$. Thus, a reasonable proposal of infinitesimal symmetries $\\vec Y\\in\\mathfrak{X}(\\mathscr{J})$ is simply\n$$\n\\pounds_Y (h_{ab} \\mathcal F_{cd} \\mathcal F_{ef})=0.\n$$\nThis can be easily shown to be equivalent to\n\\begin{equation}\\label{basicsym}\n\\pounds_Y h_{ab} =2 \\psi h_{ab} , \\hspace{1cm} \\pounds_Y D_{ab} = - \\psi D_{ab}\n\\end{equation}\nfor some function $\\psi$. That this is a good definition is justified by noting that any solution $\\vec Y$ of \\eqref{basicsym} generates a Killing vector field on the physical spacetime and viceversa. This follows from a result due to Paetz \\cite{Paetz2016}. Any solution of \\eqref{basicsym} is termed {\\em basic infinitesimal symmetry}. They satisfy\n$$\n\\pounds_Y \\bar{p}_a =-5\\psi \\bar{p}_a .\n$$\n\nNevertheless, an obvious problem arises with such basic symmetries. Observe that the first equation in \\eqref{basicsym} informs us that $\\vec Y$ must be a conformal Killing vector of $(\\mathscr{J},h_{ab})$, and of course a generic 3-dimensional Riemannian manifold does not need to possess such vector fields. Hence, there are many $(\\mathscr{J},h_{ab})$  without any {\\em basic} infinitesimal symmetries. \n\nTo remedy this situation, let me restrict the possible $(\\mathscr{J},h_{ab})$ to those which possess a vector field $m^a$, orthogonal to a foliation of cuts, such that \\eqref{DisC} holds on $\\mathscr{J}$, that is to say, $\\mathscr{J}$ is strictly equipped, and also strongly oriented,  by $m^a$. Then, we want that the symmetries preserve this structure, conformally keeping the orientation and equipment. This is achieved by the vector fields that satisfy \n\\begin{equation}\\label{sym}\n\\pounds_Y h_{ab}=2\\psi h_{ab} +2\\gamma m_a m_b, \\hspace{1cm} \\pounds_Y m_a =(\\gamma +\\psi) m_a\n\\end{equation}\nfor some functions $\\psi$ and $\\gamma$ on $\\mathscr{J}$. From this one also has\n$$\n\\pounds_Y m^a = -(\\gamma +\\psi) m^a .\n$$\nFirst of all, observe that the basic symmetries \\eqref{basicsym} are included here (for $\\gamma =0$) as long as they preserve the direction field $m^a$. Secondly, it is easy to check that the family of solutions of \\eqref{sym} constitute a Lie algebra. Thirdly, the function $\\gamma$ is gauge invariant under \\eqref{gauge} while $\\psi$ has the following behaviour\n$$\n\\tilde\\psi =\\psi +\\frac{1}{\\omega} \\pounds_Y \\omega.$$\nFourthly, equations \\eqref{sym} are equivalent to\n\\begin{equation}\\label{sym2}\n\\pounds_Y P_{ab} = 2\\psi P_{ab} , \\hspace{1cm} \\pounds_Y m_a =(\\gamma +\\psi) m_a\n\\end{equation}\nwhere \n$$\nP_{ab} := h_{ab} -m_a m_b\n$$\nis the orthogonal projector of the foliation defined by $m_a$ that projects to the leaves. In this form, and given that the projector restricted to each leaf $\\mathcal{S}$ of the foliation gives the corresponding first fundamental form $q_{AB}$, the first relation in \\eqref{sym2} states that the vector fields leave the conformal metrics invariant. Actually, \\eqref{sym2} or \\eqref{sym} are an example of the infinitesimal symmetries called bi-conformal vector fields \\cite{GarciaParrado2004} that leave two orthogonal distributions  conformally invariant.  As proved in  \\cite{GarciaParrado2004}, the solutions of \\eqref{sym2} can form an infinite-dimensional Lie algebra. \n\nIt remains the question of whether or not these new symmetries can be somehow derived as asymptotic generalized symmetries from the physical spacetime. This is certainly the case, as we briefly explain next. Start by considering a vector field $\\hat\\xi^\\mu$ on the physical spacetime $(\\hat M,\\hat g)$ such that is has a smooth extension to $\\mathscr{J}$ on $M$. Then on $\\hat M$\n$$\n\\pounds_{\\hat\\xi} g_{\\mu\\nu} = \\Omega^2 \\pounds_{\\hat\\xi}\\hat{g}_{\\mu\\nu}+ \\frac{2}{\\Omega} \\pounds_{\\hat\\xi}\\Omega \\hat{g}_{\\mu\\nu}\n$$\nand require that \n$$\nH_{\\mu\\nu} := \\Omega^2 \\pounds_{\\hat\\xi}\\hat{g}_{\\mu\\nu}\n$$\nhas a regular limit to $\\mathscr{J}$. The basic idea is to find the `minimum' possible $H_{\\mu\\nu}$ that induces the symmetries on $(\\mathscr{J},h_{ab})$. In other words, $\\hat\\xi^\\mu$ can be thought as an approximate symmetry when approaching infinity. One can easily prove \\cite{Fernandez-Alvarez_Senovilla-dS} that \n$$\n\\xi^\\mu n_\\mu \\ \\stackrel{\\scri}{=}\\  0 \\Longrightarrow \\, \\, \\hat\\xi^\\mu =Y^a e^\\mu_a\n$$\nand $Y^a$ is a vector field on $\\mathscr{J}$. It is necessary to take into account that only the class of vector fields $\\hat\\xi^\\mu$ defined modulo the addition of any term of the form $\\Omega v^\\mu$, for arbitrary $v^\\mu$, makes sense. This implies that combinations of type\n$$\nv_\\mu n_\\nu + v_\\nu n_\\mu -2v^\\rho n_\\rho g_{\\mu\\nu} + 2\\Omega (\\nabla_\\mu v_\\nu +\\nabla_\\nu v_\\mu)\n$$\ncan be added to $H_{\\mu\\nu}$ without changing the sought asymptotic symmetry. \n\nThus, in order to choose $H_{\\mu\\nu}$ one first notices that $H_{\\mu\\nu}\\propto g_{\\mu\\nu}$ (including $H_{\\mu\\nu}=0$, which mimics the case of $\\Lambda =0$ as studied in \\cite{Geroch81}) will lead to conformal Killing vectors of $(\\mathscr{J},h_{ab})$, that is, to the basic symmetries \\eqref{basicsym}. Thus, one needs a more general choice. The next `minimal' possible such choice is that $H_{\\mu\\nu}$ is a rank-1 tensor field on $\\mathscr{J}$, that is, there exists a vector field $m^\\mu$ such that $H_{\\mu\\nu} = F m_\\mu m_\\nu$, or including the redundant terms above\n$$\nH_{\\mu\\nu} = F m_\\mu m_\\nu +v_\\mu n_\\nu + v_\\nu n_\\mu -2v^\\rho n_\\rho g_{\\mu\\nu} + 2\\Omega (\\nabla_\\mu v_\\nu +\\nabla_\\nu v_\\mu)\n$$\nwhere necessarily $m^\\mu n_\\mu \\ \\stackrel{\\scri}{=}\\  0$ \\cite{Fernandez-Alvarez_Senovilla-dS}. Projection to $\\mathscr{J}$ then shows that \\cite{Fernandez-Alvarez_Senovilla-dS}\n$$\n\\pounds_Y h_{ab}= 2\\psi h_{ab} +2\\gamma m_a m_b \n$$\nwhere $\\gamma =F|_\\mathscr{J}$ and $\\psi = -(2v^\\rho n_\\rho +\\xi^\\mu n_\\mu\/\\Omega)|_\\mathscr{J}$. This is precisely the first in \\eqref{sym}, and the Lie algebra property requires the second one. \n\nThe precise structure of the Lie algebra of the symmetries \\eqref{sym} depends on the specific situation, that is, on the particular properties of the foliation determined by the vector field $m^a$ that equips and orientates $\\mathscr{J}$. For instance, in the case that the orientation and the equipment are both strong (so that the foliation is by umbilical cuts), the structure is the product of conformal transformations of the cuts times an ideal which commutes with the previous and depends on arbitrary functions, so that the algebra is infinite dimensional  \\cite{Fernandez-Alvarez_Senovilla-dS}.\n\n\n\\section{Closing comments with examples}\\label{sec:fin}\nCriteria \\ref{crit1} and \\ref{crit2} have been tested in a variety of spacetimes \\cite{Fernandez-Alvarez_Senovilla20b,Fernandez-Alvarez_Senovilla-dS} that admit a conformal completion and so far they agree with the expected results concerning existence of gravitational radiation, as well as in relation to other concepts introduced in this paper. Herein I provide a summary of the known results and add a couple of new ones.\n\nFirst of all, take spherically symmetric spacetimes that, as we know, do not contain any kind of gravitational radiation. If they admit a conformal completion this can be assumed to have spherical symmetry too, and then $C_{ab}$ and $\\mathcal F_{ab}$ inherit the symmetry. This readily proves that $C_{ab}$ and $\\mathcal F_{ab}$ must be proportional to each other, so that their commutator vanishes and using \\eqref{superp} this leads to $\\bar{p}_a=0$ in agreement with the absence of radiation in such situations according to our criteria. This includes, in particular, de Sitter spacetime which actually has both $C_{ab}$ and $\\mathcal F_{ab}$ vanishing, where one can identify the 10 asymptotically basic infinitesimal symmetries, four possible strong equipments (all of them equivalent) with umbilical foliations by $\\mathbb{S}^2$ cuts, and find the structure of the group of symmetries of type \\eqref{sym} for any of the strong equipments. This is composed of the conformal Killing vectors of the sphere together with a vector field of type $F(\\chi) \\partial_\\chi$, for {\\em arbitrary} function $F$, where $\\chi$ is a typical latitud coordinate on the 3-dimensional sphere \\cite{Fernandez-Alvarez_Senovilla-dS}.\n\nNext, consider the ``Kerr-de Sitter-like spacetimes'' as defined in \\cite{Mars2016}. Basically, these are the $\\Lambda $-vacuum spacetimes with a Killing vector filed whose `Mars-Simon' tensor vanishes \\cite{MS2015} and admit  a conformal completion. They include in particular the Kerr-de Sitter solution as well as many others \\cite{MS2015,Mars2016,Mars2017,MPS1}.\nKerr-de Sitter-like spacetimes are characterized by initial data $(\\mathscr{J},h_{ab},\\mathcal F_{ab})$ with\n$$\nC_{ab}= \\frac{A}{|Y|^5} \\left(Y_a Y_b -\\frac{1}{3} |Y|^2 h_{ab} \\right), \\hspace{3mm} \n\\mathcal F_{ab}= \\frac{B}{|Y|^5} \\left(Y_a Y_b -\\frac{1}{3} |Y|^2 h_{ab} \\right)\n$$\nfor some constants $A,B$ where $Y^a\\in \\mathfrak{X}(\\Sigma)$ is a conformal Killing vector on $(\\mathscr{J},h)$ with no fixed points. $Y^a$ is the conformal Killing vector induced by the Killing vector of the physical spacetime with vanishing Mars-Simon tensor. From the expressions above we check that again $C_{ab}$ and $\\mathcal F_{ab}$ are proportional to each other so that \\eqref{superp} imples $\\bar p^a=0$ and criterion \\ref{crit2} states that there is no gravitational radiation. This is also an expected result.\nIn the particular case of Kerr-de Sitter spacetime, including the Kottler solution for zero angular momentum, the constant $A=0$ ($(\\mathscr{J},h_{ab})$ is conformally flat), there are two strong orientations but neither of them leads to a strong equipment. The corresponding symmetries \\eqref{sym} coincide with the basic asymptotic symmetries \\eqref{basicsym} and are induced by the two Killing vectors of the spacetime. Still, there exists a `natural' strong equipment by umbilical cuts and the corresponding algebra of symmetries \\eqref{sym} is again infinite dimensional depending on an arbitrary function of one variable \\cite{Fernandez-Alvarez_Senovilla-dS}.\n\n\nIn \\cite{Mars2016} a more general class of spacetimes, termed {\\em asymptotically Kerr-de Sitter-like spacetimes}, was introduced. They also have a Killing vector but now the Mars-Simon tensor is only required to vanish asymptotically. Their characterization at infinity is given by data $(\\mathscr{J},h_{ab},\\mathcal F_{ab})$ such that\n$$\nC_{ab} Y^b = \\delta Y_a, \\hspace{1cm} \\mathcal F_{ab} Y^b =\\beta Y_a\n$$\nfor some functions $\\delta,\\beta$ on $\\mathscr{J}$, where $Y^a$ is the conformal Killing vector on $(\\mathscr{J},h_{ab})$ induced by the Killing vector of the physical spacetime. In other words, $C_{ab}$ and $\\mathcal F_{ab}$ have $Y^a$ as a common eigenvector field. Obviously, the Kerr-de Sitter-like spacetimes are included here, but there are many other possibilities. In this case, gravitational radiation may be present. An interesting possibility is the analysis of asymptotically Kerr-de Sitter-like spacetimes which also comply with \\eqref{DisC} for some $m^a$. If in this case $Y^a$ points into the direction $m^ a$ that equips $\\mathscr{J}$, that is to say, $Y^a= |Y| m^a$ then the eigenvalues of the common eigendirection are\n$$\n\\delta = C_{ab}m^am^b, \\hspace{1cm} \\beta =\\mathcal F_{ab}m^am^b\n$$\nand  also $\\mathcal F_A=0$ and $C_A=0$. Equation \\eqref{pA2} tells us that $p_A=0$ and thus from \\eqref{pm2}\n$$\n\\left(\\frac{3}{\\Lambda}\\right)^{3\/2} \\bar{p}_a = -\\hat{C}_{AB} \\hat{C}^{AB} m_a.\n$$\n\nNext, a very interesting spacetime to be used as example is the $C$-metric \\cite{Stephani2003,Griffiths-Podolsky2009}, both in the $\\Lambda>0$ and $\\Lambda =0$ cases, see \\cite{Fernandez-Alvarez_Senovilla20b,Fernandez-Alvarez_Senovilla-dS}, because this is known to have gravitational radiation in the asymptotically flat case \\cite{Ashtekar-Dray81}.  The existence of gravitational radiation according to our criterion \\ref{crit2} for $\\Lambda\\geq 0$ was proven in \\cite{Fernandez-Alvarez_Senovilla20b}. For the $C$-metric there are two possible strong orientations, both of them providing strong equipments, and the Lie algebra of symmetries \\eqref{sym} is infinite dimensional once more, but in this case depending of multiple arbitrary functions \\cite{Fernandez-Alvarez_Senovilla-dS}.\n\nAnother interesting family of spacetimes usable as examples are the Robinson-Trautman metrics \\cite{Stephani2003,Griffiths-Podolsky2009}, for $\\Lambda \\geq 0$. Generically, they have one strong orientation which defines a strong equipment, and the corresponding asymptotic symmetries \\eqref{sym} form an infinite-dimensional Lie algebra that depends on an arbitrary function of one variable. They generically contain gravitational radiation according to criterion \\ref{crit2}, the particular case of Petrov type N Robinson-Trautman metrics is analyzed in detail in \\cite{Fernandez-Alvarez_Senovilla-dS}.\n\n\n\n\n\n\n\\vspace{6pt} \n\n\n\n\n\\section*{Funding}\nResearch supported by Basque Government grant numbers IT956-16 and IT1628-22, and by Grant FIS2017-85076-P funded by the Spanish MCIN\/AEI\/10.13039\/501100011033 and by \"ERDF A way of making Europe\".\n\n\n\n\n\n\\section*{Acknowledgments} \nDiscussions with Fran Fern\\'andez-\\' Alvarez are gratefully acknowledged.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Introduction}\n\nMatrix models have been very useful to generate and study random geometries in two dimensions. At large matrix size $N$, the $1\/N$ expansion is a topological expansion, labeled by the genus of the random discrete surfaces. In the large $N$ limit, only planar maps on the sphere survive. These maps encode discrete geometries of fluctuating surfaces, making them very important in physics. A famous application is two-dimensional gravity coupled to conformal matter (central charge $c<1$) \\cite{mm-review-difrancesco}.\n\nTensor models allow to extend those ideas to random geometries with more than two dimensions \\cite{ambjorn-3d-tensors, sasakura-tensors, gross-tensors}. Their Feynman expansion is a sum over discretized (pseudo-)manifolds in dimension $d$ and it possesses a $1\/N$ expansion \\cite{Gur4, uncoloring}. A continuum limit exists, first found in \\cite{critical-colored}, which can be coupled to (non-unitary) critical matter \\cite{harold-hard-dimers, multicritical-dimers}, leading to different universality classes.\n\nThe progress obtained in the past few years on tensor models are due to the discovery that tensor models with a $U(N)^d$ symmetry naturally generate regular, edge-colored graphs (dual to triangulations of pseudo-manifolds) \\cite{uncoloring}. Those graphs, in contrast with the stranded graphs initially considered in \\cite{ambjorn-3d-tensors}, are amenable to analytical investigation. A combinatorial classification has been recently obtained, \\cite{GurauSchaeffer}. In the same time, tensor models with quartic interactions have been re-formulated as matrix models, \\cite{DSQuartic, GenericQuartic}. Both approaches have led to a double-scaling limit and more generally to a good understanding of the singularities at fixed order in the $1\/N$ expansion. The double scaling limit has been extended to models beyond the quartic interactions in \\cite{DSSD} using a typical tool of matrix models, the loop equations.\n\nIt thus appears that matrix model techniques can be useful in tensor models. Formulating tensor models as matrix models also opens the possibility of using the combinatorial techniques (or even maybe already existing results) on maps. However a precise study of the relationships between tensor and matrix models has not appeared yet. This is the program we start in the present article.\n\nIt was not obvious at first that matrix models techniques would be of any use. In particular, diagonalization and eigenvalues (together with the saddle point method or orthogonal polynomials) are among the most effective tools in random matrix models and they are not available for random tensors. Also the fact that $U(N)^d$-invariant random tensors become Gaussian at large $N$ \\cite{universality}, and are thus very different from large $N$ matrix models, tends to establishing a clear distinction between matrices and tensors.\n\nHowever those arguments are no longer relevant thanks to the intermediate field method which turns quartic models into multi-matrix models. In addition, there are two simple ideas which establish a direct connection between matrices and tensors, which we present and exploit in this article. They enable to understand the position of tensor models with respect to matrix models. Following those two ideas one after the other, we offer a novel presentation of random tensor models, in which results from tensor models are applied to matrix models and the other way around.\n\nThe first part \\ref{sec:loops} is based on the observation that a collection of matrices $M_1,\\dotsc, M_\\tau$ may be packaged into a tensor of rank three and size $N\\times N\\times \\tau$, whose first two indices are matrix indices while the third one is the label of the matrix. When the joint probability distribution on the matrices is of the form $e^{-V}$ for a polynomial potential $V$ that is $U(\\tau)$-invariant, then we have a tensor model in disguise. We therefore introduce a family of $U(\\tau)$ models which is shown to generate random surfaces dressed with configurations of oriented loops. We describe the bijection between the observables and the Feynman graphs of those $U(\\tau)$ models and their corresponding tensor models.\n\nAll known $1\/N$ expansions in tensor models rely on the \\emph{degree} of the Feynman graphs dual to the triangulations. It was originally introduced in \\cite{Gur3} to exhibit a $1\/N$ expansion for tensor models for the first time. The degree was defined as a sum of genera of ribbon sub-graphs which are generated by matrix models embedded in the tensor theory \\cite{JacketsMatrixModels}. It controls the balance between the number of faces and the number of vertices and reduces to the genus in two dimensions. The dominant triangulations of tensor models at large $N$ are those with vanishing degree and are known as \\emph{melonic} triangulations \\cite{critical-colored, uncoloring}, which have a specific, highly curved, geometry. They have been recently matched to random branched polymers \\cite{melons-BP}, meaning that the continuous geometry is that of the continuous random tree. Melonic graphs are the ones which maximize the number of faces at fixed number of vertices \\cite{critical-colored}.\n\nIn the $U(\\tau)$ models, it turns out that the number of loops at fixed genus of the random surfaces, fixed numbers of edges and vertices, is counted by the degree of the 4-colored graph representative of the Feynman graph. This provides a new combinatorial interpretation of the degree. In particular, melonic graphs are those which maximize the number of loops. It also makes clear how the large $N$, melonic behavior of tensor models arise from a matrix model when $\\tau\\to\\infty$. We then apply the classification of edge-colored graphs from \\cite{GurauSchaeffer} to the quartic $U(\\tau)$ model to get a classification of its loop configurations. Finally, the double scaling limit of tensor models is found to resum consistently the most critical loop configurations.\n\nThe intermediate field transformation is also performed on the quartic $U(\\tau)$ models, leading to a two-matrix model. To our knowledge, this two-matrix model has never been studied in the matrix model literature and we do not even know its large $N$ free energy. It generates graphs formed by two maps glued together at their vertices (at least at one of them for the whole graph to be connected). Those graphs, also called \\emph{nodal surfaces} do have already appeared in the literature \\cite{EynardBookMulticut} and they may be well suited for a combinatorial analysis.\n\nIn a companion paper \\cite{AngularIntegralsGaussian}, another simple relationship between matrix and tensor models is studied. It relies on re-packaging the set of $d$ indices into two disjoint sets which are interpreted as indices of range $N^p$ and $N^{d-p}$ so that $T$ becomes a (typically rectangular) matrix. The singular value decomposition then enables to perform partial integration over the angular degrees of freedom. The results in a notion of effective observables which actually allows the calculation of new expectations in the Gaussian distribution.\n\nIn addition to exploring relationships between random tensors and matrices, those approaches clarify the difficulties faced by random tensor theory in the light of familiar matrix models. We also hope that it sets a frame in which those challenges may be dealt with.\n\nFinally, the section \\ref{sec:interpolating} in appendix investigates the possibility of interpolating matrix and tensor models, a question often asked, or more generally interpolating various tensor models. We use for instance a tensor of size $N\\times N \\times \\dotsb \\times N^\\beta$ where $\\beta$ runs in $[0,1]$. This completes the analysis of \\cite{new1\/N} of tensor models with distinct index ranges. It is found that there are only two large $N$ behaviors in those models, $\\beta=0$ and $\\beta\\in (0,1]$. The reason is that for $\\beta=0$ we have a tensor model of rank $d-1$ but as soon as $\\beta>0$ each face of colors $(0,d)$ contributes to the large $N$ scaling. However, the $1\/N$ corrections are typically found to depend on $\\beta$, but we do not know if this affects the continuum limits.\n\n\n\n\\section{The degree expansion in completely packed loop models on random surfaces} \\label{sec:loops}\n\n\n\\subsection{Loop model on random surfaces}\n\nMatrix models are known to generate discretized random 2D surfaces. Each term of the action has the form $\\tr (AA^\\dagger)^n$, where $A$ is a complex matrix, and creates ribbon vertices of degree $2n$. A matrix model generates random surfaces through the Wick theorem which connects these ribbon vertices together via ribbon lines. Following the recipe of \\cite{difrancesco-folding}, the random surfaces can be decorated with oriented loops in the following way: let $\\{A_i, A_i^\\dagger, i = 1, \\dotsc, \\tau\\}$ be a set of decorated matrices, and rewrite the terms $\\tr (AA^\\dagger)^n$ with various matrix labelings. We allow terms of the form\n\\begin{equation}\n\\label{eq:TraceInvariant}\n\tV_{n,\\sigma}(\\{A_i,A_i^\\dagger\\}) = \\sum_{\\substack{\\alpha_1, \\dotsc\\alpha_n\\\\\\beta_1,..,\\beta_n}} \\tr\\left( A_{\\alpha_1}A^\\dagger_{\\beta_1} A_{\\alpha_2} A^\\dagger_{\\beta_2}\\,\\dotsm\\,A_{\\alpha_n}A^\\dagger_{\\beta_n}\\right) \\prod_{k=1}^n \\delta_{\\alpha_k,\\beta_{\\sigma(k)}},\n\\end{equation}\nwhere $\\sigma$ is a permutation of $\\{1,\\dotsc,n\\}$ (there are obviously redundancies in this parametrization).\nSuch terms can be interpreted as $n$ lines meeting, and possibly crossing, at a $2n$-valent ribbon vertex. The incoming line in position $k$ (corresponding to $A_{\\alpha_k}$) crosses the vertex and go out in position $\\sigma(k)$ (corresponding to $A^\\dagger_{\\alpha_{\\sigma(k)}}$). This is pictured in figure \\ref{fig:InterpretationAsLoops}. We call the drawing associated to $\\sigma$ the \\emph{link pattern labeled by $\\sigma$}.\n\n\\begin{figure}\n\t\\center\n\t\t\\subfigure[Interpretation as crossing loops on the ribbon vertex generated by the term $\\tr A_a A_b^\\dagger A_b A_c^\\dagger A_d A_a^\\dagger A_c A_d^\\dagger$]{\\makebox[6cm]{\\includegraphics[width=4cm]{CrossingLoops.pdf}}}\n\t\\hspace{1cm}\n\t\t\\subfigure[Interpretation as non-crossing loops of the term $\\tr A_a A_b^\\dagger A_b A_a^\\dagger A_c A_c^\\dagger A_d A_d^\\dagger$\\label{subfig:NonCrossingLoop}]{\\makebox[6cm]{\\includegraphics[width=4cm]{NonCrossingLoops.pdf}}}\n\t\\caption{Interpretation in terms of loops on ribbon vertices of the labeled matrix model. The loops are naturally oriented from $A_\\alpha$ towards $A_\\alpha^\\dagger$. \\label{fig:InterpretationAsLoops}}\n\\end{figure}\n\nIn this model, the most general action reads\n\\begin{equation} \\label{matrixaction}\n\tS(\\{A_i,A_i^\\dagger\\}) = \\sum_{i=1}^\\tau \\tr A_i A_i^\\dagger + \\sum_{(n,\\sigma)} V_{n,\\sigma}(\\{A_i,A_i^\\dagger\\}),\n\\end{equation}\nwhere the sums typically run over a finite set of terms only. In the Feynman expansion, propagators connect ribbon vertices so as to form random (orientable) surfaces, as usual in matrix models. Moreover, each half-line of a ribbon vertex carries a (incoming or outgoing) line with index $i=1,\\dotsc,\\tau$, and these half-lines are connected by propagators to create loops. Each propagator between two vertices identifies their label $i=1,\\dotsc,\\tau$. As a result, there is a free sum per loop, giving rise to a factor $\\tau$, hence a factor $\\tau^L$ for the whole ribbon graph, $L$ being its number of loops.\n\nThe free energy of the model admits the following expansion,\n\\begin{equation} \\label{matrixF}\n\tF = N^2f = -\\ln \\int \\prod_{i=1}^{\\tau}\\diff A_i\\diff A_i^\\dagger \\exp\\left(-\\frac{N}{\\lambda}S(\\{A_i,A_i^\\dagger\\})\\right) = \\sum_{\\substack{\\text{connected}\\\\\\text{ribbon graphs G}}} \\frac{1}{s(G)}N^{2-2g(G)}\\lambda^{E-V}\\tau^{L},\n\\end{equation}\nwhere $s(G)$ is a symmetry factor, $E$ is the number of edges, $V$ of vertices, $F$ of faces and $L$ of loops. The $1\/N$ expansion of the free energy is, as usual, the genus expansion, where the genus $g$ is\n\\begin{equation}\n\\label{eq:Genus}\n\t2 - 2g(G) = F - E + V.\n\\end{equation}\n\nIt is worth noting that two kinds of configurations may happen:\n\\begin{itemize}\n\t\\item CPL configurations, where all loops are self and mutually avoiding. The name `CPL' comes from the Completely Packed Loop model. In what follows, we will see that these CPL configurations have a dominant role. They are generated by gluings of link patterns with no crossing, like on figure \\ref{subfig:NonCrossingLoop}, i.e. \\emph{planar} patterns up to rotations and reflection,\n\t\\item Configurations with crossings, where at least one loop crosses itself or another loop.\n\\end{itemize}\n\n\n\n\\subsection{Mapping to colored graphs and the degree expansion of tensor models}\n\nWe will map the Feynman graphs of our matrix model to a family of edge-colored graphs which we now introduce.\n\n\\subsubsection{Colored graphs and their degree}\n\n\\begin{definition}\nA $\\Delta$-colored graph is a regular bipartite graph (say, with black and white vertices) where each edge carries a color from the set $\\{1,\\dotsc,\\Delta\\}$ and such that the vertices have degree $\\Delta$ and the edges incidnet to a vertex all have distinct colors.\n\\end{definition}\n\nSome graphs are given in figure \\ref{fig:bubbles}. If $2p$ denotes the number of vertices in such a graph, the total number of edges is $\\Delta p$, and the number of edges of any given color is simply $p$. Furthermore, coloring gives an additional structure, which provides in particular a natural notion of faces. A \\emph{face with colors} $(a,b) \\in \\{1,\\dotsc,\\Delta\\}$ is a closed path with alternating colors $a$ and $b$. The total number of faces of a graph $G$ is $F(G) = \\sum_{a<b} F_{ab}$, where $F_{ab}$ is the number of faces with colors $a,b$.\n\n\\begin{figure}\n\\includegraphics[scale=.5]{tensobs.pdf}\n\\caption {\\label{fig:bubbles} Graphs on six vertices with 3 colors.}\n\\end{figure}\n\n\\begin{definition}\n Let $\\Delta\\geq 3$ be an integer, $G$ be a connected, $\\Delta$-colored graph with $2p$ vertices and $\\sigma$ be a cycle on $\\{1,\\dotsc,\\Delta\\}$. The jacket $J$ associated to $\\sigma$ is the connected ribbon graph which contains all the faces of colors $(\\sigma^q(1),\\sigma^{q+1}(1))$ for $q=0,\\dots,\\Delta-1$ in $G$. Therefore the number of faces in $J$ is given by $f_J = 2-2g_J + \\Delta p -2p$, where $g_J$ is the genus of $J$. We define the degree $\\omega(G)\\in{\\mathbb  N}$ of $G$ as the sum of the genera of the jackets.\n\\end{definition}\n\nOne gets an (over-)counting of faces by summing the formulas of the genus over all jackets, leading to the following theorem.\n\n\\begin{theorem} \\label{thm:degree}\nLet $G$ be a $\\Delta$-colored graph with $2p$ vertices. The number of faces and vertices are related to the degree as follows,\n\\begin{equation} \\label{degree formula}\nF - \\frac{(\\Delta-1)(\\Delta-2)}{2}\\,p = \\Delta-1 -\\frac2{(\\Delta-2)!}\\,\\omega(G).\n\\end{equation}\n\\end{theorem}\n\nFor a 3-colored graph, $2-2\\omega(G) = F-p = F - 3p +2p$, where $3p$ is the total number of lines. Therefore the degree then reduces to the well-known formula of the genus. The degree was introduced for 4-colored graphs in \\cite{Gur3}, and generalized in \\cite{Gur4}.\n\nThe colored graphs generated by a model of a single random tensor of rank $d$, $T_{a_1 \\dotsb a_d}$, are obtained from the following Feynman rules. A \\emph{bubble} is a connected $d$-colored graph, with colors $1,\\dotsc,d$, like in figure \\ref{fig:bubbles}. It generalizes the notion of ribbon vertex used in two dimensions \\cite{uncoloring}. Propagators then create edges which connect black vertices to white vertices. We assign the color $0$ to these edges. Each vertex thus receives an edge of color 0 in addition to the $d$ other edges of its bubble. Therefore the Feynman graphs are $(d+1)$-colored graphs with colors $\\{0,\\dotsc,d\\}$ built by gluing some bubbles, as in figure \\ref{fig:feynmangraph}.\n\n\\begin{figure}\n\\includegraphics[scale=.5]{tensobsgraph.pdf}\n\\caption{\\label{fig:feynmangraph} This is a $(3+1)$-colored graphs, obtained by connecting bubbles (in solid lines) via propagators (dashed lines, with the color 0).}\n\\end{figure}\n\nBubbles are colored graphs and therefore have a degree. Applying the degree formula \\eqref{degree formula} to a Feynman graph $G$ with $d+1$ colors and to all its bubbles $\\{B_\\rho\\}$, it comes\n\\begin{equation} \\label{graph degree}\n\\sum_{a=1}^d F_{0a} - (d-1)(p-b) = d - 2\\left[ \\frac{1}{(d-1)!}\\,\\omega(G) - \\frac{1}{(d-2)!}\\sum_\\rho \\omega(B_\\rho)\\right].\n\\end{equation}\nThe quantity into square brackets is a positive integer, which is zero if and only if $\\omega(B_\\rho)=0$ for each bubble together with $\\omega(G)=0$.\n\nThe large $N$ limit of tensor models is dominated by graphs which maximize the number of faces at fixed number of vertices and bubbles. They are therefore the graphs whose degrees vanish as well as the degrees of their bubbles. Such bubbles and Feynman graphs are called \\emph{melonic}.\n\\begin{definition}\n\tA closed melonic graph with $\\Delta$ colors is built by recursive insertions of $(\\Delta-1)$-dipoles, i.e. two vertices connected by $\\Delta-1$ lines inserted on any line, starting from the closed graph on two vertices. The $(\\Delta-1)$-dipole is represented in figure \\ref{fig:dipole}, as well as a melonic graph.\n\\end{definition}\n\n\\begin{theorem}\nThe colored graphs of degree $\\omega=0$ are the melonic graphs.\n\\end{theorem}\nThis theorem was proved in \\cite{critical-colored}.\n\n\\begin{figure}\n\\includegraphics[scale=0.5]{D-1Dipole.pdf}\n\\hspace{3cm}\n\\includegraphics[scale=.35]{Melonic2pt.pdf}\n\\caption{\\label{fig:dipole} On the left is a $(D-1)$-dipole with external color $c$. A 2-point (i.e. with two open half-edges) melonic graph on 4 colors is represented on the right. It is built by recursive insertions of 3-dipoles. A closed graph is obtained by connecting the two external open half-edges.}\n\\end{figure}\n\nWe say that a melonic graph has melons only on the colors $a_1,\\dotsc, a_k$ if it can be constructed by dipole insertions on edges of colors $a_1,\\dotsc,a_k$ only.\n\nEdge-colored graphs have been recently classified according to their degree in \\cite{GurauSchaeffer}.\n\n\n\\subsubsection{The corresponding tensor model}\n\nIn this section, we explain why our matrix model can be written as a tensor model. Note that not all multi-matrix models are tensor models in disguise, since the action of a tensor model for a tensor of size $N_1\\times \\dotsm \\times N_d$ is required to be $U(N_1)\\times \\dotsm U(N_d)$-invariant, as we explain.\n\n\nLet $T_{a_1 \\dotsb a_d}$ be the entries of a tensor $T$, with $a_i = 1,\\dotsc,N_i$ for $i=1,\\dotsc,d$, and $\\overline{T}_{a_1\\dotsb a_d}$ for its complex conjugate. The algebra of polynomials in the entries of $T$ and $\\overline{T}$ which are invariant under the fundamental action of $U(N_1)\\times \\dotsm \\times U(N_d)$, that is\n\\begin{equation} \\label{U(N)Transfo}\nT_{a_1\\dotsb a_d} \\mapsto \\sum_{b_1, \\dotsc, b_d} U^{(1)}_{a_1 b_1}\\,\\dotsm\\, U^{(d)}_{a_d b_d}\\ T_{b_1\\dotsb b_d},\n\\end{equation}\nwhere $U^{(1)},\\dotsc,U^{(d)}$ are $d$ independent unitary matrices (of different sizes), and similarly for the complex conjugate $\\overline{T}$, is generated by a set of polynomials labeled by bubbles. Recall that bubbles here are connected, edge-colored graphs with $d$ colors. The correspondence between invariant polynomials and bubbles works as follows. Let $B$ be a bubble. To each white (respectively black) vertex of $B$ we associate a $T$ (respectively $\\overline{T}$). An edge of color $c\\in\\{1,\\dotsc,d\\}$ between a white vertex and a black vertex means that the indices in the position $c$ of the corresponding $T$ and $\\overline{T}$ are identified and summed over (from $1$ to $N_c$). The polynomial labeled by $B$ can be written explicitly. Let $\\mathcal{W}$ be the set of white vertices, $\\mathcal{B}$ the set of black vertices and $\\mathcal{E}$ the set of edges. We identify an edge $e\\in\\mathcal{E}$ via the black vertex and the white vertex it connects and its color, thus $e = (w,b,c)$ with $w\\in\\mathcal{W}, b\\in\\mathcal{B}$. Then the polynomial, denoted $B(T,\\overline{T})$, is\n\\begin{equation}\nB(T, \\overline{T}) = \\prod_{k\\in\\mathcal{W}} \\sum_{i_1^k, \\dotsc, i_d^k}\\ \\prod_{l\\in\\mathcal{B}} \\sum_{j_1^l, \\dotsc, j_d^l}T_{i_1^k \\dotsb i_d^k}\\ \\overline{T}_{j_1^l \\dotsb j_d^l}\\ \\prod_{e=(w,b,c)\\in\\mathcal{E}}{\\delta_{i_c^w,j_c^b}}.\n\\end{equation}\n\nIn order to write the matrix action \\eqref{matrixaction}, defined from the set of matrices $\\{A_i,A^\\dagger_i\\}_{i=1,\\dotsc,\\tau}$, as a tensor action, we make the quite obvious ansatz,\n\\begin{equation}\nT_{a_1 a_2 a_3} = \\left(A_{a_3}\\right)_{a_1 a_2},\\qquad \\overline{T}_{a_1 a_2 a_3} = \\left(A^\\dagger_{a_3}\\right)_{a_2 a_1}.\n\\end{equation}\nIt remains to check that the matrix potential has the required invariance, that is that each $V_{n,\\sigma}(\\{A_i,A_i^\\dagger\\})$ defined in \\eqref{eq:TraceInvariant}, is actually an invariant polynomial labeled by a 3-colored bubble. The above definitions of $T$ and $\\overline{T}$ shows that the matrix element of a product $(A_a A_b^\\dagger)_{a_1 b_1}$ is exactly $\\sum_{a_2}T_{a_1 a_2 a} \\overline{T}_{b_1 a_2 b}$, i.e. a contraction along the second index. Similarly, $(A_b^\\dagger A_a)_{b_2 a_2}=\\sum_{a_1} \\overline{T}_{a_1 b_2 b} T_{a_1 a_2 a}$ is a contraction on the first index. Finally, $\\sum_{i=1}^\\tau (A_i)_{a_1 a_2} (A_i^\\dagger)_{b_2 b_1} = \\sum_{a_3} T_{a_1 a_2 a_3} \\overline{T}_{b_1 b_2 a_3}$ creates the contraction along the third index.\nThe action \\eqref{matrixaction} is thus really a sum over invariant polynomials labeled by bubbles, and the quadratic part is obviously $\\sum_{i=1}^\\tau \\tr A_iA_i^\\dagger = \\sum_{a_1,a_2,a_3} T_{a_1 a_2 a_3} \\overline{T}_{a_1 a_2 a_3}$.\n\n\\begin{figure}\n\t\\center\n    \\includegraphics[height=4cm]{LoopToTensor1.pdf}\n    \\hspace{0.5cm}\n    \\raisebox{2cm}{$\\to$}\n    \\hspace{0.5cm}\n    \\includegraphics[height=4cm]{LoopToTensor2.pdf}\n    \\hspace{0.5cm}\n    \\raisebox{2cm}{$\\to$}\n    \\hspace{0.5cm}\n    \\includegraphics[height=4cm]{LoopToTensor3.pdf}\n    \\caption{The map from ribbon vertices with loop lines to 3-colored bubbles. \\label{fig:LoopToTensor}}\n    \\end{figure}\n\nIt is also interesting to proceed graphically. There is a straightforward mapping between the link patterns, i.e. the ribbon vertices of the matrix model, and bubbles with 3 colors and a single face of colors $(1,2)$, as shown in figure \\ref{fig:LoopToTensor}. One draws an (unknotted) circle around the ribbon vertex, such that the intersections between the circle and the loop lines (labeled $1,\\dotsc, \\tau$) give rise to the vertices of the bubble (say an outgoing line gives a white vertex, and an incoming line gives a black vertex). The segments on the circle are given alternating colors 1 and 2. There are two possible choices to do that, and we choose the color 1 when going clockwise from a black to a white vertex. The loop lines which cross the ribbon vertex are then given the color 3, and they indeed connect white to black bubble vertices.\n\nNotice that our ribbon vertices do not generate all 3-colored bubbles, but only those with a single face of colors $(1,2)$. This is because they come from single trace invariants in the matrix model.\n\nConversely, given a 3-colored bubble with a single face of colors $(1,2)$, one gets a unique ribbon vertex with loop lines. The ribbon vertex is determined by the face with colors $(1,2)$: there is one open ribbon line per vertex of the bubble. Each line of color 3 connects a black and a white bubble vertex and corresponds to an oriented loop line going through the ribbon vertex.\n\nThe planar patterns, used to build CPL configurations, are exactly the bubbles which are melonic with melonic insertions on the colors 1 and 2 only. This set of bubbles has been studied in \\cite{SDEs} where it is shown to be in one-to-one correspondence with non-crossing partitions of $\\{1,\\dotsc,p\\}$ up to rotations and reflections.\n\nAs an example, those on 4 vertices correspond to the terms $V_{n,\\sigma}$ with $n=2$. There are only two permutations on two elements, hence two 3-colored graphs with a single face of colors $(1,2)$ which we denote $B_1$ and $B_2$, and corresponding to the identity $\\sigma=(1)(2)$ and the transposition $\\sigma=(12)$ (in cycle notation). Graphically, we have\n\\begin{equation} \\label{QuarticBubbles}\n\\begin{aligned}\n&V_{n=2,\\sigma=(1)(2)} = \\sum_{i,j=1}^\\tau \\tr( A_i A_i^\\dagger A_j A_j^\\dagger) = \\begin{array}{c}\\includegraphics[scale=.25]{4PointLinkPattern1.pdf}\\end{array} = \\begin{array}{c}\\includegraphics[scale=.45]{4PointBubbleColor1.pdf}\\end{array},\\\\\n&V_{n=2,\\sigma=(12)} = \\sum_{i,j=1}^\\tau \\tr( A_i A_j^\\dagger A_j A_i^\\dagger) = \\begin{array}{c}\\includegraphics[scale=.25]{4PointLinkPattern2.pdf}\\end{array} = \\begin{array}{c}\\includegraphics[scale=.45]{4PointBubbleColor2.pdf}\\end{array}.\n\\end{aligned}\n\\end{equation}\n\nTherefore, we get a tensor model whose free energy expansion is \\eqref{matrixF}. This implies that if we could solve exactly the matrix models defined by the potentials \\eqref{eq:TraceInvariant}, we would in fact get the exact solution of some rank-three tensor models, with tensors of size $N\\times N\\times \\tau$ (for which the $1\/N$ expansion is organized according to the genus of subgraphs with colors 0,1,2 while the degree partially controls the $1\/\\tau$ expansion, as we are going to see)\n\n\\subsubsection{The mapping}\n\nThe fact that the initial matrix model fits in the frame of tensor models suggests the existence of a bijection between the random surfaces decorated with loops and the $(3+1)$-colored Feynman graphs of the tensor model. From the correspondence between the ribbon vertices $V_{n,\\sigma}$ and the bubbles established above, this bijection is quite trivial: only propagators have to be added. The edges of the ribbon graphs are simply mapped to edges of color 0, which indeed connect black to white vertices.\n\nThis allows to evaluate the number of faces, edges, vertices and loops of the matrix Feynman graphs in terms of faces, vertices and bubbles of the corresponding 4-colored graphs, as summarized in the Table \\ref{tab:MatrixToTensorCorrespondence}.\n\n\\begin{table}\n\t\\begin{center}\n\t\t\\begin{tabular}{|c c c|}\n\t\t\t\\hline\n\t\t\t\tLoops model graphs  & \\vline & 4-colored graphs \\\\\n\t\t\t\\hline\n\t\t\t\tFaces: $F$\t\t& $\\to$ & Faces of colors (0,1), (0,2): $F_{01}+F_{02}$\\\\\n\t\t\t\tEdges: $E$\t\t& $\\to$ & Vertexes: $p$\\\\\n\t\t\t\tVertexes: $V$\t& $\\to$ & Bubbles of color (1,2,3): $b$\\\\\n\t\t\t\tLoops: $L$\t\t& $\\to$ & Faces of colors (0,3): $F_{03}$\\\\\n\t\t\t\\hline\n\t\t\\end{tabular}\n\t\\end{center}\n\t\\caption{The correspondence between the characteristics of the random surfaces with loops and the characteristics of the corresponding colored graphs. \\label{tab:MatrixToTensorCorrespondence}}\n\\end{table}\n\nThe slightly non-trivial is to find a relation between the genus of the random surfaces and the degree of the colored graphs. Being 3-colored, the bubbles themselves can be interpreted as ribbon graphs. This is done by clockwise-ordering the edges of colors 1,2,3 around each white vertex, and counterclockwise-ordering them around each black vertex (then thickening the edges if desired). The degree of a bubble is then the genus of that discrete surface, $\\omega(B) = g(B)$, and it is a measure of the amount of crossing of the loop lines at a given ribbon vertex. Obviously, the melonic bubbles are the planar ones.\n\nUsing this correspondence, the degree of a $(3+1)$-colored graph, equation \\eqref{graph degree}, reads\n\\begin{equation}\n\tF + L - 2(E - V) = 3 - \\omega(G) +2\\sum_\\rho g(B_\\rho),\n\\end{equation}\nSince the genus of the ribbon graph fixes the number of faces at fixed number of edges and vertices, through equation \\eqref{eq:Genus}, the number of loops $L$ can be extracted as a function of the degree, of the genus of the subgraph of colors (0,1,2), the genera of the bubbles, and of the numbers of edges and vertices\n\\begin{equation}\n\\label{eq:LoopCounting}\n\tL = E - V + 1 + 2g(G) - \\omega(G) + 2\\sum_\\rho g(B_\\rho).\n\\end{equation}\nThis \\emph{loop counting formula} is the main outcome of the mapping. To complete the loop counting, we prove that the quantity $\\omega(G) - 2\\sum_\\rho g(B_\\rho) - 2g(G) \\geq 0$, and identify the configurations for which it vanishes. We use the obvious bound $L<F$, which together with \\eqref{eq:LoopCounting} implies that\n\\begin{equation}\n\\omega(G) - 2\\sum_\\rho g(B_\\rho) - 2g(G) > -1+2g(G).\n\\end{equation}\nThis means that if $g(G)\\geq 1$, then the left hand side is strictly positive. Only the case $g(G)=0$ remains. Then we find\n\\begin{equation}\n\\omega(G) - 2\\sum_\\rho g(B_\\rho) = \\frac13\\,\\omega(G) +\\frac23 \\bigl(\\omega(G) - 3\\sum_\\rho g(B_\\rho)\\bigr).\n\\end{equation}\nIn addition to $\\omega(G)\\geq 0$ as part of the Theorem \\ref{thm:degree}, it can be proved that $\\omega \\geq 3\\sum_\\rho g(B_\\rho)$, which is a particular case of the Lemma 7 in \\cite{tree-algebra}.\n\nWe summarize the consequences of this analysis in the following proposition.\n\\begin{proposition} \\label{prop:LoopCounting}\nThe number of loops on a random discrete surface $G$ decorated with oriented loops visiting all edges once (and such that the orientations on the edges around each vertex alternate), made of the gluing of $V$ link patterns $\\{B_\\rho\\}_\\rho$ via $E$ edges satisfies\n\\begin{equation*}\n\tL = 1 + E - V + 2g(G) - \\omega(G) + 2\\sum_\\rho g(B_\\rho),\n\\end{equation*}\nwhere $\\omega(G)$ is the degree of the corresponding $(3+1)$-colored graph. Furthermore\n\\begin{itemize}\n\\item The graphs of degree zero are those which maximize the number of loops at fixed number of edges and vertices (and the link pattern at each vertex is planar).\n\\item They are planar, $\\omega(G)=0\\ \\Rightarrow\\ g(G)=0$.\n\\item At fixed genus, fixed numbers of edges and vertices of each allowed type, the degree measures how far $G$ is from the configuration which maximizes the number of loops.\n\\end{itemize}\n\\end{proposition}\n\n\nWe also note that the formula \\eqref{eq:LoopCounting} can be used to get a bound on the maximal degree of the colored graphs built from 3-colored bubbles with a single face of colors 1,2. Since $L\\geq1$, it comes (using the notation of colored graphs)\n\\begin{equation}\n\\omega(G)\\leq p-b +2g(G) + 2\\sum_\\rho g(B_\\rho).\n\\end{equation}\n\nLet us illustrate a bit the melonic sector in the special case where only the terms with $n=2$ are kept in the action \\eqref{matrixaction}. This leaves only the two link patterns, or the two 3-colored graphs (both planar), of the equation \\eqref{QuarticBubbles}. In this model, all melonic insertions come from inserting appropriately one of these two bubbles on any edge of color 0. Assuming the edge of color 0 correponds to a ribbon edge with a loop going up north, the two possibilities are\n\\begin{equation}\n\\begin{array}{c}\\includegraphics[scale=.5]{MelonicInsertionBubble1.pdf} \\end{array} = \\begin{array}{c}\\includegraphics[scale=.25]{MelonicInsertion1.pdf} \\end{array}\\qquad \\text{and}\\qquad \n\\begin{array}{c}\\includegraphics[scale=.5]{MelonicInsertionBubble2.pdf} \\end{array} = \\begin{array}{c}\\includegraphics[scale=.25]{MelonicInsertion2.pdf} \\end{array}.\n\\end{equation}\nWe see that a melonic insertion adds one loop, one ribbon vertex and one ribbon edge (and does not change the genus). By contrast, a non-melonic insertion on an edge of color 0 would be\n\\begin{equation}\n\\begin{array}{c}\\includegraphics[scale=.5]{NonMelonicInsertionBubble2.pdf} \\end{array} = \\begin{array}{c}\\includegraphics[scale=.25]{NonMelonicInsertion2.pdf} \\end{array}\n\\end{equation}\nwhich would not create a new loop.\n\nIn the section which follows, we solve explicitly the melonic sector ($\\omega=0$) and further use the recent classification of colored graphs \\cite{GurauSchaeffer} to organize the $1\/\\tau$ expansion.\n\n\\subsection{Scaling limits} \\label{sec:ScalingLimits}\n\nUsing the counting of loops obtained in equation \\eqref{eq:LoopCounting}, the free energy writes\n\\begin{equation}\n\tN^2 f = \\sum_{\\substack{\\text{connected}\\\\ \\text{ribbon graphs}}} \\left(\\frac{N}{\\tau}\\right)^{2-2g(G)}\\ \\left(\\lambda \\tau\\right)^{E-V}\\ \\tau^{3-\\omega(G)}\\ \\frac1{s(G)}.\n\\end{equation}\nThis shows three contributions to the exponent of $\\tau$. Those with the genus and with $E-V$ are not relevant since these quantities are controled by $N$ and $\\lambda$. Consequently, the degree $\\omega$ labels the expansion in the number of loops.\n\n\\subsubsection{Large $\\tau$ limit}\n\nFurthermore, it is possible to build a scaling limit which projects the loop model onto the melonic sector. To project onto the melonic family, the limit $\\tau \\to \\infty$ is required. To ensure the limit is well-defined, we must scale $\\lambda$ with $\\tau$ as follows: $\\lambda\\tau = \\tilde{\\lambda}$, where $\\tilde{\\lambda}$ is kept finite. We also scale $N$ with $\\tau$, and for convenience set their ratio to 1, $\\tau = N$. The rescaled free energy $\\tilde{f} = \\frac{f}{\\tau} =  \\frac{f}{N}$ then reads\n\\begin{equation} \\label{FreeEnergyTensor}\n\tN^3 \\tilde{f} = N^2 f = \\sum_{\\substack{\\text{4-colored}\\\\\\text{connected graphs}}} N^{3-\\omega(G)} \\tilde{\\lambda}^{E-V} \\frac1{s(G)},\n\\end{equation}\nIt is finite in the large $N$, large $\\tau$ limit, and its leading order in the $1\/N$ expansion consists of melonic graphs.\n\nIt is interesting to perform the rescaling directly in the matrix integral (and setting $\\tau$ to $N$ everywhere),\n\\begin{equation} \\label{tensorscaling}\n\tN^3 \\tilde{f} = -\\ln \\int \\prod_{i=1}^N \\diff A_i \\diff A_i^\\dagger \\exp\\left( -\\frac{N^2}{\\tilde{\\lambda}} S(\\{A_i,A_i^\\dagger\\})\\right)\n\\end{equation}\nThe factor $N^2$ in front of the action is exactly the standard scaling for a random tensor of rank-three and size $N^3$. This is natural in this scaling limit, since there are $\\tau=N$ matrices, each of size $N\\times N$.\n\nWe can write the solution quite explicitly in the large $N$ limit \\cite{universality, uncoloring}. Indeed, large random tensors in a unitary-invariant distribution (invariant under \\eqref{U(N)Transfo}) are subjected to a universality theorem, stating that all large $N$ expectations are Gaussian, with the covariance being the large $N$ 2-point function. For an invariant polynomial $B(T,\\overline{T})$ of degree $p_B$ in $T$, this gives\n\\begin{equation}\n\\frac1N\\,\\langle B(T,\\overline{T})\\rangle = N^{-\\omega^*(B)} \\Bigl(C_B\\ G_2^{p_B}+\\mathcal{O}(1\/N)\\Bigr),\n\\end{equation}\nwhere $G_2 = \\lim_{N\\to\\infty} \\langle T\\cdot\\overline{T}\\rangle\/N = \\lim_{N\\to\\infty} \\langle \\sum_{i=1}^N \\tr (A_iA_i^\\dagger)\\rangle\/N$. $\\omega^*(B) \\geq 0$ and vanishes if and only if $B$ is melonic, meaning that melonic bubbles of the action are the only relevant ones at large $N$. Therefore, only the terms of the type $V_{n,\\sigma}$ in \\eqref{matrixaction} with $\\sigma$ corresponding to a planar link pattern survive. Moreover, $C_B$ is the leading order number of Wick contractions and for a melonic bubble turns out to be 1 only. This way,\n\\begin{equation}\n\\frac1N\\,\\langle V_{n,\\sigma \\rm{ planar}}(\\{A_i,A_i^\\dagger\\}) \\rangle = G_2^n.\n\\end{equation}\nAll large $N$ calculations thus boil down to the leading order 2-point function. It is found thanks to the Schwinger-Dyson equation\n\\begin{equation}\n\\sum_{a_1,a_2,i=1,\\dotsc,N} \\int \\prod_i dA_i\\,dA_i^\\dagger\\ \\frac{\\partial}{\\partial (A_i)_{a_1 a_2}} \\Bigl((A_i)_{a_1 a_2}\\ e^{-N^2 S(\\{A_i,A_i^\\dagger\\})\/\\tilde{\\lambda}}\\Bigr) = 0,\n\\end{equation}\nwhich after making the derivatives explicit and using the universality to close the system leads to the equation\n\\begin{equation}\n\\tilde{\\lambda} - G_2 - \\sum_{n, \\text{planar }\\sigma} n\\,G_2^n = 0,\n\\end{equation}\nwhich is polynomial as long as the action contains a finite collection of planar link patterns. It is a standard result that one then gets a square-root singularity for $G_2$ when approaching the critical value of $\\tilde{\\lambda}$, i.e. $G_2 \\sim (\\tilde{\\lambda}_c - \\tilde{\\lambda})^{1\/2}$. Therefore the singular part of the free energy behaves as $\\tilde{f} \\sim (\\tilde{\\lambda}_c - \\tilde{\\lambda})^{2-\\gamma}$ with $\\gamma=1\/2$. The regime where $\\tilde{\\lambda}$ is close to $\\tilde{\\lambda}_c$ is called the \\emph{continuum limit}.\n\n\\subsubsection{The $1\/\\tau$ expansion}\n\nThanks to the $1\/N$ expansion, we can work at fixed genus. We can then take advantage of the recent classification of edge-colored graphs according to their degree \\cite{GurauSchaeffer} to organize the the $1\/\\tau$ expansion.\n\nThis classification relies on the fact that only 2-point subgraphs and 4-point subgraphs can generate infinite family of graphs of constant degree\\footnote{We remind the reader that tensor model at large $N$ are dominated by Gaussian contributions, i.e. 2-point functions, while the first $1\/N$ correction only involves 2-point and 4-point functions, \\cite{NLO, DSSD}.}. Once replaced by ``reduced'' 2-point and 4-point functions, there exists only a finite number of graphs of given degree. Those reduced graphs are called \\emph{schemes} in \\cite{GurauSchaeffer}.\n\nIn the following, we restrict the potential to $n=2$, leaving only room for the bubbles $B_1, B_2$ introduced in \\eqref{QuarticBubbles}. (This reduces the source of 4-point functions; otherwise we would have for instance a 6-point bubble with an arbitrary 2-point function between two of its vertices also play the role of an effective 4-point bubble, and so on\\footnote{The reference \\cite{GurauSchaeffer} studies the whole set of colored graphs, which is somewhat simpler than focusing on the set of graphs generated by a given but arbitrary set of bubbles, except if this set is simple enough. This is the case for graphs built from quartic interactions ($n=2$ here), and this is the choice made in \\cite{DSQuartic}.}). We recall the loop counting formula specialized to this case (i.e. with $E=2V$ and planar link patterns),\n\\begin{equation}\nL = V +(3-\\omega(G))-(2-2g(G)).\n\\end{equation}\nTherefore the degree measures how far a loop configuration is from the one which maximizes the number of loops for a fixed number of vertices and a fixed genus.\n\nThe Schwinger-Dyson equation on $G_2$ simply reads $\\lambda - G_2 - 4 G_2^2=0$, hence\n\\begin{equation}\nG_2 = \\frac{\\sqrt{1+16\\lambda}-1}{8},\n\\end{equation}\nwith $G_2\\sim \\lambda$ for small $\\lambda$. The critical point which defines the continuum limit is $\\lambda_c=-1\/16$.\n\nGiven an arbitrary $(3+1)$-colored graph built from the bubbles of type $B_1, B_2$ glued along edges of color 0, one first reduces the purely melonic 2-point subgraphs, as the one in figure \\ref{fig:dipole} (this is done recursively by identifying 2-cut edges; the order does not matter, as proved in \\cite{GurauSchaeffer}). Arbitrary melonic insertions do not change the degree, and so does this reduction. This way, we have to consider only \\emph{melon-free} graphs, while all melonic insertions are completely accounted for by simply using $G_2(\\lambda)$ as the new propagator, i.e. $G_2$ becomes the weight associated to edges of color 0.\n\nSecond, one identifies \\emph{chains}. In our model, chains are simply sequences of quartic bubbles glued in a chain-like manner,\n\\begin{equation*}\n\\begin{array}{c}\\includegraphics[scale=.45]{GenericChain.pdf}\\end{array}\n\\end{equation*}\nwhere $c,c'$ are 1 and\/or 2. Those chains have to be maximal, so they have 4 half-edges of color 0 as external edges. There are two types of chains.\n\\begin{itemize}\n\\item Those built from a sequence of a single bubble, either $B_1$ or $B_2$, and called \\emph{unbroken chains}. In terms of ribbon graphs and loops, an unbroken chain takes the form\n\\begin{equation*}\n\\begin{array}{c}\\includegraphics[scale=.4]{UnbrokenRibbonChain.pdf}\\end{array}\n\\end{equation*}\nwith two possible orientations. It is clearly planar.\n\nThe generating function of unbroken chains with a weight $-2\/\\lambda$ on each bubble and $G_2(\\lambda)$ on each edge of color 0 is\n\\begin{equation}\nC_u(\\lambda) = \\sum_{n\\geq 1} \\frac{(-2)^n\\,(G_2(\\lambda))^{2(n-1)}}{\\lambda^n} = -\\frac{2}{\\lambda + 2(G_2(\\lambda))^2}.\n\\end{equation}\n\n\\item Those which contain both bubbles $B_1, B_2$ and called \\emph{broken chains}. The generating function of broken chains is found by considering the one of arbitrary chains, with arbitrary 4-point bubbles (bubbles hence receiving the weight $-2\\times 2\/\\lambda$, to account for the two possible types of bubbles at each time), and substracting the generating functions of the two unbroken chains,\n\\begin{equation}\n\\begin{aligned}\nC_b(\\lambda) &= \\sum_{n\\geq 1} \\frac{(-4)^n\\,(G_2(\\lambda))^{2(n-1)}}{\\lambda^n} -2C_u(\\lambda) = -\\frac{4}{\\lambda + 4(G_2(\\lambda))^2} + \\frac{4}{\\lambda + 2(G_2(\\lambda))^2} \\\\\n&= \\frac{8(G_2(\\lambda))^2}{\\lambda + 2(G_2(\\lambda))^2}\\ \\frac1{\\lambda + 4  (G_2(\\lambda))^2}.\n\\end{aligned}\n\\end{equation}\n\\end{itemize}\n\nChains can be arbitrarily long with no change in the degree of the melon-free graphs. We have to make sure that does not change the genus of the random surface neither. It is clear for the unbroken chains. As for the broken ones, if a bubble $B_i$ is inserted somewhere in an unbroken subchain of type $i$, including at an end, this does not change anything. So we are left with the case where a bubble $B_1$, for instance, is inserted in a subchain of bubbles $B_2$. Because the full chain is broken, there is another bubble $B_1$ somewhere, say on the right of the chain,\n\\begin{equation}\n\\begin{array}{c}\\includegraphics[scale=.5]{BrokenChain.pdf}\\end{array}\n\\end{equation}\nwhere we have marked the added bubble in a bounding box. We have to evaluate the variation of the genus of the subgraph with colors 0,1,2 between before and after the insertion. Clearly, the number of ribbon edges changes by 2 and the number of ribbon vertices (i.e. bubbles) by 1. Therefore $\\Delta (E-V) = 1$. To find the variation of the number of faces, it is more convenient to use the representation as an edge-colored graph rather than as a ribbon graph. The number of faces of the surface is $F=F_{01}+F_{02}$. The face of colors $(0,2)$ which arrives from the top left leaves on the bottom left, and that was already the case before the insertion because the chain is broken. There is however a new face of colors $(0,2)$, which goes around the bubble of type $B_2$ on the right of the bounding box. Therefore $\\Delta F_{02}=1$. Moreover, there is no new face of colors $(0,1)$, so $\\Delta F_{01}=0$. The variation of the genus is thus $-2\\Delta g= \\Delta (F-E+V) = 0$, meaning that the genus is independent of the length of the chain.\n\nAs a consequence, it is safe to simply contract chains into ``boxes'' called broken or unbroken chain-vertices with two incident edges of color 0 on one side of the box and two on the opposite side, and weight them with the generating functions $C_b(\\lambda)$ or $C_u(\\lambda)$ respectively. We obtain this way the set of \\emph{schemes}, i.e. melon-free graphs with chain-vertices representing arbitrarily long chains. The key result of \\cite{GurauSchaeffer} is then the finiteness of the number of schemes at any fixed degree.\n\nLet $s$ be a scheme with $p\\geq2$ black vertices, $\\alpha$ unbroken chain-vertices and $\\beta$ broken chain-vertices. Then the generating function of colored graphs, rooted on an edge of color 0, with scheme $s$ is\n\\begin{equation}\nG_s(\\lambda) = (G_2(\\lambda))^{p}\\,(C_u(\\lambda))^\\alpha\\,(C_b(\\lambda))^\\beta.\n\\end{equation}\nTo get the free energy of the model (or rather its 2-point function), one substitutes $\\lambda \\tau$ instead of $\\lambda$. Then the 2-point function at genus $g$ has the expansion\n\\begin{equation}\nG_2^{(g)}(\\lambda) = G_2(\\lambda\\tau)\\,\\delta_{g,0} + \\sum_{\\omega\\geq1} \\tau^{3-\\omega} \\sum_{\\substack{\\text{schemes $s$} \\\\ \\omega(s)=\\omega, g(s)=g}}  (-2)^\\alpha 8^\\beta \\frac{(G_2(\\lambda\\tau))^{p+2\\beta+1}}{(\\lambda\\tau + 2(G_2(\\lambda\\tau))^2)^{\\alpha+\\beta}\\ (\\lambda\\tau+4(G_2(\\lambda\\tau))^2)^\\beta},\n\\end{equation}\nwhere the sum over schemes at fixed genus and degree is finite. Here we have isolated the purely melonic part, which corresponds to the empty scheme with no vertices.\n\nOf course, to complete the analysis, it is necessary to know how the degree of a scheme behaves as a function of the number of chains. Again, this was done in \\cite{GurauSchaeffer}. If a chain-vertex is separating, i.e. if after its removal and after connecting the half-edges of color 0 together on each side of the chain vertex we get two connected components, then the degree of the graph is simply the sum of the degrees of both connected components. For an non-separating, unbroken chain of type $i$, such that there are two different faces of colors $(0,i)$ going through the chain, the degree is the degree of the graph with the chain removed plus one, meaning such a chain contribute to a factor $1\/\\tau$. In all other situations, a chain brings in a factor $1\/\\tau^3$.\n\nA corollary of this analysis is the double scaling regime. First notice that the critical point defining the continuum limit is $\\lambda_\\tau = -1\/(16\\tau)$. The generating function $C_u(\\lambda\\tau)$ of unbroken chains is finite at criticality. However, the generating function of broken chains is singular. Indeed, its denominator contains\n\\begin{equation}\n\\lambda\\tau + 4 (G_2(\\lambda\\tau))^2 = -\\lambda\\tau\\ G_2(\\lambda\\tau)\\ \\sqrt{1-\\lambda\/\\lambda_\\tau}.\n\\end{equation}\nTherefore a scheme with $\\beta$ broken chains diverges as $(1-\\lambda\/\\lambda_\\tau)^{\\beta\/2}$ at criticality. The idea of the double scaling limit is to pick up the terms of arbitrary degree which maximize the divergence. \n\nThe answers provided in \\cite{GurauSchaeffer} for generic edge-colored graphs and in \\cite{DSQuartic} in the special case of quartic melonic interactions coincide. We consider a rooted, binary tree with a single loop attached to every leaf. For each such tree, we get an edge-colored graph of the model by replacing the edges with broken chains which are glued together at the vertices in the obvious way, while the loops on the leaves represent unbroken chains. Those loops break melonicity and were called cherries in \\cite{DSQuartic}. Since all broken chains are separating, the degree is simply the number of unbroken ones, $\\omega=n$. Moreover, the binary-tree structure of broken chains is the way to maximize the divergence at fixed number of cherries. The number of broken chains grows linearly as two times the number of cherries, so each such graph receives a factor $\\tau^{-n}\/\\sqrt{1-\\lambda\/\\lambda_\\tau}^{2n}$. This shows that the optimal balance is reached by introducing $x = \\tau (1-\\lambda\/\\lambda_\\tau)$ and sending $\\tau\\to\\infty, \\lambda\\to\\lambda_\\tau$ while $x$ is kept fixed.\n\nUsing the same technique as in \\cite{GurauSchaeffer} to extract the behavior of the degree as a function of the chain-vertices, the graphs of the double-scaling regime can be shown to be planar. Indeed, one breaks up the cherry trees into isolated vertices, edges which represent broken chains, and loops attached to the leaves and analyze the genus of each piece (found to vanish in all cases).\n\nThe resummation of this family is quite simple to perform\\footnote{Compared to \\cite{DSQuartic}, one has to set $D=3$ when $D$ enters the degree, but $D=2$ in the equations for criticality since we have only two quartic bubbles and not three.}. It has a square-root singularity in $x$ that is likely to lead to a branched polymer phase.\n\n\\subsection{Another bijection and the intermediate field method} \\label{sec:Bimaps}\n\nWe have shown a bijection between the ribbon graphs with oriented loops generated by the matrix model \\eqref{matrixF} and the edge-colored graphs of tensor models whose interactions are labeled by bubbles with a single cycle of colors $(1,2)$. In the case the potential is restricted to the two quartic terms in equation \\eqref{QuarticBubbles}, there is a bijection between the graphs of the tensor model and a family of maps. It was first observed in \\cite{BeyondPert} in quartic melonic models, and generalized to tensor models with arbitrary quartic interactions in \\cite{GenericQuartic}. Algebraically, this bijection corresponds to the intermediate field method. Here, we first present the bijection, then the corresponding intermediate field theory.\n\nNotice that the bubbles used in the quartic case, equation \\eqref{QuarticBubbles}, have four vertices with a canonical partition in pairs. A canonical pair of vertices consists of those connected by a multiple edge (here two edges including the one of color 3). The two pairs are connected by two edges of color $i$ (here 1 or 2) and we have labeled the bubble by that color ($B_1$ and $B_2$). We are now going to represent $B_i$ as an edge of color $i$, as if the canonical pairs of vertices were contracted to single points.\n\nFurthermore, each white\/black vertex has an incident edge of color 0. This means that every edge of color 0 belongs to a single closed cycle made of alternating edges of color 0 and multiple edges. We map those cycles to vertices, while preserving the cyclic ordering of the bubbles. In our model, those vertices correspond to the faces of color $(0,3)$, i.e. the loops. Through this process, we represent every colored graph as a map, since the ordering around each vertex matters, with edges of colors 1 or 2.\n\nFor such a map, there are two canonical submaps, ${\\cal M}_1$ and ${\\cal M}_2$, which respectively correspond to the submaps  containing only the edges of color 1 and 2. The faces of colors $(0,i)$ in the Feynman graph of the tensor model are mapped to the faces of the map ${\\cal M}_i$. Moreover, the bubbles are mapped to edges and the loops to vertices.\n\nRemarkably, many problems in tensor models become quite simple when formulated in this way. For instance, the dominance of the melonic sector: the question is how to maximize the number of loops at fixed number of ribbon vertices. After the mapping, it becomes how to maximize the number of vertices at fixed number of edges; the answer clearly being trees, which indeed are the representatives of the melonic edge-colored graphs. Further, the double scaling regime presented in the previous section is dominated by Motzkin trees (i.e. trees whose nodes can have zero, one or two children), such that there always is at least one change of color between two vertices of degree three, and with loops of arbitrary length and of a fixed color attached to the leaves.\n\nSince we exhibited a bijection to maps, there may be a matrix model which generates them with the correct amplitudes. This works through the intermediate field method which transforms the initial matrix model \\eqref{matrixF} with $n=2$ into a two-Hermitian-matrix model. Here it is useful to introduce independent coupling constants $\\lambda_1, \\lambda_2$ and consider\n\\begin{equation}\nZ_{N,\\tau}(\\lambda_1,\\lambda_2) = \\int \\prod_{i=1}^\\tau dA_i\\,dA_i^\\dagger\\ e^{-N\\left(\\sum_i \\tr A_iA_i^\\dagger + \\lambda_1 \\tr \\sum_{i,j} A_i A_i^\\dagger A_j A_j^\\dagger + \\lambda_2 \\tr \\sum_{i,j} A_i A_j^\\dagger A_j A_i^\\dagger\\right)}.\n\\end{equation}\nWe can re-write each quartic term via a Gaussian integral over an auxiliary, Hermitian matrix,\n\\begin{equation}\ne^{-N\\lambda_1\\tr \\sum_{i,j} A_i A_i^\\dagger A_j A_j^\\dagger} = \\int dM_1\\ e^{-N \\tr M_1^2 -2iN\\sqrt{\\lambda_1} \\tr \\sum_i M_1 A_i A_i^\\dagger},\n\\end{equation}\nup to irrelevant constants, and similarly for the other quartic term. The partition function is then\n\\begin{equation}\nZ_{N,\\tau}(\\lambda_1,\\lambda_2) = \\int dM_1\\,dM_2\\,\\prod_{i=1}^\\tau dA_i\\,dA_i^\\dagger\\ e^{-N\\tr(M_1^2+M_2^2+\\sum_i A_iA_i^\\dagger) - 2iN \\sqrt{\\lambda_1} \\tr M_1 \\sum_i A_i A_i^\\dagger - 2iN\\sqrt{\\lambda_2} \\tr M_2 \\sum_i A_i^\\dagger A_i}.\n\\end{equation}\nPerforming the Gaussian integral on the $\\tau$ matrices $A_i$, one gets,\n\\begin{equation} \\label{TwoMatrixModel}\nZ_{N,\\tau}(\\lambda_1,\\lambda_2) = \\int dM_1\\,dM_2\\ e^{-N\\tr(M_1^2+M_2^2) - \\tau \\tr \\ln \\left(\\mathbb{I}\\otimes \\mathbb{I} - 2i\\sqrt{\\lambda_1} M_1\\otimes \\mathbb{I} -2i\\sqrt{\\lambda_2}\\mathbb{I}\\otimes M_2\\right)}.\n\\end{equation}\nIf the logarithm is expanded onto powers of $M_1, M_2$, it is clear that we have a generating function for the maps described above.\n\nWe are not aware of a solution of this model for arbitrary $\\lambda_1,\\lambda_2,\\tau$ in the literature. Nevertheless, setting $\\lambda_2=0$, one gets\n\\begin{equation} \\label{SingleColorQuartic}\nZ_{N,\\tau}(\\lambda_1,0) = \\int \\prod_{i=1}^\\tau dA_i\\,dA_i^\\dagger\\ e^{-N\\left(\\sum_i \\tr A_iA_i^\\dagger + \\lambda_1 \\tr \\sum_{i,j} A_i A_i^\\dagger A_j A_j^\\dagger\\right)} = \\int dM\\ e^{-N\\tr M^2 - N\\tau \\tr \\ln \\left(\\mathbb{I} - 2i\\sqrt{\\lambda_1} M_1\\right)}.\n\\end{equation}\nOne recognizes here the generalized Penner model with a quadratic potential. We refer to \\cite{PennerAllGenera} for an analysis with an arbitrary polynomial, at all genera, using the loop equations. In the case of the quadratic potential, the Penner model with coupling $\\tau$ on the logarithmic part is equivalent to the quartic matrix model with $\\tau$ matrices, which can actually be solved directly. For instance, a rectangular matrix of size $N\\times \\tau N$ can be formed, $C_{a_1 \\alpha} = (A_i)_{a_1 a_2}$ with the ``fat'' index $\\alpha=(a_2,i)$. Then the action is simply $\\tr CC^\\dagger + \\lambda_1 \\tr (CC^\\dagger)^2$, and the partition function can be evaluated using techniques developed for rectangular matrix models, like the orthogonal polynomials in \\cite{MyersTripleScaling} (and see \\cite{toy-doublescaling} for an application to tensor models). \n\n\nAs far as we know, the quartic case with $\\lambda_2=0$ is the only situation where a model of the generic class we have introduced has been solved. However, it should be emphasized that already the quartic model with $\\lambda_2\\neq 0$ is very different. In particular, for $\\lambda_2=0$, the distinction between broken and unbroken chains disappears and all chains become singular at the critical point.\n\n\\section*{Conclusion}\n\nThe motivation of this article is to connect tensor models and its challenges to the more familiar framework of matrix models\n\nWith this in mind, the present article has been devoted to a novel presentation of random tensor models, from the view of matrix models. It is based on a really simple observations: that a tensor of size $N\\times N \\times \\tau$ can be seen as a set of $\\tau$ matrices.\n\nIn section \\ref{sec:loops}, this observation allows to interpret models for tensors of size $N\\times N\\times \\tau$ whose interactions have a single cycle of colors $(1,2)$ as $U(\\tau)$-invariant matrix models. We describe this correspondence through a bijection between edge-colored graphs and random surfaces decorated with oriented loops and show that the degree, which organizes the $1\/N$ expansion of tensor models, here organizes the expansion with respect to the number of loops on the random surfaces, via the equation \\eqref{eq:LoopCounting}. That provides a new, combinatorial interpretation of the degree.\n\nWe have taken this as an opportunity to review the most recent results on tensor models applied in the context of the loop models. This approach also unravels the challenges faced by random tensor theory. It is emphasized that to our knowledge there is no known solution to those models (e.g. for the large $N$ free energy at finite $\\tau$), beyond a very particular case which corresponds to a Penner model. Beyond this case, the most generic and explicit result is the classification of edge-colored graphs according to their degree, due to Gurau and Schaeffer \\cite{GurauSchaeffer} which as we have explained in section \\ref{sec:ScalingLimits} classifies the loop configurations at fixed genus and number of edges according to the number of loops. There is moreover a double-scaling limit which sums consistenly the most singular (at criticality) loop configurations. We hope that the relationship we have established between tensor models and loop models can lead to fruitful cross-fertilization.\n\nWhile we have focused in section \\ref{sec:ScalingLimits} on the scaling limits, further connections between matrix and tensor models have been reviewed in section \\ref{sec:Bimaps}, based on the Hubbard-Stratanovich (intermediate field) transformation. It reveals that melonic quartic tensor models generate maps formed by maps with different edge colors glued together at vertices (or by duality, at the center of their faces), \\cite{DSQuartic} (see also \\cite{BeyondPert} for a constructive analysis (Borel summability) of this model and \\cite{GenericQuartic} for an extension of those ideas to arbitrary quartic models). In the case of two edge colors, those maps have already appeared under the name of \\emph{nodal surfaces} in \\cite{EynardBookMulticut} as multicut solutions of the one-matrix model\\footnote{Obviously the multicut solution satisfies the loop equations of the 1-matrix model and is not a solution of quartic tensor models which correspond to a different evaluation of the generating function of nodal surfaces.}. It has been further observed that such maps can be generated in a Givental-like fashion\\footnote{We are indebted to Bertrand Eynard for pointing this out and we would like to thank St\\'ephane Dartois for sharing his progress on such a re-formulation of tensor models.} \\cite{GiventalDartois}.\n\nWe believe that viewing tensor models as matrix models constitutes an interesting research road and places tensor models in a frame where powerful tools are available. In particular, the intermediate field method turns quartic tensor models into matrix models which generate generalizations of nodal surfaces. Either techniques developed for matrix models, such as the topological recursion \\cite{TopRec}, or combinatorial approaches, could lead to new results. Among the combinatorial approaches, bijective methods akin to Schaeffer's bijection for planar quadrangulations could be useful to solve the large $N$ limit (i.e. the planar sector) of the two-matrix model \\eqref{TwoMatrixModel}, while algebraic methods have proved helpful to probe maps at arbitrary genus \\cite{KPGouldenJackson, CarrellChapuyRecursion}. Preliminary calculations suggest that the large $N$ limit of \\eqref{TwoMatrixModel} is a generalization of the $O(\\tau)$ model (where the eigenvalues of $M_1$ are attracted to the mirror image of those of $M_2$) \\cite{ExactO(n)Eynard, O(n>2)Eynard}.\n\n\\section*{Acknowledgements}\n\nResearch at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research and Innovation.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}