diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzdvqe" "b/data_all_eng_slimpj/shuffled/split2/finalzzdvqe" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzdvqe" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction and main result}\\label{sec1}\n\n\\noindent\nWe study H\\\"older regularity up to the boundary for the weak solutions of the Dirichlet problem\n\\begin{equation}\\label{dir}\n\\begin{cases}\n(-\\Delta)_p^s u=f & \\text{in $\\Omega$} \\\\\nu=0 & \\text{in $\\Omega^c$}.\n\\end{cases}\n\\end{equation}\nHere $\\Omega\\subset{\\mathbb R}^N$ ($N>1$) is a bounded domain with a $C^{1,1}$ boundary $\\partial\\Omega$, $\\Omega^c={\\mathbb R}^N\\setminus\\Omega$, $s\\in(0,1)$ and $p\\in(1,\\infty)$ are real numbers and $f\\in L^\\infty(\\Omega)$. The $s$-fractional $p$-Laplacian operator is the gradient of the functional\n\\[J(u):=\\frac{1}{p}\\int_{{\\mathbb R}^N\\times{\\mathbb R}^N}\\frac{|u(x)-u(y)|^p}{|x-y|^{N+ps}}\\, dx\\, dy,\\]\ndefined on\n\\[W^{s,p}_0(\\Omega):=\\{u\\in L^p({\\mathbb R}^N): J(u)<\\infty, \\,\\, u=0 \\text{ in $\\Omega^c$}\\},\\]\nwhich is a Banach space with respect to the norm $J(u)^{1\/p}$. Under suitable smoothness conditions on $u$ the operator can be written as\n\\begin{equation*}\n(- \\Delta)_p^s\\, u(x) = 2 \\lim_{\\varepsilon \\searrow 0} \\int_{B_\\varepsilon^c(x)} \\frac{|u(x) - u(y)|^{p-2}\\, (u(x) - u(y))}{|x - y|^{N+sp}}\\, dy, \\quad x \\in {\\mathbb R}^N.\n\\end{equation*}\nA weak solution $u\\in W^{s,p}_0(\\Omega)$ of problem \\eqref{dir} satisfies,\nfor every $\\varphi\\in W^{s,p}_0(\\Omega)$, \n\\[\\int_{{\\mathbb R}^N\\times{\\mathbb R}^N}\\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))(\\varphi(x)-\\varphi(y))}{|x-y|^{N+ps}} \\,dx\\,dy=\\int_\\Omega f(x)\\varphi(x) \\,dx.\\]\nProblem \\eqref{dir} is thus well posed and, in the case $p=2$, it corresponds to an inhomogeneous fractional Laplacian equation with Dirichlet boundary condition. \nFor the sake of completeness we recall that in the literature the fractional Laplacian is often defined by\n\\begin{equation*}\n \\langle (-\\Delta)^su,\\varphi\\rangle\n = \\frac{c(N,s)}{2} \\int_{{\\mathbb R}^N\\times{\\mathbb R}^N}\\frac{(u(x)-u(y))(\\varphi(x)-\\varphi(y))}{|x-y|^{N+2s}} \\,dx\\,dy,\\quad \\varphi \\in W^{s,2}_0(\\Omega),\n\\end{equation*}\nwhere $c(N,s) = s 2^{2s} \\, \\Gamma((N+2s)\/2) \/ (\\pi^{N\/2} \\Gamma(1-s))$, in order to be coherent with the\nFourier definition of $(-\\Delta)^s$ (see Remark 3.11 of \\cite{CS1}). We point out that, in the current literature, there are several\nnotions of fractional Laplacian, all of which agree when the problems\nare set on the whole ${\\mathbb R}^N$, but some of them disagree in a bounded\ndomain. We refer the reader to~\\cite{SV2} for a discussion on the comparison\nbetween the integral fractional laplacian and the regional (or spectral) notion\nobtained by taking the $s$-powers of the Laplacian operator $-\\Delta$ with zero Dirichlet\nboundary conditions.\n\\vskip2pt\n\\noindent\nIn the case $p\\neq 2$, problem \\eqref{dir} is a non-local and non-linear one. \nIts leading term $(-\\Delta)^s_p$ is furthermore degenerate when $p>2$ and singular when $11\/(1-s)$. In \\cite{BCF} the fully non-linear approach is used to study the non-local analogue of the $p$-Laplacian equation in non-divergence form\n\\[ \\Delta u +(p-2)\\frac{\\nabla u}{|\\nabla u|}D^2u\\frac{\\nabla u}{|\\nabla u|}=0,\\]\narising from non-local `tug of war' games. Interior $C^{1,\\alpha}$ estimates and H\\\"older continuity up to the boundary is proved under rather general assumptions.\n\\vskip2pt\n\\noindent\nOur main result is the following:\n\n\\begin{theorem}\\label{main}\nThere exist $\\alpha\\in(0,s]$ and $C_\\Omega>0$, depending only on $N$, $p$, $s$, with $C_\\Omega$ also depending on $\\Omega$, such that, for all weak solution $u\\in W^{s,p}_0(\\Omega)$ of problem \\eqref{dir}, $u\\in C^\\alpha(\\overline\\Omega)$ and\n\\begin{equation}\\label{thm57tesi}\n\\|u\\|_{C^\\alpha(\\overline\\Omega)}\\le C_\\Omega\\|f\\|_{L^\\infty(\\Omega)}^\\frac{1}{p-1}.\n\\end{equation}\n\\end{theorem}\n\n\\noindent\nNotice that, regarding regularity {\\em up to the boundary}, one cannot expect more than $s$-H\\\"older continuity due to explicit examples (see Section \\ref{sec3} below). On the other hand, the optimal H\\\"older exponent up to the boundary seems to be $s$ for any $p>1$, while we prove $C^\\alpha$ regularity for an unspecified small $\\alpha$, the issue being a lack of higher (at least $C^s$) regularity results in the interior of the domain.\n\\vskip2pt\n\\noindent\nLet us describe the strategy to prove Theorem \\ref{main}. We choose to use the notion of weak rather than viscosity solution, since we feel that the equation is more naturally seen as a variational one. However, we will frequently use barrier arguments, rather than De Giorgi-Nash-Moser techniques. Indeed, the proof of Theorem \\ref{main} is performed in the spirit of Krylov's approach to boundary regularity, see \\cite{krylov}, and uses two main ingredients:\n\\begin{itemize}\n\\item[$(a)$]\na uniform H\\\"older control (see Theorem \\ref{estid}) on how $u$ reaches its boundary values, which amounts to \n\\begin{equation}\\label{deltasintro}\n|u(x)|\\leq C\\|f\\|_\\infty^{\\frac{1}{p-1}}{\\rm dist}^s(x,\\Omega^c);\n\\end{equation}\n\\item[$(b)$]\na local regularity estimate (see Theorem \\ref{osc}) in terms of quantities which may blow up in general when reaching \nthe boundary, but remain bounded for functions satisfying \\eqref{deltasintro}.\n\\end{itemize}\nPoint $(a)$ is obtained through a barrier argument, and stems from the fact that $(-\\Delta)^s_p (x_+)^s=0$ in the half line ${\\mathbb R}_+$. Notice that for $p\\neq 2$ we do not have at our disposal the fractional Kelvin transform, and the concrete calculus of the $s$-fractional $p$-Laplacian even on smooth functions is a prohibitive task, in general. \nThus constructing upper barriers can be quite technical, and is done as following:\n\\begin{itemize}\n\\item\nConsider $u_N(x)=(x_N)_+^s$: explicit calculus shows that $(-\\Delta)^s_p u_N=0$ in the half-space ${\\mathbb R}^N_+$. We locally deform the half-space to $\\Omega^c$ by a diffeomorphism $\\Phi$ close to the identity, and obtain a function $u_N\\circ \\Phi$ with small $s$-fractional $p$-Laplacian in a small ball $\\hat B$ centered at a point of $\\partial\\Omega$.\n\\item\nThe resulting function $u_N\\circ \\Phi$ can be controlled in $\\hat B\\cap \\Omega$ by distance-like functions from the boundary, and we can modify it to globalize the controls, while keeping the smallness of $(-\\Delta)^s_p(u_N\\circ\\Phi)$ in $\\hat B\\cap \\Omega$.\n\\item\nWe exploit the non-local nature of the equation to add a fixed {\\em positive} quantity to $(-\\Delta)^s_p(u_N\\circ\\Phi)$ in $\\hat B\\cap \\Omega$, by truncation away from $\\hat B$. Since $(-\\Delta)^s_p(u_N\\circ\\Phi)$ is arbitrarily small, its truncation has therefore $s$-fractional $p$-Laplacian bounded from below by a positive constant in $\\hat B\\cap \\Omega$, and provides the local upper barrier.\n\\end{itemize}\nPoint $(b)$ is a generalization, in the whole range $p>1$, to non-homogeneous equations of Theorem 1.2 from \\cite{DKP1}, and it could be deduced in the case $p>2-s\/N$ using the results of \\cite{KMS} and in the case $p>1\/(1-s)$ using \\cite{Ling}. However we choose to prove it with a different approach. Much in the spirit of \\cite{S}, rather than considering the non-locality of the equation as an additional technical difficulty to the implementation of the De Giorgi-Moser regularity theory, we use it at our advantage to construct a more elementary proof. It should be noted that we do not employ Caccioppoli-like inequalities, or estimates on $\\log u$ (which are the elementary counterpart of John-Nirenberg's lemma). Actually we don't even need a Poincar\\'e or Sobolev inequality, which are usually looked at as basic tools for (variational) regularity theory. This feature seems typical of the non-local framework and it should be noted that the proof doesn't seem to immediately ``pass to the limit to local equations'' as the obtained estimates blow up for $s\\to 1$. \n\\vskip2pt\n\\noindent\nRegarding possible developments and generalizations, a first remark regards the choice of the kernel in the non-local operator\n\\[L(u)={\\rm PV}\\int_{{\\mathbb R}^N}|u(x)-u(y)|^{p-2}(u(x)-u(y))K(x,y)\\, dy.\\]\nRegarding interior regularity, a bound from above and below in terms of the model kernel $|x-y|^{-N-ps}$ seems to suffice to obtain H\\\"older regularity, due to the results of \\cite{DKP1,KMS}. For non-local, fully non-linear, uniformly elliptic equation, higher interior regularity (up to $C^{2, \\alpha}$) is proved in \\cite{CafSil1,CafSil2,Serra} when the kernel satisfies additional structural and regularity assumption, but no such result is known for the $s$-fractional $p$-Laplacian. \nRegarding regularity up to the boundary things are more subtle. In the uniformly elliptic case ($p=2$), the optimal regularity is $C^s(\\overline{\\Omega})$ due to the results of \\cite{RS3}, but only for a subclass of rough symmetric kernels arising from stable L\\'evy processes, of the form\n\\[K(x, y)=H(x-y),\\quad H(z)=\\frac{a\\big(z\/|z|\\big)}{|z|^{N+2s}}, \\quad 0<\\lambda\\leq a\\leq \\Lambda.\\] \nCounterexamples show that this is the largest kernel's class where to expect such regularity up to the boundary. However, for any $p>1$, one still expects $C^\\alpha(\\overline{\\Omega})$ regularity for arbitrarily rough symmetric kernels, for a small $\\alpha0$ we denote by $B_r(x)$, $\\overline B_r(x)$, and $\\partial B_r(x)$, respectively, the open ball, the closed ball and the sphere centered at $x$ with radius $r$. When the center is not specified, we will understand that it's the origin, e.g. $B_1=B_1(0)$. For all measurable $A\\subset{\\mathbb R}^N$ we denote by $|A|$ the $N$-dimensional Lebesgue measure of $A$. If $u$ is a measurable function and $A$ is a measurable subset of ${\\mathbb R}^N$, we will set for brevity\n\\[\\inf_A u=\\essinf_A u,\\quad \\sup_A u=\\essinf_A u.\\]\nFor all measurable $u:{\\mathbb R}^N\\to{\\mathbb R}$ we define\n\\[[u]_{s,p}=\\Big(\\int_{{\\mathbb R}^N\\times{\\mathbb R}^N}\\frac{|u(x)-u(y)|^p}{|x-y|^{N+ps}} \\,dx\\,dy\\Big)^{\\frac{1}{p}},\\]\n\\[\\|u\\|_{W^{s,p}(\\Omega)}=\\|u\\|_{L^p(\\Omega)}+\\Big(\\int_{\\Omega\\times\\Omega}\\frac{|u(x)-u(y)|^p}{|x-y|^{N+ps}}\\, dx\\, dy\\Big)^{\\frac{1}{p}}\\]\nand we will consider the following spaces (see \\cite{DPV} for details):\n\\begin{align*}\n&W^{s,p}(\\Omega)=\\big\\{u\\in L^p(\\Omega):\\|u\\|_{W^{s,p}(\\Omega)}<\\infty\\big\\}, \\\\\n&W^{s,p}_0(\\Omega) =\\big\\{u\\in W^{s,p}({\\mathbb R}^N):\\,u=0 \\ \\text{in $\\Omega^c$}\\big\\},\\\\\n&W^{-s,p'}(\\Omega)=(W^{s,p}_0(\\Omega))^*,\n\\end{align*}\nwhere the last one is the Banach dual, whose pairing with $W^{s,p}_0(\\Omega)$ will be denoted by $\\langle\\cdot,\\cdot\\rangle_{s,p,\\Omega}$.\nWe will extensively make use of the following space:\n\\begin{definition}\\label{defwtilde}\nLet $\\Omega\\subseteq {\\mathbb R}^N$ be bounded. We set\n\\[\\widetilde{W}^{s,p}(\\Omega):=\\Big\\{u\\in L^p_{\\rm loc}({\\mathbb R}^N):\\,\\exists\\,U\\Supset\\Omega \\ \\text{s.t.}\\,\\|u\\|_{W^{s,p}(U)}+\\int_{{\\mathbb R}^N}\\frac{|u(x)|^{p-1}}{(1+|x|)^{N+ps}}\\, dx<\\infty\\Big\\}.\\]\nIf $\\Omega$ is unbounded, we set\n\\[\\widetilde{W}^{s,p}_{\\rm loc}(\\Omega):=\\big\\{u\\in L^p_{\\rm loc}({\\mathbb R}^N):\\,u\\in \\widetilde{W}^{s,p}(\\Omega')\\, \\text{for any bounded $\\Omega'\\subseteq\\Omega$}\\big\\}.\\]\n\\end{definition}\n\n\\noindent\nWe notice that the condition \n\\[\\int_{{\\mathbb R}^N}\\frac{|u(x)|^{p-1}}{(1+|x|)^{N+ps}}\\, dx<\\infty\\]\nholds if $u\\in L^\\infty({\\mathbb R}^N)$ or $[u]_{C^s({\\mathbb R}^N)}<\\infty$. The spaces $\\widetilde{W}^{s,p}(\\Omega)$, $\\widetilde{W}^{s,p}_{\\rm loc}(\\Omega)$ can be endowed with a topological vector space structure as inductive limit, but we will not use it.\nFor all $\\alpha\\in(0,1]$ and all measurable $u:\\overline\\Omega\\to{\\mathbb R}$ we set\n\\[[u]_{C^\\alpha(\\overline\\Omega)}=\\sup_{x,y\\in\\overline\\Omega,\\,x\\neq y}\\frac{|u(x)-u(y)|}{|x-y|^\\alpha},\\]\n\\[C^\\alpha(\\overline\\Omega)=\\big\\{u\\in C(\\overline\\Omega):\\,[u]_{C^\\alpha(\\overline\\Omega)}<\\infty\\big\\},\\]\nthe latter being a Banach space under the norm $\\|u\\|_{C^\\alpha(\\overline\\Omega)}=\\|u\\|_{L^\\infty(\\overline\\Omega)}+[u]_{C^\\alpha(\\overline\\Omega)}$. A similar definition is given for $C^{1,\\alpha}(\\overline\\Omega)$. When no misunderstanding is possible, we set for all measurable $D\\subset{\\mathbb R}^N$, $x\\in D$, and all measurable $\\psi:D\\times D\\to{\\mathbb R}$\n\\[{\\rm PV}\\int_{D}\\psi(x, y)\\,dy=\\lim_{\\varepsilon\\to 0^+}\\int_{D\\setminus B_\\varepsilon(x)}\\psi(x, y)\\,dy.\\]\nFor all measurable $u:{\\mathbb R}^N\\to{\\mathbb R}$ we recall that the {\\em non-local tail} centered at $x\\in{\\mathbb R}^N$ with radius $R>0$, introduced in \\cite{DKP1}, is defined as\n\\begin{equation}\\label{deftail}\n{\\rm Tail}(u;x,R)=\\Big(R^{ps}\\int_{B_R^c(x)}\\frac{|u(y)|^{p-1}}{|x-y|^{N+ps}}\\,dy\\Big)^\\frac{1}{p-1}.\n\\end{equation}\nWe will also set ${\\rm Tail}(u; 0, R)={\\rm Tail}(u; R)$. Unless otherwise stated, the numbers $p>1$ and $s\\in(0,1)$ will be fixed as the order of summability and the order of differentiability. By a {\\em universal} constant we mean a constant $C=C(N,p,s)$. This dependence will always be omitted, even when other dependencies are present, in which case they are the only ones explicitly stated: for example $C_\\Omega$ will denote a constant depending on $N, p, s$, and $\\Omega$. During chains of inequalities, universal constants will be denoted by the same letter $C$ even if their numerical value may change from line to line. The same treatment will be used for constants which retain their dependencies from line to line. When needed, we will denote a specific universal constant with a number, e.g.\\ $C_1$, $C_2$ {\\em et cetera}.\n\n\\subsection{Some elementary inequalities} \n\nFor all $a\\in{\\mathbb R}$, $q>0$, we set\n\\[a^{q}=|a|^{q-1}a.\\]\nThis notation has great advantages in readability and, for future reference, we recall here some more or less known elementary inequalities about the function $a\\mapsto a^q$. We will provide a sketch of proof for the less frequent ones.\n\\vskip2pt\n\\noindent\nWe begin with the well known inequalities\n\\begin{equation}\\label{in3}\n(a+b)^q\\le 2^{q-1}(a^q+b^q) \\quad a,b\\ge 0,\\,q\\ge 1;\n\\end{equation}\n\\begin{equation}\\label{in2}\n(a+b)^q\\le a^q+b^q \\quad a,b\\ge 0,\\,q\\in(0,1];\n\\end{equation}\n\\begin{equation}\\label{in4}\n|a^q-b^q|\\le q(|a|^{q-1}+|b|^{q-1})|a-b| \\quad a,b\\in{\\mathbb R},\\,q\\ge 1,\n\\end{equation}\nthe last one being a trivial consequence of Taylor's formula. We will also use\n\\begin{equation}\\label{in6}\na^q-(a-b)^q\\le C_M\\max\\{b,b^q\\} \\quad |a|\\le M,\\,b\\ge 0,\\,q>0,\n\\end{equation}\nwhich follows immediately from \\eqref{in2} if $q\\in (0,1]$. If $q>1$ we can prove it distinguishing the cases $b\\leq M$, where we use \\eqref{in4}, and the case $b\\geq M$, where we use $a^q-(a-b)^q\\leq M^q+2M^q\\leq 3b^q$.\nWe now prove\n\\begin{equation}\\label{in7}\n(a+b)^q-a^q\\le\\theta a^q+C_\\theta b^q \\quad a,b\\ge 0,\\,q\\ge 1,\\,C_\\theta\\to\\infty \\ \\text{as} \\ \\theta\\to 0^+.\n\\end{equation}\nLetting $C_q= 1$ if $q\\leq 1$ and $C_q=2^{q-1}$ if $q\\geq 1$, \\eqref{in3} and \\eqref{in2} can be written as\n\\[(a+b)^q\\le C_q(a^q+b^q) \\quad a,b\\ge 0,\\,q>0.\\]\nNow \\eqref{in7} can be proved using Taylor's formula and Young's inequality:\n\\begin{align*}\n(a+b)^q-a^q &\\leq C_q(a^{q-1}+b^{q-1})b= (\\theta q'a)^{q-1} \\frac{C_q b}{(\\theta q')^{q-1}}+C_qb^q\\\\\n&\\leq\\theta a^q+\\frac{1}{q}\\Big(\\frac{C_q}{(\\theta q')^{q-1}}\\Big)^q b^q +C_qb^q.\n\\end{align*}\nWe prove the following inequality:\n\\begin{equation}\\label{in1}\na^{q}-(a-b)^{q}\\ge 2^{1-q}b^{q} \\quad a\\in{\\mathbb R},\\,b\\ge 0,\\,q\\ge 1.\n\\end{equation}\nWe can suppose $b>0$ and consider the function\n\\[f(t)=t^{q}-(t-b)^{q},\\quad f'(t)=q(|t|^{q-1}-|t-b|^{q-1}).\\]\nTherefore $f$ is positive, increasing for $t>b$ and decreasing for $t<-b$ and thus it's coercive. Since $f'(t)=0$ if and only if $t=b\/2$, its global minimum is $f(b\/2)=2^{1-q}b^{q}$.\n\\vskip2pt\n\\noindent\nFinally, we will use the following inequality, holding for all $A,B\\subset{\\mathbb R}^N$ with $A$ bounded and ${\\rm dist}(A, B^c)=d>0$:\n\\begin{equation}\\label{lkj}\n|x-y|\\geq C(A, B)(1+|y|), \\quad x\\in A,\\,y\\in B^c.\n\\end{equation}\n\n\\subsection{Weak and strong solutions}\n\n\\noindent\nWe compare in the following different notions of solutions for equations driven by $(-\\Delta)^s_p$.\n\n\\begin{definition}\nLet $\\Omega$ be bounded, $u\\in \\widetilde{W}^{s,p}(\\Omega)$ and $f\\in W^{-s,p'}(\\Omega)$. We say that $u$ is a {\\em weak solution} of $(-\\Delta)^s_p u=f$ in $\\Omega$ if for all $\\varphi\\in W^{s,p}_0(\\Omega)$\n\\[\n\\int_{{\\mathbb R}^N\\times{\\mathbb R}^N}\\frac{(u(x)-u(y))^{p-1}(\\varphi(x)-\\varphi(y))}{|x-y|^{N+ps}}\\,dx\\,dy=\\langle f, \\varphi\\rangle_{s,p,\\Omega}\n\\]\nIf $\\Omega$ is unbounded, we say that $u\\in \\widetilde{W}^{s,p}_{\\rm loc}(\\Omega)$ solves $(-\\Delta)^s_p u=f$ (with $f\\in W^{-s, p'}(\\Omega)$) weakly in $\\Omega$ if it does so in any bounded open set $\\Omega'\\subseteq\\Omega$.\n\\end{definition}\n\n\\noindent\nThe inequality $(-\\Delta)^s_p u\\leq f$ weakly in $\\Omega$ will mean that\n\\[\\int_{{\\mathbb R}^N\\times{\\mathbb R}^N}\\frac{(u(x)-u(y))^{p-1}(\\varphi(x)-\\varphi(y))}{|x-y|^{N+ps}}\\,dx\\,dy\\le\\langle f, \\varphi\\rangle_{s,p,\\Omega}\\]\nfor all $\\varphi\\in W^{s,p}_0(\\Omega)$, $\\varphi\\ge 0$, and similarly for $(-\\Delta)^s_p u\\ge f$. Noticing that $\\pm K\\in W^{-s,p'}(\\Omega)$ for any $K>0$ and any bounded $\\Omega$, by $|(-\\Delta)^s_p u|\\le K$ weakly in $\\Omega$ we mean that both $-K\\le(-\\Delta)^s_p u\\le K$ weakly in $\\Omega$.\n\\vskip2pt\n\\noindent\nIn the following proposition we will prove that $(-\\Delta)^s_p u\\in W^{-s,p'}(\\Omega)$ if $u\\in\\widetilde{W}^{s,p}(\\Omega)$, which implies that the previous definition makes sense.\n\n\\begin{lemma}\n\\label{remws}\nLet $\\Omega$ be bounded and $u\\in \\widetilde{W}^{s, p}(\\Omega)$. Then the functional \n\\[W^{s,p}_0(\\Omega)\\ni \\varphi\\mapsto ( u, \\varphi):= \\int_{{\\mathbb R}^N\\times{\\mathbb R}^N}\\frac{(u(x)-u(y))^{p-1}(\\varphi(x)-\\varphi(y))}{|x-y|^{N+ps}} \\,dx\\,dy\\]\nis finite and belongs to $W^{-s,p'}(\\Omega)$.\n\\end{lemma}\n\n\\begin{proof}\nLet $U\\Supset\\Omega$ be such that\n\\begin{equation}\\label{Uwtilde}\n\\|u\\|_{W^{s,p}(U)}+\\int_{{\\mathbb R}^N}\\frac{|u(x)|^{p-1}}{(1+|x|)^{N+ps}}\\, dx<\\infty,\n\\end{equation}\nand write\n\\begin{equation}\\label{<>}\n\\begin{split}\n(u, \\varphi)&=\\int_{U\\times U}\\frac{(u(x)-u(y))^{p-1}(\\varphi(x)-\\varphi(y))}{|x-y|^{N+ps}} \\,dx\\,dy\\\\\n&\\quad +\\int_{U\\times U^c}\\frac{(u(x)-u(y))^{p-1}\\varphi(x)}{|x-y|^{N+ps}} \\,dx\\,dy\\\\\n&\\quad -\\int_{U^c\\times U}\\frac{(u(x)-u(y))^{p-1}\\varphi(y)}{|x-y|^{N+ps}}\\,dx\\,dy\\\\\n&=\\int_{U\\times U}\\frac{(u(x)-u(y))^{p-1}(\\varphi(x)-\\varphi(y))}{|x-y|^{N+ps}} \\,dx\\,dy\\\\\n&\\quad +2\\int_{\\Omega\\times U^c}\\frac{(u(x)-u(y))^{p-1}\\varphi(x)}{|x-y|^{N+ps}} \\,dx\\,dy,\n\\end{split}\n\\end{equation}\nsince ${\\rm supp}(\\varphi)\\subset\\overline\\Omega$.\nThe integral in $U\\times U$ is finite and continuous with respect to strong convergence of $\\varphi\\in W^{s,p}_0(\\Omega)$ since $u\\in W^{s,p}(U)$. For the second term, observe that for a.e.\\ $x\\in \\Omega$ it holds\n\\begin{equation}\\label{hhh}\n\\begin{split}\n&\\int_{U^c}\\frac{|u(x)-u(y)|^{p-1}}{|x-y|^{N+ps}}\\,dy\\\\\n&\\leq C\\Big(|u(x)|^{p-1}\\int_{U^c}\\frac{1}{|x-y|^{N+ps}}\\,dy+\\int_{U^c}\\frac{|u(y)|^{p-1}}{(|x-y|)^{N+ps}} \\,dy\\Big)\\\\\n&\\leq C\\Big(|u(x)|^{p-1}+\\int_{{\\mathbb R}^N}\\frac{|u(y)|^{p-1}}{(1+|y|)^{N+ps}} \\,dy\\Big),\n\\end{split}\n\\end{equation}\nwhere we used \\eqref{lkj}\nwith $A=\\Omega$ and $B=U$. The right hand side of \\eqref{hhh} belongs to $L^{p'}(\\Omega)$ since $\\Omega$ is bounded and $u\\in L^p(\\Omega)$. Thus the second term in \\eqref{<>} is continuous with respect to $L^p(\\Omega)$-convergence of $\\varphi$. Therefore it is also continuous in $W^{s,p}_0(\\Omega)$.\n\\end{proof}\n \n\\begin{definition}[Point-wise and strong solutions]\nLet $u\\in \\widetilde{W}^{s,p}_{\\rm loc}(\\Omega)$ and $f:\\Omega\\to {\\mathbb R}$ be measurable. We say that $u$ is an {\\em a.e.\\ point-wise} solution of $(-\\Delta)^s_p u=f$ in $\\Omega$ if for a.a.\\ Lebesgue point $x\\in \\Omega$ of $u$ it holds\n\\begin{equation}\n\\label{psl-strong.1}\n2\\, {\\rm PV}\\int_{{\\mathbb R}^N}\\frac{(u(x)-u(y))^{p-1}}{|x-y|^{N+ps}}\\, dy=f(x).\n\\end{equation}\nMoreover, for $f\\in L^1_{\\rm loc}(\\Omega)$ we say that $u$ is a {\\em strong} solution of $(-\\Delta)^s_p u=f$ if\n\\begin{equation}\n\\label{psl-strongg}\n2\\int_{B_\\varepsilon^c(x)}\\frac{(u(x)-u(y))^{p-1}}{|x-y|^{N+ps}}\\, dy\\to f\\quad \\text{strongly in $L^1_{\\rm loc}(\\Omega)$, as $\\varepsilon\\to 0^+$.}\n\\end{equation}\n\\end{definition}\n\n\\noindent\nSimilar definitions are given for sub- and supersolutions. \n\\vskip2pt\n\\noindent\nNow we prove that a strong solution is also a weak solution. First, we introduce a more general result, which will be used in the following: we denote by ${\\bf D}$ the diagonal of ${\\mathbb R}^N\\times{\\mathbb R}^N$.\n\n\\begin{lemma}\\label{symmset}\nLet $u\\in\\widetilde{W}^{s,p}_{\\rm loc}(\\Omega)$. For all $\\varepsilon>0$ let $A_\\varepsilon\\subset{\\mathbb R}^N\\times{\\mathbb R}^N$ be a neighborhood of ${\\bf D}$ which satisfies \n\\begin{enumroman}\n\\item\\label{symmset1}\n$(x,y)\\in A_\\varepsilon$ for all $(y,x)\\in A_\\varepsilon$;\n\\item\\label{symmset2}\n${\\rm dist}_H(A_\\varepsilon, {\\bf D})\\to 0$ as $\\varepsilon\\to 0^+$.\n\\end{enumroman}\nFor all $x\\in{\\mathbb R}^N$ we set $A_\\varepsilon(x)=\\{y\\in{\\mathbb R}^N:(x,y)\\in A_\\varepsilon\\}$ and\n\\[g_\\varepsilon(x)=\\int_{A_\\varepsilon^c(x)}\\frac{(u(x)-u(y))^{p-1}}{|x-y|^{N+ps}}\\,dy.\\]\nIf $2g_\\varepsilon\\to f$ in $L^1_{\\rm loc}(\\Omega)$, then $u$ is a weak solution of $(-\\Delta)^s_p u=f$ in $\\Omega$.\n\\end{lemma}\n\\begin{proof}\nWe can suppose that $\\Omega$ is bounded and let $U\\Supset\\Omega$ be such that \\eqref{Uwtilde} holds for $u$, fix $\\varphi\\in C^\\infty_c(\\Omega)$ and let $K={\\rm supp}(\\varphi)$. First we prove that $g_\\varepsilon\\in L^1(K)$. For all $x\\in K$ there exists $\\rho>0$ such that $B_\\rho(x)\\subset A_\\varepsilon(x)$, and by a covering argument we may choose $\\rho$ independent of $x$ (while $\\rho$ depends on $\\varepsilon$). Moreover, for all $x\\in K$ and $y\\in A_\\varepsilon^c(x)$ we have $|x-y|\\ge C(1+|y|)$ (see \\eqref{lkj}). So we can compute\n\\begin{align*}\n\\int_K|g_\\varepsilon(x)|\\,dx &\\le C\\int_K\\int_{A_\\varepsilon^c(x)}\\frac{|u(x)|^{p-1}}{|x-y|^{N+ps}}\\,dy\\,dx+C\\int_K\\int_{A_\\varepsilon^c(x)}\\frac{|u(y)|^{p-1}}{|x-y|^{N+ps}}\\,dy\\,dx \\\\\n&\\le C\\int_K|u(x)|^{p-1}\\,dx\\int_{B_\\rho^c}\\frac{1}{|z|^{N+ps}}\\,dz+C\\int_K\\int_{A^c_\\varepsilon(x)}\\frac{|u(y)|^{p-1}}{(1+|y|)^{N+ps}}\\,dy\\\\\n&\\le C_\\varepsilon\\int_U|u(x)|^{p-1}\\,dx+C|K|\\int_{{\\mathbb R}^N}\\frac{|u(y)|^{p-1}}{(1+|y|)^{N+ps}}\\,dy < \\infty.\n\\end{align*}\nLemma \\ref{remws} shows that \n\\[\\frac{(u(x)-u(y))^{p-1}(\\varphi(x)-\\varphi(y))}{|x-y|^{N+ps}}\\in L^1({\\mathbb R}^N\\times {\\mathbb R}^N)\\]\nand thus, through \\ref{symmset1}, \\ref{symmset2}, and Fubini's theorem we have\n\\begin{align*}\n&\\int_{{\\mathbb R}^N\\times{\\mathbb R}^N}\\frac{(u(x)-u(y))^{p-1}(\\varphi(x)-\\varphi(y))}{|x-y|^{N+ps}}\\,dx\\,dy\\\\\n&\\underset{(ii)}{=} \\lim_{\\varepsilon\\to 0^+}\\int_{A_\\varepsilon^c}\\frac{(u(x)-u(y))^{p-1}(\\varphi(x)-\\varphi(y))}{|x-y|^{N+ps}}\\,dy\\,dx\\\\\n&\\underset{(i)}{=}\\lim_{\\varepsilon\\to 0^+}2\\int_K\\int_{A_\\varepsilon^c(x)}\\frac{(u(x)-u(y))^{p-1}}{|x-y|^{N+ps}}\\varphi(x)\\,dx\\,dy\\\\\n&=\\lim_{\\varepsilon\\to 0^+}2\\int_Kg_\\varepsilon(x)\\varphi(x)\\,dx.\n\\end{align*}\nSince $2g_\\varepsilon\\to f$ in $L^1(K)$, the density of $C^\\infty_c(\\Omega)$ in $W^{s,p}_0(\\Omega)$ and Lemma \\ref{remws} give the assertion.\n\\end{proof}\n\n\\begin{remark}\nAs the proof shows, it suffices to assume that the convergence in \\eqref{psl-strongg} be in $L^1_{\\rm loc}(\\Omega)$ weakly. We deliberately choose to assume strong $L^1_{\\rm loc}$-convergence since in all subsequent applications this is enough.\n\\end{remark}\n\n\\begin{corollary}\n\\label{simplw}\nLet $u\\in \\widetilde{W}^{s,p}_{\\rm loc}(\\Omega)$ be a strong solution of $(-\\Delta)^s_p u=f$ in $\\Omega$, with $f\\in L^1_{\\rm loc}(\\Omega)$. Then $u$ is a weak solution of $(-\\Delta)^s_p u=f$ in $\\Omega$.\n\\end{corollary}\n\\begin{proof}\nIt follows from Lemma \\ref{symmset} with $A_\\varepsilon=\\{(x,y)\\in{\\mathbb R}^N\\times{\\mathbb R}^N:\\,|x-y|<\\varepsilon\\}.$\n\\end{proof}\n\n\\subsection{Some basic properties of $(-\\Delta)^s_p$}\n\nThe following result describes a fundamental non-local feature of $(-\\Delta)^s_p$.\n\\begin{lemma}[Non-local behavior of $(-\\Delta)^s_p$]\n\\label{psadd}\nSuppose $u\\in \\widetilde{W}^{s,p}_{\\rm loc}(\\Omega)$ solves $(-\\Delta)^s_p u=f$ weakly, strongly or point-wisely in $\\Omega$ for some $f\\in L^1_{\\rm loc}(\\Omega)$. Let $v\\in L^1_{\\rm loc}({\\mathbb R}^N)$ be such that \n\\begin{equation}\n\\label{hyppertv}\n{\\rm dist}({\\rm supp}(v),\\Omega)>0,\\quad \\int_{\\Omega^c}\\frac{|v(x)|^{p-1}}{(1+|x|)^{N+ps}}\\, dx<\\infty,\n\\end{equation}\nand define for a.e.\\ Lebesgue point $x\\in \\Omega$ of $u$\n\\[h(x)=2\\int_{{\\rm supp}(v)}\\frac{(u(x)-u(y)-v(y))^{p-1}-(u(x)-u(y))^{p-1}}{|x-y|^{N+ps}}\\,dy.\\]\nThen $u+v\\in \\widetilde{W}^{s,p}_{\\rm loc}(\\Omega)$ and it solves $(-\\Delta)^s_p(u+v)=f+h$ weakly, strongly or pointwisely respectively in $\\Omega$.\n\\end{lemma}\n\n\\begin{proof}\nAs usual, it suffices to consider the case $\\Omega$ bounded, and we first prove that $u+v\\in \\widetilde{W}^{s,p}(\\Omega)$. \nLet $K={\\rm supp}(v)$ and $U$ be such that \\eqref{Uwtilde} holds for $u$, and suppose without loss of generality that $\\Omega\\Subset U\\Subset K^c$. Clearly $u+v=u$ in $U$, and thus it belongs to $W^{s,p}(U)$. Moreover\n\\[\\int_{{\\mathbb R}^N}\\frac{|u(x)+v(x)|^{p-1}}{(1+|x|)^{N+ps}}\\, dx\\leq C\\Big(\\int_{{\\mathbb R}^N}\\frac{|u(x)|^{p-1}}{(1+|x|)^{N+ps}}\\, dx+\\int_{K}\\frac{|v(x)|^{p-1}}{(1+|x|)^{N+ps}}\\, dx\\Big),\\]\nand the last term is finite due to \\eqref{hyppertv}.\nWith a similar estimate, we see that the integral defining $h$ is finite (due also to \\eqref{hyppertv} and \\eqref{lkj}).\nConsider now the case where $(-\\Delta)^s_p u=f$ weakly.\nChoose $\\varphi\\in C^\\infty_c(\\Omega)$ and compute\n\\begin{align*}\n&\\int_{{\\mathbb R}^N\\times{\\mathbb R}^N}\\frac{(u(x)+v(x)-u(y)-v(y))^{p-1}(\\varphi(x)-\\varphi(y))}{|x-y|^{N+ps}}\\,dx\\,dy \\\\\n&=\\int_{\\Omega\\times \\Omega}\\frac{(u(x)-u(y))^{p-1}(\\varphi(x)-\\varphi(y))}{|x-y|^{N+ps}}\\,dx\\,dy \\\\\n&\\quad + \\int_{\\Omega\\times \\Omega^c}\\frac{(u(x)-u(y)-v(y))^{p-1}\\varphi(x)}{|x-y|^{N+ps}}\\,dx\\,dy\\\\\n&\\quad -\\int_{\\Omega^c\\times \\Omega}\\frac{(u(x)+v(x)-u(y))^{p-1}\\varphi(y)}{|x-y|^{N+ps}}\\,dx\\,dy \\\\\n&= \\int_{{\\mathbb R}^N\\times {\\mathbb R}^N}\\frac{(u(x)-u(y))^{p-1}(\\varphi(x)-\\varphi(y))}{|x-y|^{N+ps}}\\,dx\\,dy\\\\\n&\\quad -\\int_{\\Omega \\times \\Omega^c}\\frac{(u(x)-u(y))^{p-1}\\varphi(x)}{|x-y|^{N+ps}}\\,dx\\,dy \\\\\n&\\quad +\\int_{\\Omega^c\\times \\Omega}\\frac{(u(x)-u(y))^{p-1}\\varphi(y)}{|x-y|^{N+ps}}\\,dx\\,dy+2\\int_{\\Omega\\times \\Omega^c}\\frac{(u(x)-u(y)-v(y))^{p-1}\\varphi(x)}{|x-y|^{N+ps}}\\,dx\\,dy \\\\\n&= \\int_\\Omega f(x)\\varphi(x)\\,dx+2\\int_{\\Omega\\times \\Omega^c}\\frac{(u(x)-u(y)-v(y))^{p-1}-(u(x)-u(y)))^{p-1}}{|x-y|^{N+ps}}\\varphi(x)\\,dx\\,dy \\\\\n&= \\int_\\Omega(f(x)+h(x))\\varphi\\,dx,\n\\end{align*}\nwhere in the end we have used Fubini's theorem. The density of $C^\\infty_c(\\Omega)$ in $W^{s,p}_0(\\Omega)$ allows to conclude.\n\\vskip2pt\n\\noindent\nSuppose now that $(-\\Delta)^s_p u=f$ strongly or pointwisely in $\\Omega$. Let for $x\\in V\\Subset\\Omega$ and $\\varepsilon<{\\rm dist}(V, \\Omega^c)$\n\\[g_\\varepsilon(x)=\\int_{B^c_\\varepsilon(x)}\\frac{(u(x)+v(x)-u(y)-v(y))^{p-1}}{|x-y|^{N+ps}}\\,dy.\\] \nUsing \\eqref{hyppertv} we get\n\\begin{align*}\n&g_\\varepsilon(x)=\\int_{\\Omega\\setminus B_\\varepsilon(x)}\\frac{(u(x)-u(y))^{p-1}}{|x-y|^{N+ps}}\\,dy+\\int_{\\Omega^c}\\frac{(u(x)-u(y)-v(y))^{p-1}}{|x-y|^{N+ps}}\\,dy\\\\\n&= \\int_{B_\\varepsilon^c(x)}\\frac{(u(x)-u(y))^{p-1}}{|x-y|^{N+ps}}\\,dy+\\int_{K}\\frac{(u(x)-u(y)-v(y))^{p-1}-(u(x)-u(y))^{p-1}}{|x-y|^{N+ps}}\\,dy.\n\\end{align*}\nTaking the limit for $\\varepsilon\\to 0^+$ gives the claim in the pointwise case. To show that $(-\\Delta)^s_p (u+v)=f+h$ strongly it suffices to show that\n\\[x\\mapsto \\int_{K}\\frac{(u(x)-u(y)-v(y))^{p-1}-(u(x)-u(y))^{p-1}}{|x-y|^{N+ps}}\\,dy\\]\nbelongs to $L^1(K)$, which can be done proceeding as in \\eqref{hhh} and using \\eqref{hyppertv} for the term involving $v$. \n\\end{proof}\n\n\\noindent\nWe also recall the well known homogeneity, scaling, and rotational invariance properties of $(-\\Delta)^s_p$. For all $\\rho>0$, $M\\in O_N$ (the orthogonal group), $v$ measurable, $\\Omega\\subseteq{\\mathbb R}^N$, set\n\\begin{align*}\nv_\\rho(x)& =v(\\rho x),\\quad \\rho^{-1}\\Omega=\\{x\/\\rho:x\\in \\Omega\\}, \\\\\nv_M(x)&=v(Mx),\\quad M^{-1}\\Omega =\\{M^{-1}x:x\\in \\Omega\\}.\n\\end{align*}\n\\begin{lemma}\\label{hs}\nLet $u\\in \\widetilde{W}^{s,p}_{\\rm loc}(\\Omega)$ satisfy $(-\\Delta)^s_p u=f$ weakly in $\\Omega$ for some $f\\in L^{1}_{\\rm loc}(\\Omega)$. Then we have\n\\begin{enumroman}\n\\item\\label{hs.1} for all $h>0$, $(-\\Delta)^s_p (hu)=h^{p-1}f$ weakly in $\\Omega$;\n\\item\\label{hs.2} for all $\\rho>0$, $u_\\rho\\in \\widetilde{W}^{s,p}(\\rho^{-1}\\Omega)$ and $(-\\Delta)^s_p u_\\rho=\\rho^{ps}f_\\rho$ weakly in $\\rho^{-1}\\Omega$;\n\\item\\label{hs.3} for all $M\\in O_N$, $u_M\\in \\widetilde{W}^{s,p}(M^{-1}\\Omega)$ and $(-\\Delta)^s_p u_M=f_M$ weakly in $M^{-1}\\Omega$.\n\\end{enumroman}\n\\end{lemma}\n\n\\noindent\nFinally, from Lemma 9 of \\cite{LL} we have the following comparison principle for $(-\\Delta)^s_p$.\n\n\\begin{proposition}[Comparison Principle]\\label{comp}\nLet $\\Omega$ be bounded, $u,v\\in \\widetilde{W}^{s,p}(\\Omega)$ satisfy $u\\le v$ in $\\Omega^c$ and, for all $\\varphi\\in W^{s,p}_0(\\Omega)$, $\\varphi\\ge 0$ in $\\Omega$,\n\\begin{align*}\n&\\int_{{\\mathbb R}^N\\times{\\mathbb R}^N}\\frac{(u(x)-u(y))^{p-1}(\\varphi(x)-\\varphi(y))}{|x-y|^{N+ps}}\\,dx\\,dy\\\\\n&\\le \\int_{{\\mathbb R}^N\\times{\\mathbb R}^N}\\frac{(v(x)-v(y))^{p-1}(\\varphi(x)-\\varphi(y))}{|x-y|^{N+ps}}\\,dx\\,dy.\n\\end{align*}\nThen $u\\le v$ in $\\Omega$.\n\\end{proposition}\n\n\\begin{proof}\nThe proof follows by the arguments of \\cite{LL}. It is sufficient to know that both sides \nof the inequality are finite and $(u-v)_+\\in W^{s,p}_0(\\Omega)$, which is used there as a test function.\nBy Lemma \\ref{remws}, both sides are finite. We claim that $w:=(u-v)_+\\in W^{s,p}_0(\\Omega)$. Let $U\\Supset\\Omega$ be as in Definition \\ref{defwtilde} for both $u$ and $v$. We split the Gagliardo norm in ${\\mathbb R}^N$ as\n\\begin{align*}\n&\\int_{{\\mathbb R}^N\\times{\\mathbb R}^N}\\frac{|w(x)-w(y)|^p}{|x-y|^{N+ps}}\\, dx\\, dy\\\\\n&= \\int_{U\\times U}\\frac{|w(x)-w(y)|^p}{|x-y|^{N+ps}}\\, dx\\, dy+2\\int_{\\Omega\\times U^c}\\frac{|w(x)|^p}{|x-y|^{N+ps}}\\, dx\\, dy\n\\end{align*}\nwhere we used that $w=0$ in $\\Omega^c$ by assumption.\nThe first term is bounded since $u, v\\in W^{s,p}(U)$, which is a lattice. The second term is non-singular since ${\\rm dist}(\\Omega, U^c)>0$ and using \\eqref{lkj} we get\n\\begin{align*}\n\\int_{\\Omega\\times U^c}\\frac{|w(x)|^p}{|x-y|^{N+ps}}\\,dx\\,dy\n&\\leq C_{\\Omega, U}\n\\int_\\Omega (|u(x)|^{p}+ |v(x)|^{p})\\, dx\\int_{{\\mathbb R}^N}\\frac{1}{(1+|y|)^{N+ps}}\\, dy \\\\\n& \\leq C_{\\Omega, U}\\int_\\Omega (|u(x)|^{p}+ |v(x)|^{p})\\, dx,\n\\end{align*}\nwhich proves the claim. \n\\end{proof}\n\n\\subsection{$(-\\Delta)^s_p$ on smooth functions}\n\nNext we show that in the class of sufficiently smooth functions, the $s$-fractional $p$-Laplacian exists strongly (and thus weakly) and is locally bounded. First we recall the following definition of $(-\\Delta)^s_p$, equivalent to \\eqref{psl-strong.1} (by a simple change of variable):\n\\begin{equation}\\label{psl-strong}\n(-\\Delta)^s_p u(x) = {\\rm PV}\\int_{{\\mathbb R}^N}\\frac{(u(x)-u(x+z))^{p-1}+(u(x)-u(x-z))^{p-1}}{|z|^{N+ps}}\\,dz.\n\\end{equation}\nOur first lemma displays an estimate which allows us to remove the singularity at $0$, when $u$ is smooth enough:\n\n\\begin{lemma}\\label{zero}\nIf $u\\in C^{1,\\gamma}_{\\rm loc}(\\Omega)$, $\\gamma\\in [0,1]$, and $K\\subset\\Omega$ is compact, then there exist $C_{K, u},R_K>0$ \nsuch that for all $x\\in K$, $z\\in B_{R_K}$\n\\begin{equation*}\n\\big|(u(x)-u(x+z))^{p-1}+(u(x)-u(x-z))^{p-1}\\big| \n\\leq \n\\begin{cases}\n C_{K, u}|z|^{\\gamma+p-1} & \\text{if $p\\geq 2$}\\\\\n C_{K, u}|z|^{(\\gamma+1)(p-1)} & \\text{if $p< 2$}.\n\\end{cases}\n\\end{equation*}\n\\end{lemma}\n\\begin{proof}\nSince $K$ is compact, we can find $R_K>0$ such that\n\\[\\Omega_K:=\\{x\\in{\\mathbb R}^N:\\,{\\rm dist}(x,K)\\le R_K\\}\\subset\\Omega.\\]\nConsider first the case $p\\geq 2$. Since $u\\in C^{1,\\gamma}(\\Omega_K)$, for all $x\\in K$, $z\\in B_{R_K}$ there exist $\\tau_1,\\tau_2\\in[0,1]$ with\n\\begin{align*}\n&\\big|(u(x)-u(x+z))^{p-1}+(u(x)-u(x-z))^{p-1}\\big| \\\\\n&= \\big|(Du(x+\\tau_1 z)\\cdot z)^{p-1}-(Du(x-\\tau_2 z)\\cdot z)^{p-1}\\big| \\\\\n&\\le (p-1)\\sup_{B_{R_K}(x)}|Du|^{p-2}|z|^{p-2}\n\\big|\\big(Du(x+\\tau_1 z)-Du(x-\\tau_2 z)\\big)\\cdot z\\big| \\\\\n&\\le C\\|Du\\|_{C^{0,\\gamma}(\\Omega_{K})}^{p-1}|z|^{\\gamma+p-1}.\n\\end{align*}\nIf $1\n\\begin{cases}\n1-p(1-s) & \\text{if $p\\ge 2$} \\\\\n\\displaystyle\\frac{1-p(1-s)}{p-1} & \\text{if $p<2$}.\n\\end{cases}\n\\end{equation}\nThen $(-\\Delta)^s_p u=f$ strongly in $\\Omega$ for some $f\\in L^\\infty_{\\rm loc}(\\Omega)$\n\\end{proposition}\n\\begin{proof}\nLet $U$ be as in Definition \\ref{defwtilde} for $u$, fix a compact set $K\\subset\\Omega$ and let $R_K,C_K>0$ be as in Lemma \\ref{zero}. Define, for $x\\in K$, $\\varepsilon>0$, \n\\begin{align*}\ng_\\varepsilon(x)&:=\\int_{B_\\varepsilon^c}\\frac{(u(x)-u(x+z))^{p-1}+(u(x)-u(x-z))^{p-1}}{|z|^{N+ps}}\\, dz\\\\\n&=2\\int_{B_\\varepsilon^c}\\frac{(u(x)-u(x-z))^{p-1}}{|z|^{N+ps}}\\, dz.\n\\end{align*}\nWe claim that $g_\\varepsilon$ converges as $\\varepsilon\\to 0^+$ in a dominated way to some $f\\in L^\\infty(K)$. We split the integral in one for $z\\in B_{R_K}$ and one over $B_{R_K}^c$. For the first one, the previous lemma gives\n\\[\\Big|\\frac{(u(x)-u(x+z))^{p-1}+(u(x)-u(x-z))^{p-1}}{|z|^{N+ps}}\\Big| \\le \\frac{C_{K, u}}{|z|^{N+ps-\\sigma}},\\]\nwhere $\\sigma=\\gamma+p-1$ if $p\\geq 2$ and $\\sigma=(\\gamma+1)(p-1)$ if $10$. For the integral over $z\\in B^c_{R_K}$ we have, as in \\eqref{hhh},\n\\begin{align*}\n|f_2(x)|&:=\\Big|2\\int_{B^c_{R_K}}\\frac{(u(x)-u(x+z))^{p-1}}{|z|^{N+ps}}\\, dz\\Big|\\\\\n&\\leq C_{K, U}\\Big(\\|u\\|_{L^\\infty(K)}^{p-1}+\\int_{{\\mathbb R}^N}\\frac{|u(y)|^{p-1}}{(1+|y|)^{N+ps}}\\, dy\\Big).\n\\end{align*}\nGathering togheter the two estimates, we get\n\\[|g_\\varepsilon(x)|\\leq C_{K, u, U}\\quad \\text{$\\forall x\\in K$, $\\varepsilon>0$},\\]\n\\[\\lim_{\\varepsilon\\to 0^+}g_\\varepsilon(x)= f_1(x)+f_2(x)\\quad \\forall x\\in K,\\]\nand thus by the dominated convergence theorem $g_\\varepsilon\\to f_1+f_2$ in $L^1(K)$. \n\\end{proof}\n\n\\begin{remark}\nIt is useful to outline the dependence of $\\|(-\\Delta)^s_p u\\|_\\infty$ on $s$ in the previous proposition. Suppose, to fix ideas, that $p\\geq 2$ and $u\\in C^\\infty_c({\\mathbb R}^N)$, so that the domain $\\Omega$ has no role. Then, following the proof, we can find a constant $c_N$ depending only on $N$ such that\n\\[\\|(-\\Delta)^s_p u\\|_{\\infty}\\leq c_N\\frac{\\|u\\|_{C^2({\\mathbb R}^N)}^{p-1}}{1-s}.\\]\nThis is in accordance with the well known fact that $(1-s)(-\\Delta)^s_p \\to -\\Delta_p$ as $s\\to 1^-$ (see e.g.\\ \\cite{ponce}).\n\\end{remark}\n\n\\begin{remark}\n\\label{remarkL}\nConsider the class of functions\n\\[{\\mathcal L}(\\Omega)=\\{u\\in \\widetilde{W}^{s,p}(\\Omega):(-\\Delta)^s_p u=f\\,\\,\\text{strongly for some $f\\in L^\\infty_{\\rm loc}(\\Omega)$}\\}.\\]\nThe previous theorem asserts that if $p\\geq 2$, then $C^2(\\Omega)\\subseteq {\\mathcal L}(\\Omega)$. However, if $10$ consider the function\n\\begin{equation}\n\\label{defg}\ng^{(1)}_\\varepsilon(x)=\\int_{B_\\varepsilon^c(x)}\\frac{(x^s-y_+^s)^{p-1}}{|x-y|^{1+ps}}\\, dy.\n\\end{equation}\nWe claim that $g^{(1)}_\\varepsilon\\to 0$ uniformly on $K$, as $\\varepsilon\\to 0^+$. Note that for any $\\varepsilon\\varepsilon>0,\n\\end{equation}\nwhere $C$ is a universal constant.\nSince $\\psi(x, \\varepsilon)\\to 0$ uniformly on $[\\rho, \\rho^{-1}]\\supseteq K$, as $\\varepsilon\\to 0^+$, the claim follows. Finally we prove that $u_1\\in \\widetilde{W}^{s,p}(a,b)$ for any $a<00\\}.\\]\n\n\\begin{lemma}\\label{solnd}\nSet for any $A\\in GL_N$ and $x\\in {\\mathbb R}^N_+$,\n\\[g_{\\varepsilon}(x, A)=\\int_{B_\\varepsilon^c}\\frac{(u_1(x_N)-u_1(x_N+z_N))^{p-1}}{|Az|^{N+ps}}\\, dz\\]\nand $u(x)=u_1(x_N)$.\nThen $g_\\varepsilon\\to 0$ uniformly in any compact $K\\subseteq {\\mathbb R}^N_+\\times GL_N$ and $u\\in \\widetilde{W}^{s,p}_{\\rm loc}({\\mathbb R}^N)$ solves $(-\\Delta)^s_p u=0$ strongly and weakly in ${\\mathbb R}^N_+$.\n\\end{lemma}\n\\begin{proof}\nIt suffices to prove the statement for $K=H\\times H'$, where $H\\subseteq {\\mathbb R}^N_+$ and $H'\\subseteq GL_N$ are compact (recall that $GL_N$ is open in ${\\mathbb R}^{N^2}$). \nTo estimate $g_{\\varepsilon}$ we use elliptic coordinates. A consequence of the singular value decomposition is that $A S^{N-1}$ is an ellipsoid whose semiaxes are the singular values of $A$, and thus its diameter is $2\\|A\\|_2$, where the latter is the spectral norm of $A$.\nThe corresponding elliptic coordinates are uniquely defined by\n\\[y=\\rho\\omega, \\quad \\omega\\in AS^{N-1},\\quad \\rho>0,\\]\nfor $y\\in {\\mathbb R}^N\\setminus\\{0\\}$. It holds $dy=\\rho^{N-1}d\\omega\\, d\\rho$ where $d\\omega$ is the surface element of $A S^{N-1}$. Setting \n\\[e_A:=A^{-1}e_N,\\quad E_A:=\\{x\\in {\\mathbb R}^N:\\,x\\cdot e_A\\geq 0\\},\\]\nwe compute, through the change of variable $z=A^{-1}y$,\n\\begin{align*}\n&g_{\\varepsilon}(x, A)\\\\\n&=\\int_{B^c_\\varepsilon}\\frac{(u(x)-u(x+z))^{p-1}}{|Az|^{N+ps}}\\, dz=|{\\rm det} A|^{-1}\\int_{AB^c_\\varepsilon}\\frac{(u(x)-u(x+A^{-1}y))^{p-1}}{|y|^{N+ps}}\\, dy\\\\\n&=\\int_{AS^{N-1}}\\frac{1}{|{\\rm det} A||\\omega|^{N+ps}}\\int_\\varepsilon^{\\infty}\\frac{(u_1(x_N) - u_1(x_N+\\omega\\cdot e_A\\rho))^{p-1}}{\\rho^{1+ps}}\\, d\\rho\\, d\\omega\\\\\n&=\\int_{AS^{N-1}\\cap E_A}\\!\\!\\frac{|\\omega\\cdot e_A|^{1+ps}}{|{\\rm det} A||\\omega|^{N+ps}}\\int_{(-\\varepsilon, \\varepsilon)^c}\\frac{(u_1(x_N)-u_1(x_N+\\omega\\cdot e_A\\rho))^{p-1}}{|\\omega\\cdot e_A\\rho|^{1+ps}}\\, d (\\omega\\cdot e_A\\rho)\\, d\\omega\\\\\n&=\\int_{AS^{N-1}\\cap E_A}\\frac{|\\omega\\cdot e_A|^{1+ps}}{|{\\rm det} A||\\omega|^{N+ps}}\\,g^{(1)}_{\\omega\\cdot e_A\\varepsilon}(x_N) \\, d\\omega,\n\\end{align*}\nwhere $g^{(1)}_{\\omega\\cdot e_A\\varepsilon}$ is defined as in \\eqref{defg}. Since $|\\omega\\cdot e_A|\\leq \\|A\\|_2\\|A^{-1}\\|_2$, the condition\n\\[\\omega\\cdot e_A\\varepsilons>0$ it follows\n\\[\\frac{\\partial \\psi(x_N, t)}{\\partial t}=s\\frac{(x_N-t)^{s-1}t-x_N^s+(x_N-t)^s}{t^{1+s}}\\geq 0, \\quad \\text{for $0< t\\le x_N$}.\\]\nTherefore $\\psi(x_N, t)$ is non-decreasing in $t$, thus we get\n\\begin{align*}\n|g_{\\varepsilon}(x, A)|&\\leq Cx_N^{-s}\\psi(x_N, \\|A\\|_2\\|A^{-1}\\|_2\\varepsilon)\\int_{AS^{N-1}}\\frac{|\\omega\\cdot e_A|^{1+ps}}{|{\\rm det} A||\\omega|^{N+ps}}\\, d\\omega\\\\\n&\\leq Cx_N^{-s}\\psi(x_N, \\|A\\|_2\\|A^{-1}\\|_2\\varepsilon)\\int_{S^{N-1}}\\frac{|\\omega\\cdot e_N|^{1+ps}}{|A\\omega|^{N+ps}}\\, d\\omega\\\\\n&\\leq Cx_N^{-s}\\psi(x_N, \\|A\\|_2\\|A^{-1}\\|_2\\varepsilon)\\|A^{-1}\\|_2^{N+ps}.\n\\end{align*}\nNow $ \\|A\\|_2$ and $\\|A^{-1}\\|_2$ are bounded on $H'$ from below and above, as well as $x_N$ on $H$, and the uniform convergence follows. As in the previous proof, it is readily checked that $u\\in \\widetilde{W}^{s,p}(V)$ for any bounded $V$, and the second statement follows as before. \n\\end{proof}\n\n\\begin{remark}\\label{rotate}\nDue to rotational invariance, Lemma \\ref{solnd} easily extends to any half-space\n\\[H_e=\\{x\\in{\\mathbb R}^N:\\,x\\cdot e\\ge 0\\} \\quad (e\\in S^{N-1}),\\]\nsimply considering the solution $u(x)=(x\\cdot e)_+^s$.\n\\end{remark}\n\n\\noindent\nThe following lemma gives a control on the behaviour of $(-\\Delta)^s_p (x_N)^s_+$ under a smooth change of variables.\n\\begin{lemma}[Change of variables]\n\\label{lemmadiffeo}\nLet $\\Phi$ be a $C^{1,1}$ diffeomorphism of ${\\mathbb R}^N$ such that $\\Phi=I$ in $B_r^c$, $r>0$. Then the function $v(x)=(\\Phi^{-1}(x)\\cdot e_N)_+^s$ belongs to $\\widetilde{W}^{s,p}_{\\rm loc}({\\mathbb R}^N)$ and is a weak solution of $(-\\Delta)^s_p v=f$ in $\\Phi({\\mathbb R}^N_+)$, with\n\\begin{equation}\n\\label{stimaf}\n\\|f\\|_\\infty\\leq C(\\|D\\Phi\\|_\\infty, \\|D\\Phi^{-1}\\|_\\infty, r)\\|D^2\\Phi\\|_\\infty.\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nFirst we recall that, since $D\\Phi$ is globally Lipschitz in ${\\mathbb R}^N$ with constant $L>0$, \nthen $D^2\\Phi(x)$ exists in the classical sense for a.a. $x\\in{\\mathbb R}^N$, and $\\|D^2\\Phi\\|_{L^\\infty({\\mathbb R}^N)}\\le L$. Let $J_\\Phi(\\cdot)=|{\\rm det}\\,D\\Phi(\\cdot)|$, $u_1(t)=t_+^s$. Due to Lemma \\ref{symmset}, applied with $A_\\varepsilon=\\{|\\Phi^{-1}(x)-\\Phi^{-1}(y)|<\\varepsilon\\}$ it suffices to show that\n\\[g_\\varepsilon(x)=\\int_{\\{|\\Phi^{-1}(x)-\\Phi^{-1}(y)|\\ge\\varepsilon\\}}\\frac{(v(x)-v(y))^{p-1}}{|x-y|^{N+ps}}\\, dy\\]\nconverges in $L^1(K)$ for any compact $K\\subseteq \\Phi({\\mathbb R}^N_+)$. Changing variables $x=\\Phi(X)$, this is equivalent to claiming that\n\\begin{equation}\n\\label{claimgepsilon}\nX\\mapsto \\int_{B_\\varepsilon^c(X)}\\frac{(u_1(X_N)-u_1(Y_N))^{p-1}}{|\\Phi(X)-\\Phi(Y)|^{N+ps}}J_\\Phi(Y)\\, dY\n\\end{equation}\nconverges as $\\varepsilon\\to 0$ in $L^1_{\\rm loc}({\\mathbb R}^N_+)$. To prove this claim, we write\n\\begin{equation}\\label{diffeo1}\n\\begin{split}\ng_\\varepsilon(x) &= \\int_{B_\\varepsilon^c(X)}\\frac{(u_1(X_N)-u_1(Y_N))^{p-1}}{|D\\Phi(X)(X-Y)|^{N+ps}}h(X, Y)\\, dY\\\\\n&\\quad +\\int_{B_\\varepsilon^c(X)}J_\\Phi(X)\\frac{(u_1(X_N)-u_1(Y_N))^{p-1}}{|D\\Phi(X)(X-Y)|^{N+ps}}\\, dY,\n\\end{split}\n\\end{equation}\nwhere \n\\[h(X, Y)=\\frac{|D\\Phi(X)(X-Y)|^{N+ps}}{|\\Phi(X)-\\Phi(Y)|^{N+ps}} J_\\Phi(Y)-J_\\Phi(X), \\quad X\\neq Y.\\]\nWe will now prove the following estimate, from which convergence of \\eqref{claimgepsilon} will follow:\n\\begin{equation}\n\\label{diffeo2}\n|h(X, Y)|\\leq C_\\Phi\\|D^2\\Phi\\|_\\infty\\min\\{|X-Y|, 1\\},\n\\end{equation}\nwhere $C_\\Phi$ depends on $N$, $p$, $s$ as well as on $\\|D\\Phi\\|_\\infty$, $\\|D\\Phi^{-1}\\|_\\infty$ and $r$. Write\n\\begin{align*}\nh(X, Y) &=\\frac{|D\\Phi(X)(X-Y)|^{N+ps}}{|\\Phi(X)-\\Phi(Y)|^{N+ps}} (J_\\Phi(Y)-J_\\Phi(X))\\\\\n&\\quad +J_\\Phi(X)\\Big(\\frac{|D\\Phi(X)(X-Y)|^{N+ps}}{|\\Phi(X)-\\Phi(Y)|^{N+ps}} -1\\Big)\\\\\n&=:J_1+J_2.\n\\end{align*}\nFirst observe that using Taylor formula yields\n\\[\\frac{|D\\Phi(X)(X-Y)|}{|\\Phi(X)-\\Phi(Y)|}\\leq C\\|D\\Phi\\|_\\infty\\|D\\Phi^{-1}\\|_\\infty,\\]\ntherefore\n\\[|J_1| \\le \\tilde{C}\\|D^2\\Phi\\|_{L^\\infty({\\mathbb R}^N)}|X-Y|.\\]\nTo estimate $J_2$, we note that the mapping $t\\mapsto t^{(N+ps)\/2}$ is smooth in a neighborhood of $1$ and that\n\\[\\lim_{Y\\to X}\\frac{|D\\Phi(X)(X- Y)|^2}{|\\Phi(X)-\\Phi(Y)|^2}=1,\\]\nhence\n\\begin{equation}\n\\label{j2}\n|J_2|\\leq C_\\Phi\\Big(\\frac{|D\\Phi(X)(X-Y)|^{2}}{|\\Phi(X)-\\Phi(Y)|^{2}} -1\\Big).\n\\end{equation}\nBesides, for all $Y\\in{\\mathbb R}^N$ there exist $\\tau_1,\\ldots,\\tau_N\\in[0,1]$ such that\n\\[\\Phi^i(X)-\\Phi^i(Y)=D\\Phi^i(\\tau_i X+(1-\\tau_i)Y)\\cdot(X-Y), \\quad i=1,\\ldots,N,\\]\nwhere $\\Phi^i$ denotes the $i$-th component of $\\Phi$. So we have (still allowing $C_\\Phi>0$ to depend on $\\|D\\Phi\\|_{L^\\infty({\\mathbb R}^N)}$)\n\\begin{align*}\n&\\big||\\Phi(X)-\\Phi(Y)|^2-|D\\Phi(X)(X-Y)|^2\\big| \\\\\n&= \\big|(\\Phi(X)-\\Phi(Y)+D\\Phi(X)(X-Y))\\cdot(\\Phi(X)-\\Phi(Y)-D\\Phi(X)(X-Y))\\big| \\\\\n&\\le C_\\Phi|X-Y|\\sum_{i=1}^N|\\Phi^i(X)-\\Phi^i(Y)-D\\Phi^i(X)(X-Y)| \\\\\n&\\le C_\\Phi|X-Y|^2\\sum_{i=1}^N|D\\Phi^i(\\tau_i X+(1-\\tau_i)Y)-D\\Phi^i(X)| \\\\\n&\\le C_\\Phi\\|D^2\\Phi\\|_\\infty|X-Y|^3.\n\\end{align*}\nInserting into \\eqref{j2} we obtain\n\\begin{equation*}\n|J_2| \\le C_\\Phi\\frac{\\big||D\\Phi(X)(X-Y)|^2-|\\Phi(X)-\\Phi(Y)|^2\\big|}{|\\Phi(X)-\\Phi(Y)|^2} \n\\le C_\\Phi\\|D^2\\Phi\\|_\\infty|X-Y|,\n\\end{equation*}\nwhich yields\n\\[|h(X, Y)|\\leq C_\\Phi\\|D^2\\Phi\\|_{L^\\infty({\\mathbb R}^N)}|X-Y|,\\quad \\text{for all $X, Y\\in {\\mathbb R}^N$},\\]\nand thus \\eqref{diffeo2} for $|X-Y|\\leq 2r$. Assume now $|X-Y|>2r$, then at least one of $X$, $Y$ lies in $\\overline B_r^c$. Clearly, if $X,Y\\in\\overline B_r^c$, then $h(X,Y)=0$. If $X\\in\\overline B_r$, $Y\\in\\overline B_r^c$, then for any $1\\le i\\le N$ we define a mapping $\\eta_i\\in C^{1,1}([0,1])$ by setting\n\\[\\eta_i(t)=\\Phi^i(X+t(Y-X)).\\]\nIt is readily checked that $|\\eta''_i|\\leq C\\|D^2\\Phi\\|_\\infty|X-Y|^2$ for a.e.\\ $t\\in (0,1)$.\nMoreover, if $t\\geq 2r\/|X-Y|$ then $X+t(Y-X)\\in B_r^c$, and since $\\Phi=I$ outside $B_r$ it holds \n\\[\\eta_i(t)=(X+t(Y-X))\\cdot e_i\\quad \\text{for $t\\geq \\frac{2r}{|X-Y|}$}.\\]\nTherefore $\\eta_i''(t)\\equiv 0$ for $t\\geq 2r\/|X-Y|$ and applying the Taylor formula with integral remainder we have\n\\begin{align*}\n&|\\Phi^i(Y)-\\Phi^i(X)+D\\Phi^i(X)(X-Y)|\\\\\n&=|\\eta_i(1)-\\eta_i(0)-\\eta'_i(0)|\\leq \\int_0^1|\\eta''_i(t)|(1-t)\\,dt\\\\\n&\\leq \\int_0^{2r\/|X-Y|}|\\eta''_i(t)|(1-t)\\,dt\\le C_\\Phi\\|D^2\\Phi\\|_\\infty|X-Y|.\n\\end{align*}\nSo we have\n\\begin{align*}\n&|h(X,Y)|\\\\\n&\\le \\Big|\\frac{|D\\Phi(X)(X-Y)|^{N+ps}}{|\\Phi(X)-\\Phi(Y)|^{N+ps}}-1\\Big|+|1-J_\\Phi(X)| \\\\\n&\\le C_\\Phi\\Big|\\frac{|D\\Phi(X)(X-Y)|^2-|\\Phi(X)-\\Phi(Y)|^2}{|\\Phi(X)-\\Phi(Y)|^2}\\Big|+C_\\Phi\\|D^2\\Phi\\|_\\infty \\\\\n&\\le C_\\Phi\\frac{\\big|D\\Phi(X)(X-Y)+\\Phi(X)-\\Phi(Y)\\big|}{|\\Phi(X)-\\Phi(Y)|^2}\\\\\n&\\quad \\cdot\\big|D\\Phi(X)(X-Y)-\\Phi(X)+\\Phi(Y)\\big|+C_\\Phi\\|D^2\\Phi\\|_\\infty \\\\\n&\\le \\frac{C_\\Phi}{|X-Y|}\\sum_{i=1}^N\\big|D\\Phi^i(X)(X-Y)-\\Phi^i(X)+\\Phi^i(Y)\\big|+C_\\Phi\\|D^2\\Phi\\|_\\infty \\\\\n&\\le C_\\Phi\\|D^2\\Phi\\|_\\infty.\n\\end{align*}\nIf $X\\in\\overline B_r^c$, $Y\\in\\overline B_r$, we argue in a similar way. Thus \\eqref{diffeo2} is achieved for all $X,Y\\in{\\mathbb R}^N$.\n\\vskip2pt\n\\noindent\nLet us go back to \\eqref{diffeo1}. The first integral can be estimated as follows:\n\\begin{equation}\n\\label{hsf}\n\\begin{split}\n&\\int_{B_\\varepsilon^c(X)}\\Big|\\frac{(u_1(X_N)-u_1(Y_N))^{p-1}}{|D\\Phi(X)(X-Y)|^{N+ps}}h(X, Y)\\Big|\\, dY \\\\\n&\\leq C_\\Phi\\|D^2\\Phi\\|_\\infty\\int_{B_\\varepsilon^c(X)}\\frac{\\min\\{|X-Y|,1\\}}{|X-Y|^{N+s}}\\, dY\\\\\n&\\leq C_\\Phi\\|D^2\\Phi\\|_\\infty\\Big(\\int_\\varepsilon^1\\frac{1}{t^{s}}\\, dt+\\int_1^{\\infty}\\frac{1}{t^{1+s}}\\, dt\\Big)\\\\\n\\noalign{\\vskip2pt}\n&\\leq C_\\Phi\\|D^2\\Phi\\|_\\infty(\\varepsilon^{1-s}+1).\n\\end{split}\n\\end{equation}\nThe second integral in \\eqref{diffeo1} vanishes for $\\varepsilon\\to 0$, and is estimated through Lemma \\ref{solnd}: since $D\\Phi({\\mathbb R}^N)$ is a compact subset of $GL_N$, the integral vanishes uniformly in any compact $\\Phi^{-1}(K)\\subseteq {\\mathbb R}^N_+$, and therefore uniformly in any compact $K \\subseteq \\Phi({\\mathbb R}^N_+)$. Lemma \\ref{symmset} thus gives that $(-\\Delta)^s_p v= f$ weakly in any open bounded $U\\subseteq \\Phi({\\mathbb R}^N_+)$, \nwhere \n\\[f(x):=2\\lim_{\\varepsilon\\to 0}g_\\varepsilon(x).\\]\nTaking the limit for $\\varepsilon \\to 0$ in estimate \\eqref{hsf} gives \\eqref{stimaf}. \n\\end{proof}\n\n\\noindent\nFinally, we consider a general bounded domain $\\Omega$ with a $C^{1,1}$ boundary. First we recall some geometrical properties, which can be found e.g. in \\cite{AKSZ} (see figure \\ref{geometry}):\n\\begin{figure}\n\\centering\n\\begin{tikzpicture}[scale=1.5]\n\\clip (-3.7,-0.1) rectangle (3.4,4.1);\n\\shade [left color=lightgray, right color=white, shading=axis, shading angle=180] (-3.7,0.3) to [out=45, in=180] (0,2) to [out=0, in=200] (3.4,2) -- (3.4,4) -- (-3.7,4);\n\\draw (-3.4, 3) node[right]{$\\Omega$};\n\\draw[thick] (-3.7,0.3) to [out=45, in=180] (0,2) to [out=0, in=200] (3.4,2);\t\t\n\\draw (0,1) node[right]{$x_2$} circle (1cm);\n\\filldraw (0,1) circle (0.5pt);\n\\draw (1.3,0.5) node {$B_{\\rho}(x_2)$};\n\n\\draw (1.3,2.5) node{$B_{\\rho}(x_1)$};\n\\draw (0,3) node[right]{$x_1$} circle (1cm);\n\\filldraw (0,3) circle (0.7pt);\n\n\\filldraw (0,2) circle (0.5pt);\n\\draw (0,2) node[below right]{$x_0$};\n\n\\draw[thick] (0,2) -- (0, 3);\n\n\\end{tikzpicture}\n\\caption{The interior and exterior balls at $x_0\\in \\partial\\Omega$. For all $x\\in [x_0, x_1]$ it holds $\\delta(x)=|x-x_0|$.}\n\\label{geometry}\n\\end{figure}\n\n\\begin{lemma}\\label{geo1}\nLet $\\Omega\\subset{\\mathbb R}^N$ be a bounded domain with a $C^{1,1}$ boundary $\\partial\\Omega$. Then, there exists $\\rho>0$ such that for all $x_0\\in\\partial\\Omega$ there exist $x_1,x_2\\in{\\mathbb R}^N$ on the normal line to $\\partial\\Omega$ at $x_0$, with the following properties:\n\\begin{enumroman}\n\\item $B_{\\rho}(x_1)\\subset\\Omega$, $B_\\rho(x_2)\\subset\\Omega^c$;\n\\item $\\overline B_\\rho(x_1)\\cap\\overline B_\\rho(x_2)=\\{x_0\\}$;\n\\item $\\delta(x)=|x-x_0|$ for all $x\\in[x_0,x_1]$.\n\\end{enumroman}\n\\end{lemma}\n\n\\noindent\nAs a byproduct, we prove that $(-\\Delta)^s_p\\delta^s$ is bounded in a neighborhood of the boundary.\n\n\\begin{theorem}\\label{deltas}\nLet $\\Omega\\subset{\\mathbb R}^N$ be a bounded domain with a $C^{1,1}$ boundary. There exists $\\rho=\\rho(N, p, s, \\Omega)$ such that $(-\\Delta)_p^s\\delta^s=f$ weakly in \n\\[\\Omega_\\rho:=\\{x\\in \\Omega:\\delta(x)<\\rho\\},\\]\nfor some $f\\in L^\\infty(\\Omega_\\rho)$.\n\\end{theorem}\n\\begin{proof}\nSuppose that $\\rho$ is smaller than the one given in Lemma \\ref{geo1}. We choose a finite covering of $\\Omega_{\\rho}$ made of balls of radius $2\\rho$ and center $x_i\\in \\partial\\Omega$. Using a partition of unity, it suffices to prove the statement in any set $\\Omega \\cap B_{2\\rho}(x_i)$. To do so, we flatten the boundary near the point $x_i$, which we can suppose without loss of generality to be the origin. Choosing a smaller $\\rho$ (depending only on the geometry of $\\partial\\Omega$) if necessary, there exists a diffeomorphism $\\Phi\\in C^{1,1}({\\mathbb R}^N,{\\mathbb R}^N)$, $\\Phi(X)=x$ such that $\\Phi=I$ in $B_{4\\rho}^c$ and\n\\begin{equation}\n\\label{hasd}\n\\Omega\\cap B_{2\\rho}\\Subset \\Phi(B_{3\\rho}\\cap {\\mathbb R}^N_+),\\quad \\delta(\\Phi(X))=(X_N)_+,\\quad \\forall X\\in B_{3\\rho}.\n\\end{equation}\nWe claim that \n\\[g_\\varepsilon(x)=\\int_{\\{|\\Phi^{-1}(x)-\\Phi^{-1}(y)|\\geq \\varepsilon\\}}\\frac{(\\delta^{s}(x)-\\delta^s(y))^{p-1}}{|x-y|^{N+ps}}\\, dy\\to f(x)\n\\quad\\text{in $L^1_{\\rm loc}(\\Omega\\cap B_{2\\rho})$}.\\]\nWe change variables setting $X=\\Phi^{-1}(x)$, noting that $X\\in B_{3\\rho}\\cap {\\mathbb R}^N_+$ for any $x\\in \\Omega\\cap B_{2\\rho}$, and compute\n\\begin{align*}\ng_\\varepsilon(x)&= \\int_{\\{|X-Y|\\geq \\varepsilon\\}}\\frac{(\\delta^{s}(\\Phi(X))-\\delta^s(\\Phi(Y)))^{p-1}}{|\\Phi(X)-\\Phi(Y)|^{N+ps}}J_\\Phi(Y)\\, dY\\\\\n&=\\int_{B_\\varepsilon^c(X)\\cap B_{3\\rho}}\\frac{(\\delta^{s}(\\Phi(X))-\\delta^s(\\Phi(Y)))^{p-1}}{|\\Phi(X)-\\Phi(Y)|^{N+ps}}J_\\Phi(Y)\\, dY\\\\\n&+\\int_{B_{3\\rho}^c}\\frac{(\\delta^{s}(\\Phi(X))-\\delta^s(\\Phi(Y)))^{p-1}}{|\\Phi(X)-\\Phi(Y)|^{N+ps}}J_\\Phi(Y)\\, dY\\\\\n&=\\int_{B_\\varepsilon^c(X)}\\frac{ (u_1(X_N)-u_1(Y_N))^{p-1}}{|\\Phi(X)-\\Phi(Y)|^{N+ps}}J_\\Phi(Y)\\, dY\\\\\n&\\quad +\\int_{B_{3\\rho}^c}\\frac{(\\delta^{s}(\\Phi(X))-\\delta^s(\\Phi(Y)))^{p-1}-(u_1(X_N)-u_1(Y_N))^{p-1}}{|\\Phi(X)-\\Phi(Y)|^{N+ps}}J_\\Phi(Y)\\, dY\\\\\n&= f_{1,\\varepsilon}(X)+f_2(X),\n\\end{align*}\nfor sufficiently small $\\varepsilon$, where we used the fact that\n\\[\\delta^s(\\Phi(Z))=u_1(Z_N)\\quad \\text{for all $Z\\in B_{3\\rho}$}\\]\nthanks to \\eqref{hasd}.\nClearly $f_2\\circ \\Phi^{-1}\\in L^1(\\Omega\\cap B_{2\\rho})$, and to estimate its $L^\\infty$-norm we observe that, due to \\eqref{hasd}, \n\\[{\\rm dist}(\\Phi^{-1}(\\Omega\\cap B_{2\\rho}), B_{3\\rho}^c)>\\theta_{\\Phi, \\rho}>0.\\]\nThen, using the $s$-H\\\"older regularity of $\\delta^s\\circ\\Phi$ and $u_1$, and recalling that $\\Phi^{-1}\\in {\\rm Lip}({\\mathbb R}^N)$ and \\eqref{lkj}, we obtain\n\\begin{align*}\n|f_2(X)|&\\leq C_{\\Phi, \\rho}\\int_{B^c_{3\\rho}}\\frac{|X-Y|^{s(p-1)}}{|X-Y|^{N+ps}}\\, dY\\\\\n&\\leq C_{\\Phi, \\rho}\\int_{{\\mathbb R}^N}\\frac{1}{(1+|Y|)^{N+s}}\\, dY\\\\\n&\\leq C_{\\Phi, \\rho},\\quad \\forall X\\in \\Phi^{-1}(\\Omega\\cap B_{2\\rho}).\n\\end{align*}\nRegarding $f_{1,\\varepsilon}$, it coincides with the $g_\\varepsilon$ of \\eqref{diffeo1}. Therefore claim \\eqref{claimgepsilon} of Lemma \\ref{lemmadiffeo} shows that the limit \n\\[f_1(X):=\\lim_{\\varepsilon\\to 0}\\int_{B_\\varepsilon^c(X)}\\frac{(u_1(X_N)-u_1(Y_N))^{p-1}}{|\\Phi(X)-\\Phi(Y)|^{N+ps}}J_\\Phi(Y)\\, dY\\]\nholds in $L^1_{\\rm loc}({\\mathbb R}^N_+)$, and $\\|f_1\\|_{\\infty}\\leq C_{\\Phi, \\rho}$. Therefore $g_\\varepsilon\\to f_1\\circ \\Phi^{-1}+f_2\\circ\\Phi^{-1}$ in $L^1_{\\rm loc}(\\Omega\\cap B_{2\\rho})$, and both are bounded. Lemma \\ref{symmset} finally gives the conclusion.\n\\end{proof}\n\n\\section{Barriers}\\label{sec4}\n\n\\noindent\nIn this section we provide some barrier-type functions and prove {\\em a priori} bounds for the bounded weak solutions of problem \\eqref{dir}. We begin by considering the simple problem\n\\begin{equation}\\label{dir1}\n\\begin{cases}\n(-\\Delta)^s_p v=1 & \\text{in $B_1$} \\\\\nv=0 & \\text{in $B_1^c$.}\n\\end{cases}\n\\end{equation}\nThe following lemma displays some properties of the solution of \\eqref{dir1}:\n\n\\begin{lemma}\\label{cupola}\nLet $v\\in W^{s,p}_0(B_1)$ be a weak solution of \\eqref{dir1}. Then, $v\\in L^\\infty({\\mathbb R}^N)$ is unique, radially non-increasing, and for all $r\\in(0,1)$ it holds $\\inf_{B_r}v>0$.\n\\end{lemma}\n\\begin{proof}\nFirst we prove uniqueness. Let the functional $J:W^{s,p}_0(B_1)\\to{\\mathbb R}$ be defined by\n\\[J(u)=\\frac{1}{p}\\int_{{\\mathbb R}^N\\times{\\mathbb R}^N}\\frac{|u(x)-u(y)|^p}{|x-y|^{N+ps}}\\,dx\\,dy-\\int_{B_1}u(x)\\,dx.\\]\n$J$ is strictly convex and coercive, hence it admits a unique global minimizer $v\\in W^{s,p}_0(B_1)$, which is the only weak solution of \\eqref{dir1}. By Lemma \\ref{hs} \\ref{hs.3} we see that $v$ is radially symmetric, that is, $v(x)=\\psi(|x|)$ for all $x\\in{\\mathbb R}^N$, where $\\psi:{\\mathbb R}_+\\to{\\mathbb R}_+$ is a mapping vanishing in $[1,\\infty)$. Let $v^\\#$ be the symmetric non-increasing rearrangement of $v$. By the fractional P\\'olya-Szeg\\\"o inequality (see Theorem 3 of \\cite{B}) we have $J(v^\\#)\\le J(v)$, so by uniqueness $v=v^\\#$, that is, $\\psi$ is non-increasing and continuous from the right in ${\\mathbb R}_+$. Now let\n\\[r_0=\\inf\\{r\\in(0,1]:\\,\\psi(r)=0\\}.\\]\nClearly $r_0\\in(0,1]$. Arguing by contradiction, assume $r_0\\in(0,1)$. Then $v\\in W^{s,p}_0(B_{r_0})$ and \nit solves weakly\n\\[\\begin{cases}\n(-\\Delta)^s_p v=1 & \\text{in $B_{r_0}$} \\\\\nv=0 & \\text{in $B^c_{r_0}$.}\n\\end{cases}\\]\nReasoning as above and using uniqueness and Lemma \\ref{hs} \\ref{hs.2}, we see that $v(x)=r_0^{-ps}v(r_0^{ps}x)$ in $B_{r_0}$, so\n\\[\\psi(r_0^2)=r_0^{ps}\\psi(r_0)=0,\\]\nwith $r_0^20.\\]\nFinally, we prove that $v\\in L^\\infty({\\mathbb R}^N)$. Let $w\\in C^s({\\mathbb R}^N)\\cap \\widetilde{W}^{s,p}(B_1)$ be defined by\n\\[w(x)=\\min\\{(2-x_N)_+^s,5^s\\}.\\]\nNotice that $w(x)=(2-x_N)_+^s=u_1(2-x_N)$ for all $x\\in B_2$. Thus we can apply Lemma \\ref{psadd} in $B_{3\/2}$ to \n\\[w(x)=u_1(2-x_N)-(u_1(2-x_N)-5^s)_+\\]\nto get, by Lemma \\ref{solnd} \n\\begin{align*}\n(-\\Delta)^s_p w(x) &= 2\\int_{\\{y_N\\le-3\\}}\\frac{((2-x_N)_+^s-5^s)^{p-1}-((2-x_N)_+^s-(2-y_N)_+^s)^{p-1}}{|x-y|^{N+ps}}\\,dy\\\\\n&=: I(x)\n\\end{align*}\nweakly in $B_1$. The function $I:\\bar B_1\\to{\\mathbb R}$ is continuous and positive, so there exists $\\alpha>0$ such that\n\\[(-\\Delta)^s_p w(x)\\ge\\alpha \\ \\text{weakly in $B_1$.}\\]\nWe set $\\tilde w=\\alpha^{-1\/(p-1)}w$, so we have\n\\[\\begin{cases}\n(-\\Delta)^s_p v=1\\le(-\\Delta)^s_p\\tilde w & \\text{weakly in $B_1$} \\\\\nv=0\\le\\tilde w & \\text{in $B^c_1$,}\n\\end{cases}\\]\nand Proposition \\ref{comp} yelds\n\\[0\\le v \\le\\tilde w \\le \\frac{5^s}{\\alpha^\\frac{1}{p-1}},\\quad \\text{in ${\\mathbb R}^N$},\\]\nso $v\\in L^\\infty({\\mathbb R}^N)$, concluding the proof.\n\\end{proof}\n\n\\noindent\nNext we introduce {\\em a priori} bounds for functions with bounded fractional $p$-Laplacian.\n\n\\begin{corollary}[$L^\\infty$-bound]\\label{apb}\nLet $u\\in W^{s,p}_0(\\Omega)$ satisfy $|(-\\Delta)^s_p u|\\le K$ weakly in $\\Omega$ for some $K>0$. Then\n\\[\\|u\\|_\\infty\\le (C_d K)^\\frac{1}{p-1},\\]\nfor some $C_d=C(N,p,s, d)$, $d={\\rm diam}(\\Omega)$.\n\\end{corollary}\n\\begin{proof}\nLet $v\\in W^{s,p}_0(B_1)$ be as in Lemma \\ref{cupola}, $x_0$ such that $\\Omega\\Subset B_d(x_0)$, and set\n\\[\\tilde v(x)=(Kd^{ps})^\\frac{1}{p-1}v\\Big(\\frac{x-x_0}{d}\\Big).\\]\nBy Lemma \\ref{hs} \\ref{hs.1}, \\ref{hs.2} we have weakly\n\\[\\begin{cases}\n(-\\Delta)^s_p u\\le K=(-\\Delta)^s_p\\tilde v & \\text{in $\\Omega$} \\\\\nu=0\\le\\tilde v & \\text{in $\\Omega^c$},\n\\end{cases}\\]\nwhich, by Proposition \\ref{comp}, implies $u\\le\\tilde v$ in ${\\mathbb R}^N$. A similar argument, applied to $-u$, gives the lower bound.\n\\end{proof}\n\n\\noindent\nWe can now produce (local) upper barriers on the complements of balls.\n\n\\begin{lemma}[Local upper barrier]\\label{moon}\nThere exist $w\\in C^s({\\mathbb R}^N)$, and universal $r>0$, $a\\in (0,1)$, $c>1$ with\n\\[\\begin{cases}\n(-\\Delta)^s_p w\\ge a & \\text{weakly in $B_r(e_N)\\setminus\\overline B_1$} \\\\\nc^{-1}(|x|-1)_+^s\\le w(x)\\le c(|x|-1)_+^s & \\text{in ${\\mathbb R}^N$}.\n\\end{cases}\\]\n\\end{lemma}\n\\begin{proof}\nBy translation, rotation invariance and scaling (Lemma \\ref{hs}), it suffices to prove the statement for any fixed ball of radius $R>2$, at any fixed point $\\bar x_R$ of its boundary. To fix ideas, we set $\\tilde x_R=(0,-(R^2-4)^{1\/2})$ and $\\bar x_R=\\tilde x_R+Re_N$, so that $B_R(\\tilde x_R)$ intersects the hyperplane ${\\mathbb R}^{N-1}\\times\\{0\\}$ in the $(N-1)$-ball $\\{|x'|<2\\}$ (we use the notation $x=(x',x_N)\\in{\\mathbb R}^{N-1}\\times{\\mathbb R}$).\n\\vskip2pt\n\\noindent\nIn the following we will choose $R$ large enough, depending only on $N, p, s$. If $R>2$, we can find $\\varphi\\in C^{1,1}({\\mathbb R}^{N-1})$ such that $\\|\\varphi\\|_{C^{1,1}({\\mathbb R}^{N-1})}\\le C\/R$ and\n\\[\\varphi(x')=\\big((R^2-|x'|^2)^{1\/2}-(R^2-4)^{1\/2}\\big)_+ \\ \\text{for all $|x'|\\in[0,1]\\cup[3,\\infty)$.}\\]\nWe set\n\\[U_+=\\{x\\in{\\mathbb R}^N:\\,\\varphi(x')] (-6, 0) -- (6, 0);\n\\draw[->] (0, -5) -- (0, 1.7) ;\n\\draw (0, -4.582) circle (5cm);\n\\filldraw (0, -4.582) circle (1.2pt);\n\\draw (0, -4.582) node[above left]{$\\widetilde{x}_R$} -- node[right=1pt]{$R$} (2,0)\n\n\\filldraw (2, 0) circle (1.2pt);\n\\draw (2.1, -0.4) node{\\small $2$};\n\n\\draw (0, 0.4174) circle (1cm);\n\\draw (1.4,1.45) node{$B_1(\\bar x_R)$};\n\n\\draw (-5, 1) node{$U_+$};\n\\filldraw (0, 0.4174) circle (1.2pt);\n\\draw (0, 0.4174) node[above right]{$\\bar x_R$};\n\n\\end{tikzpicture}\n\\caption{The balls $B_R(\\widetilde{x}_R)$ and $B_1(\\bar x_R)$. The thick line is the graph of $\\varphi$, whose epigraph is $U_+$.}\n\\label{figball}\n\\end{figure}\nWe claim that for any sufficiently large $R$ there exists a diffeomorphism $\\Phi\\in C^{1,1}({\\mathbb R}^N,{\\mathbb R}^N)$ such that $\\Phi(0)=\\bar x_R$, $\\Phi=I$ in $B^c_4$, and\n\\begin{equation}\n\\label{propPhi}\n\\|\\Phi-I\\|_{C^{1,1}({\\mathbb R}^N,{\\mathbb R}^N)}+\\|\\Phi^{-1}-I\\|_{C^{1,1}({\\mathbb R}^N,{\\mathbb R}^N)}\\le\\frac{C}{R}, \\ \\Phi({\\mathbb R}^N_+)=U_+.\n\\end{equation}\nIndeed, let $\\eta\\in C^2({\\mathbb R})$ satisfy $\\eta\\in [0,1]$, $\\eta(0)=1$, ${\\rm supp}\\, \\eta\\subseteq (-1,1)$. Set for all $X=(X',X_N)\\in{\\mathbb R}^{N-1}\\times{\\mathbb R}$\n\\[\\Phi(X)=X+\\varphi(X')\\eta(X_N)e_N.\\]\nThen, for sufficiently large $R$, $\\Phi\\in C^{1,1}({\\mathbb R}^N,{\\mathbb R}^N)$ is a bijection since $\\Phi(X_1)=\\Phi(X_2)$ implies $X_1'=X_2'$, and the map $t\\mapsto t+\\varphi(X')\\eta(t)$ is increasing whenever\n\\[\\sup_{X\\in {\\mathbb R}^N}\\varphi(X')|\\eta'(X_N)|=\\frac{4\\sup_{{\\mathbb R}}|\\eta'|}{R+\\sqrt{R^2-4}}< 1.\\]\nIts inverse mapping satisfies\n\\begin{equation}\\label{moon2}\n\\Phi^{-1}(x)=x-\\varphi(x')\\eta(\\Phi^{-1}(x)\\cdot e_N)e_N \\ \\text{for all $x\\in{\\mathbb R}^N$,}\n\\end{equation}\nbesides $\\Phi(0)=\\bar x_R$. Moreover, for all $X\\in B^c_4$ we have either $|X'|\\ge 3$ or $|X_N|\\ge 1$, in both cases $\\Phi(X)=X$. The $C^{1,1}$-bounds on $\\varphi$, $\\eta$ and \\eqref{moon2} yield the required $C^{1,1}$-bounds on $\\Phi-I$ and $\\Phi^{-1}-I$. Finally, reasoning as above, the monotonicity of $t\\mapsto t+\\varphi(X')\\eta(t)$ implies that $\\Phi({\\mathbb R}^N_+)=U_+$, and \\eqref{propPhi} is proved.\n\\vskip2pt\n\\noindent\nLet $v_1(x)=u_1(\\Phi^{-1}(x)\\cdot e_N)$. Lemma \\ref{lemmadiffeo} ensures that $v_1\\in \\widetilde W^{s,p}_{{\\rm loc}}({\\mathbb R}^N)$ and\n\\begin{equation}\n\\label{pslv1}\n(-\\Delta)^s_p v_1=f\\quad \\text{weakly in $U_+$, with $\\|f\\|_\\infty\\leq C\/R$}.\n\\end{equation}\nDefine \n\\[v(x)=\\min\\{v_1(x), 5^s\\},\\]\nwhich belongs to $\\widetilde{W}^{s,p}(B_4)$. From $\\Phi=I$ in $B_4^c$ we infer $\\Phi^{-1}(B_4)=B_4$ and thus \n\\[v_1(x)=u_1(x_N)\\quad\\text{in $B_4^c$},\\quad v_1\\leq 4^s\\quad\\text{in $B_4$}. \\]\nHence\n\\[v_1(x)-v(x)= (x_N)_+^s-5^s\\quad\\text{in $\\{x_N\\geq 5\\}$},\\quad v_1-v=0\\quad\\text{in $\\{x_N\\leq 5\\}\\Supset B_4$}.\\]\nThus the function $v-v_1$ satisfies conditions \\eqref{hyppertv} in $B_4$, and Lemma \\ref{psadd} provides weakly in $B_4$\n\\[(-\\Delta)^s_p v=(-\\Delta)^s_p (v_1+(v-v_1))=f+g,\\]\nwhere \n\\begin{align*}\ng(x)&=2\\int_{B_4^c}\\frac{(v_1(x)-v(y))^{p-1}-(v_1(x)-v_1(y))^{p-1}}{|x-y|^{N+ps}}\\, dy\\\\\n&\\geq 2\\int_{\\{y_N\\geq 5\\}}\\frac{((x_N)_+^s-5^s)^{p-1}-((x_N)_+^s-(y_N)_+^s)^{p-1}}{|x-y|^{N+ps}}\\, dy\n\\end{align*}\nfor any $x\\in B_4$. As in the proof of Lemma \\ref{cupola}, there is a universal $\\alpha>0$ such that $g(x)\\geq \\alpha$ for all $x\\in B_4$, and therefore using \\eqref{pslv1} we have\n\\[(-\\Delta)^s_p v\\geq f+g\\geq \\alpha - \\frac{C}{R}\\quad \\text{weakly in $U_+\\cap B_4$}.\\]\nTaking $R$ big enough we thus find $B_2(\\bar x_R)\\Subset B_4$ and \n\\begin{equation}\n\\label{moon6}\n(-\\Delta)^s_p v\\geq \\frac{\\alpha}{2}>0,\\quad \\text{weakly in $U_+\\cap B_2(\\bar x_R)$}.\n\\end{equation}\nSet for all $x\\in{\\mathbb R}^N$\n\\[d_R(x)=(|x-\\tilde x_R|-R)_+.\\]\nWe can estimate $v$ by multiples of $d_R^s$ {\\em globally} from above but only {\\em locally} from below. Indeed, since $v=0$ in $U_+^c$, $B_R(\\tilde x_R)\\subset U^c_+$, and $v\\in C^s({\\mathbb R}^N)$, there exists $\\tilde c>1$ such that\n\\begin{equation}\\label{moon7}\nv(x)\\leq \\tilde c\\, {\\rm dist}(x,U^c_+)^s\\leq \\tilde c\\, d_R^s(x), \\,\\,\\,\\,\\, \\text{for all $x\\in {\\mathbb R}^N$.}\n\\end{equation}\nOn the other hand, for all $x\\in B_{1}(\\bar x_R)$ it holds either $x\\in B_1(\\bar x_R)\\setminus U_+\\subseteq B_R(\\tilde x_R)$, in which case $d_R^s(x)=0=\\tilde c v(x)$, or $x\\in B_1(\\bar x_R)\\cap U_+\\subseteq B_R^c(\\tilde x_R)$. In the latter case let $X=(X', X_N)$ be such that $x=\\Phi(X)$, $Z=(X', 0)$ and $z=\\Phi(Z)$. It holds $|X'|\\leq 1$ and by the construction of $\\Phi$, it follows that $z\\in \\partial B_R(\\tilde x_R)$, therefore\n\\[d_R^s(x)\\leq |x-z|^s\\le\\tilde c|X-Z|^s=\\tilde cX_N^s=\\tilde cv(x).\\]\nThus we have (taking $\\tilde c>1$ bigger if necessary)\n\\begin{equation}\\label{moon8}\nv\\ge\\frac{1}{\\tilde c}d_R^s \\ \\text{in $B_{1}(\\bar x_R)$.}\n\\end{equation}\nWe aim at extending \\eqref{moon8} to the whole ${\\mathbb R}^N$, while retaining \\eqref{moon6} and \\eqref{moon7}. For any $\\varepsilon\\in(0,1\/\\tilde c)$ set\n\\[v_\\varepsilon=\\max\\{v,\\varepsilon d_R^s\\}.\\]\nClearly $v_\\varepsilon$ satisfies estimates like \\eqref{moon7} and \\eqref{moon8} in ${\\mathbb R}^N$ with a constant $\\tilde c_\\varepsilon=\\max\\{\\tilde c+\\varepsilon,\\varepsilon^{-1}\\}$. Besides $v\\le v_\\varepsilon\\le v+\\varepsilon d_R^s$ in ${\\mathbb R}^N$, being $\\varepsilon<1\/\\tilde{c}$, $v_\\varepsilon-v=0$ in $B_1(\\bar x_R)$. So, by \\eqref{moon6}, Lemma \\ref{psadd} and \\eqref{in6} (with $M=5^s$ and $q=p-1$)\n\\begin{align*}\n(-\\Delta)^s_p v_\\varepsilon(x)\n&= (-\\Delta)^s_p v(x)-2\\int_{B^c_{1\/2}(\\bar x_R)}\\frac{(v(x)-v(y))^{p-1}-(v(x)-v_\\varepsilon(y))^{p-1}}{|x-y|^{N+ps}}\\,dy \\\\\n&\\ge \\frac{\\alpha}{2}-C\\int_{B^c_{1}(\\bar x_R)}\\frac{\\max\\{\\varepsilon d_R^s(y),(\\varepsilon d_R^s(y))^{p-1}\\}}{|\\bar x_R-y|^{N+ps}}\\,dy \\\\\n&\\ge \\frac{\\alpha}{2}-CJ(\\varepsilon)\n\\end{align*}\nweakly in $B_{1\/2}(\\bar x_R)\\cap U_+$ (in the end we have used the inequality $|x-y|\\ge 1\/2|\\bar x_R-y|$ for all $x\\in B_{1\/2}(\\bar x_R)$, $y\\in B^c_{1}(\\bar x_R)$). Notice that $J(\\varepsilon)\\to 0$ as $\\varepsilon\\to 0^+$ independently of $x$, thus, for $\\varepsilon>0$ small enough we have\n\\[(-\\Delta)^s_p v_\\varepsilon(x)\\ge\\frac{\\alpha}{4}>0 \\ \\text{weakly in $B_{1\/2}(\\bar x_R)\\setminus B_R(\\tilde x_R)$.}\\]\nTo obtain the barrier of the thesis, we set $w(x)=v_\\varepsilon(\\tilde{x}_R+Rx)$ and using Lemma \\ref{hs} we reach the conclusion for $r=1\/(2R)$, $a=\\alpha\/(4R^{ps})$, $c=R^s\\max\\{\\tilde c+\\varepsilon,\\varepsilon^{-1}\\}$.\n\\end{proof}\n\n\\noindent\nFinally, we prove that any bounded weak solution of \\eqref{dir} can be estimated by means of a multiple of $\\delta^s$.\n\n\\begin{theorem}\\label{estid}\nLet $u\\in W^{s,p}_0(\\Omega)$ satisfy $|(-\\Delta)^s_p u|\\le K$ weakly in $\\Omega$ for some $K>0$. Then\n\\begin{equation}\n\\label{thm44tesi}\n|u|\\le (C_\\Omega K)^\\frac{1}{p-1}\\delta^s \\quad \\text{a.e.\\ in $\\Omega$,}\n\\end{equation}\nfor some $C_\\Omega=C(N,p,s,\\Omega)$.\n\\end{theorem}\n\\begin{proof}\nConsidering $u\/K^{1\/(p-1)}$ and using homogeneity, we can prove \\eqref{thm44tesi} in the case $K=1$. Thanks to Corollary \\ref{apb} we may focus on a neighborhood of $\\partial\\Omega$. Let $\\rho>0$ be as in Lemma \\ref{geo1}, and let $r\\in (0,1)$ be defined in Lemma \\ref{moon}. Set\n\\[U=\\Big\\{x\\in\\Omega:\\,\\delta(x)0,\\quad \\text{in $B_{\\rho}^c(x_2)\\setminus B_{r\\rho\/2}(x_0)$}.\\]\nfor a constant $\\theta$ which depends only on $\\rho$ and $r$ (and thus on $\\Omega$ alone).\nSince $\\Omega\\subseteq B_{\\rho}^c(x_2)$, the latter inequality together with \\eqref{estid3} provides\n\\begin{equation}\n\\label{estid4}\nw\\geq c^{-1}\\theta^s=:\\alpha>0,\\quad \\text{in $\\Omega\\setminus B_{r\\rho\/2}(x_0)$}.\n\\end{equation}\nWe define the open set\n\\[V=\\Omega\\cap B_{r\\frac{\\rho}{2}}(x_0)\\subseteq B_{r\\frac{\\rho}{2}}(x_0)\\setminus B_{\\rho\/2}(x_1),\\]\nwhere we will apply the comparison principle. Suppose without loss of generality that in \\eqref{estid4} $\\alpha\\in (0,1)$ and let $C_d>1$ be as in Corollary \\ref{apb}. Set\n\\[M=\\frac{1}{\\alpha}\\Big(\\frac{C_d}{a}\\Big)^\\frac{1}{p-1}, \\quad \\bar w=Mw.\\]\nBy \\eqref{estid2} and $C_d\/\\alpha^{p-1}\\geq 1$ we have\n\\[(-\\Delta)^s_p \\bar w=M^{p-1}(-\\Delta)^s_p w\\geq \\frac{C_d}{\\alpha^{p-1}}\\geq 1\\geq (-\\Delta)^s_p u,\\quad \\text{weakly in $V$}.\\]\nMoreover $u=0\\leq \\bar w$ in $\\Omega^c$, while \\eqref{estid4}, $a<1$ and Corollary \\ref{apb} give \n\\[\\bar w\\geq M\\alpha=\\Big(\\frac{C_d}{a}\\Big)^\\frac{1}{p-1}\\geq \\sup_{\\Omega} u,\\quad \\text{in $\\Omega\\setminus B_{r\\rho\/2}(x_0)$}.\\] \nTherefore $\\bar w\\geq u$ in the whole $V^c$, and Proposition \\ref{comp} together with \\eqref{estid3} yelds\n\\[u(x)\\leq \\bar w(x)\\leq cMd^s(x)\\quad \\text{for a.e.\\ $x\\in {\\mathbb R}^N$}.\\]\nRecalling \\eqref{estid2b} we get\n\\[u(\\bar x)\\leq cMd^s(\\bar x)=cM\\delta^s(\\bar x)\\quad \\text{for all $\\bar x=x_0-tn_{x_0}$, $t\\in\\Big[0, r\\frac{\\rho}{2}\\Big]$},\\]\nwhere $n_{x_0}$ is the exterior normal to $\\partial\\Omega$ at $x_0$, which gives the thesis since $cM$ depends only on $N, p, s$, $\\rho$, $r$, and $\\Omega$. A similar argument applied to $-u$ yields the lower bound.\n\\end{proof}\n\n\\section{H\\\"older regularity}\\label{sec5}\n\n\\noindent\nIn this section we will obtain the H\\\"older regularity of solutions.\n\n\\subsection{Interior H\\\"older regularity}\n\n\\noindent\nWe now study the behavior of a weak supersolution in a ball, proving a weak Harnack inequality. Then we will obtain an estimate of the oscillation of a bounded weak solution in a ball (this can be interpreted as a first interior H\\\"older regularity result). All balls are meant to be centered at $0$, as translation invariance of $(-\\Delta)^s_p$ allows to extend the results to balls centered at any point.\n\\vskip2pt\n\\noindent\nWe begin with a curious Jensen-type inequality:\n\n\\begin{lemma}\\label{jensen}\nLet $E\\subset{\\mathbb R}^N$ be a set of finite measure and let $u\\in L^1(E)$ satisfy\n\\[\\Xint-_E u\\,dx=1.\\]\nThen, for all $r\\ge 1$ and $\\lambda\\ge 0$, it holds\n\\[\\Xint-_E(u^r-\\lambda^r)^\\frac{1}{r}\\,dx\\ge 1-2^\\frac{r-1}{r}\\lambda.\\]\n\\end{lemma}\n\\begin{proof}\nAvoiding trivial cases, we assume $r>1$ and $\\lambda>0$. Set, for all $t\\in{\\mathbb R},$\n\\[g(t)=(t^r-\\lambda^r)^\\frac{1}{r}.\\]\nThen, for all $t\\in{\\mathbb R}\\setminus\\{0,\\lambda\\}$, we have\n\\[g'(t)=|t^r-\\lambda^r|^\\frac{1-r}{r}|t|^{r-1}.\\]\nIn particular, $t_\\lambda=2^{-1\/r}\\lambda$ is the only solution of $g'(t)=1$. Elementary calculus shows that for all $t\\in{\\mathbb R}$\n\\[g(t)\\ge g(t_\\lambda)+g'(t_\\lambda)(t-t_\\lambda)=t-2t_\\lambda.\\]\nSo we have\n\\[\\Xint-_E (g\\circ u)\\,dx\\ge\\Xint-_E(u-2t_\\lambda)\\,dx=1-2^\\frac{r-1}{r}\\lambda,\\]\nwhich concludes the proof.\n\\end{proof}\n\n\\noindent\nNow we prove a weak Harnack-type inequality for non-negative supersolutions:\n\n\\begin{theorem}[Weak Harnack inequality]\\label{harnack}\nThere exist universal $\\sigma\\in(0,1)$, $\\bar C>0$ with the following property: if $u\\in \\widetilde{W}^{s,p}(B_{R\/3})$ satisfies weakly\n\\[\\begin{cases}\n(-\\Delta)^s_p u\\ge -K & \\text{in $B_{R\/3}$} \\\\\nu\\ge 0 & \\text{in ${\\mathbb R}^N$,}\n\\end{cases}\\]\nfor some $K\\ge 0$, then\n\\[\\inf_{B_{R\/4}}u\\ge\\sigma\\Big(\\Xint-_{B_R\\setminus B_{R\/2}}u^{p-1}\\,dx\\Big)^\\frac{1}{p-1}-\\bar C(KR^{ps})^\\frac{1}{p-1}.\\]\n\\end{theorem}\n\\begin{proof}\nWe first consider the case $p\\ge 2$. Let $\\varphi\\in C^\\infty({\\mathbb R}^N)$ be such that $0\\le\\varphi\\le 1$ in ${\\mathbb R}^N$, $\\varphi=1$ in $B_{3\/4}$, $\\varphi=0$ in $B^c_1$, and by Proposition \\ref{weak-strong} $|(-\\Delta)^s_p\\varphi|\\le C_1$ weakly in $B_1$. We rescale by setting $\\varphi_R(x)=\\varphi(3x\/R)$, so $\\varphi_R\\in C^\\infty({\\mathbb R}^N)$, $0\\le\\varphi_R\\le 1$ in ${\\mathbb R}^N$, $\\varphi_R=1$ in $B_{R\/4}$, $\\varphi_R=0$ in $B^c_{R\/3}$, and $|(-\\Delta)^s_p\\varphi_R|\\le C_1R^{-ps}$ weakly in $B_{R\/3}$ (taking $C_1$ bigger).\n\\vskip2pt\n\\noindent\nFor all $\\sigma\\in(0,1)$ we set\n\\[L=\\Big(\\Xint-_{B_R\\setminus B_{R\/2}}u^{p-1}\\,dx\\Big)^\\frac{1}{p-1}, \\quad w=\\sigma L\\varphi_R+\\chi_{B_R\\setminus B_{R\/2}} u\\]\n(see figure \\ref{w-barr}).\n\\begin{figure}\n\\centering\n\\begin{tikzpicture}[scale=1.8]\n\\draw[->] (-3, 0) -- (3, 0)\n\\draw[->] (0, -0.5) -- (0, 2)\n\n\\draw[thick, rounded corners=3pt] (-0.68, 0) -- (-0.6, 0) -- (-0.54, 0.4) -- node[above=2pt, fill=white, inner sep=2pt]{$\\sigma L\\varphi$} (0.54, 0.4) -- (0.6, 0) -- (0.68,0);\n\\draw (-0.47, 0) node[below]{\\small $\\frac{R}{4}$} -- (-0.47, 0.4);\n\n\\draw[thick] (-2,0) node[below]{\\small $R$} -- (-2, 0.7) parabola bend (-1.8, 1) (-1.6,0.5) parabola bend (-1.45, 0.9) (-1.4, 0.4) parabola bend (-1.2, 0.8) (-1,0.6) -- (-1,0) node[below]{\\small $\\frac{R}{2}$} -- (-0.65, 0) ;\n\\draw (-1.5,1.4) node{$\\chi_{B_R\\setminus B_{R\/2}}u$};\n\\draw[thick] (0.66,0) -- (1,0) -- (1,0.9) parabola bend (1.1, 1.25) (1.38,0.3) parabola bend (1.43, 0.6) (1.5,0.2) parabola bend (1.6,1) (1.78,0.4) parabola bend (1.85, 0.8) (2,0.6) -- (2,0);\n\\draw (1.5,1.4) node{$\\chi_{B_R\\setminus B_{R\/2}}u$};\n\n\\end{tikzpicture}\n\\caption{The lower barrier $w$.}\n\\label{w-barr}\n\\end{figure}\nSo $w\\in \\widetilde{W}^{s,p}(B_{R\/3})$ and by Lemma \\ref{psadd} and \\eqref{in1} we have, weakly in $B_{R\/3}$,\n\\begin{align*}\n&(-\\Delta)^s_p w(x)\\\\\n&= (-\\Delta)^s_p(\\sigma L\\varphi_R)(x)+2\\int_{B_R\\setminus B_{R\/2}}\\frac{(\\sigma L\\varphi_R(x)-u(y))^{p-1}-(\\sigma L\\varphi_R(x))^{p-1}}{|x-y|^{N+ps}} \\, dy\\\\\n&\\le \\frac{C_1(\\sigma L)^{p-1}}{R^{ps}}-2^{3-p}\\int_{B_R\\setminus B_{R\/2}}\\frac{u^{p-1}(y)}{|x-y|^{N+ps}}\\,dy \\\\\n&\\le \\frac{C_1(\\sigma L)^{p-1}}{R^{ps}}-\\frac{C_2L^{p-1}}{R^{ps}}.\n\\end{align*}\nIf we assume\n\\[\\sigma<\\min\\Big\\{1,\\Big(\\frac{C_2}{2C_1}\\Big)^\\frac{1}{p-1}\\Big\\},\\]\nwe get the upper estimate\n\\begin{equation}\\label{hk1}\n(-\\Delta)^s_p w(x)\\le-\\frac{C_2L^{p-1}}{2R^{ps}} \\quad \\text{weakly in $B_{R\/3}$.}\n\\end{equation}\nWe set $\\bar C=(2\/C_2)^{1\/(p-1)}$ and distinguish two cases:\n\\begin{itemize}\n\\item if $L\\le\\bar C(KR^{ps})^{1\/(p-1)}$, then clearly\n\\[\\inf_{B_{R\/4}}u\\ge 0\\ge\\sigma L-\\bar C(KR^{ps})^\\frac{1}{p-1};\\]\n\\item if $L>\\bar C(KR^{ps})^{1\/(p-1)}$, then we use \\eqref{hk1} to have \n\\[\\begin{cases}\n(-\\Delta)^s_p w\\le -K\\le (-\\Delta)^s_p u & \\text{weakly in $B_{R\/3}$} \\\\\nw=\\chi_{B_R\\setminus B_{R\/2}} u\\le u & \\text{in $B^c_{R\/3}$,}\n\\end{cases}\\]\nwhich by Proposition \\ref{comp} implies $w\\le u$ in ${\\mathbb R}^N$, in particular\n\\[\\inf_{B_{R\/4}}u\\ge \\sigma L.\\]\n\\end{itemize}\nIn any case we have\n\\[\\inf_{B_{R\/4}}u \\ge \\sigma L-\\bar C(KR^{ps})^\\frac{1}{p-1},\\]\nwhich is the conclusion.\n\\vskip2pt\n\\noindent\nNow we consider the case $p\\in(1,2)$. Due to Remark \\ref{remarkL}, in this case we cannot use the cut-off function $\\varphi$ as before to construct the barrier $w$. We use instead the weak solution $v$ of \\eqref{dir1} introduced in Lemma \\ref{cupola}, recalling that $\\inf_{B_{3\/4}}v>0$, and we set\n\\[\\varphi_R(x)=\\Big(\\inf_{B_{3\/4}}v\\Big)^{-1}v\\Big(\\frac{3x}{R}\\Big),\\] \nso that $0\\le\\varphi_R\\le\\alpha$ (for some universal $\\alpha>0$) in ${\\mathbb R}^N$, $\\varphi_R\\ge 1$ in $B_{R\/4}$, $\\varphi_R=0$ in $B^c_{R\/3}$, and $(-\\Delta)^s_p\\varphi_R= C_1R^{-ps}$ weakly in $B_{R\/3}$. Accordingly, to obtain the estimate \\eqref{hk1} we apply Lemma \\ref{jensen} to the function $(u\/L)^{p-1}$ with $E=B_R\\setminus B_{R\/2}$, $r=1\/(p-1)$, and $\\lambda=(\\sigma\\varphi_R(x))^{p-1}$, so that\n\\[\\Xint-_{B_R\\setminus B_{R\/2}}\\Big(\\frac{u(y)}{L}-\\sigma\\varphi_{R}(x)\\Big)^{p-1}\\,dy\\ge 1-2^{2-p}(\\sigma\\varphi_R(x))^{p-1},\\]\nfor a.e.\\ $x\\in B_{R\/3}$. This, in turn, implies that for a.e.\\ $x\\in B_{R\/3}$\n\\begin{align*}\n&2\\int_{B_R\\setminus B_{R\/2}}\\frac{(\\sigma L\\varphi_R(x)-u(y))^{p-1}-(\\sigma L\\varphi_R(x))^{p-1}}{|x-y|^{N+ps}}\\, dy\\\\ \n&\\le \\frac{C_2}{R^{ps}}\\Xint-_{B_R\\setminus B_{R\/2}}(\\sigma L\\varphi_R(x)-u(y))^{p-1}\\,dy \\\\\n&\\le \\frac{C_2}{R^{ps}}\\big(2^{2-p}(\\sigma L\\varphi_R(x))^{p-1}-L^{p-1}\\big) \\\\\n&\\le -\\frac{C_2L^{p-1}}{2R^{ps}},\n\\end{align*}\nwhere we have chosen $\\sigma<2^{\\frac{p-3}{p-1}}\\alpha^{-1}$. Then, by taking $\\sigma$ even smaller if necessary, we get \\eqref{hk1} and the rest of the proof follows {\\em verbatim}.\n\\end{proof}\n\n\\noindent\nWe need to extend Theorem \\ref{harnack} to supersolutions which are only non-negative in a ball. To do so, we introduce a tail term defined as in \\eqref{deftail}:\n\n\\begin{lemma}\\label{harnackloc}\nThere exist $\\sigma\\in(0,1)$, $\\tilde C>0$, and for all $\\varepsilon>0$ a constant $C_\\varepsilon>0$ with the following property: if $u\\in \\widetilde{W}^{s,p}(B_{R\/3})$ satisfies weakly\n\\[\\begin{cases}\n(-\\Delta)^s_p u\\ge -K & \\text{in $B_{R\/3}$} \\\\\nu\\ge 0 & \\text{in $B_R$,}\n\\end{cases}\\]\nfor some $K\\geq 0$, then\n\\begin{equation}\n\\label{hloctesi}\n\\inf_{B_{R\/4}}u\\ge\\sigma\\Big(\\Xint-_{B_R\\setminus B_{R\/2}}u^{p-1}\\,dx\\Big)^\\frac{1}{p-1}-\\tilde C(KR^{ps})^\\frac{1}{p-1}-C_\\varepsilon{\\rm Tail}(u_-;R)-\\varepsilon\\sup_{B_R}u.\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nFirst we consider the case $p\\ge 2$. We apply Lemma \\ref{psadd} to the functions $u$ and $v=u_-$, so that $u+v=u_+$, and $\\Omega=B_{R\/3}$: we have in a weak sense in $B_{R\/3}$\n\\begin{align*}\n&(-\\Delta)^s_p u_+(x) \\\\\n&= (-\\Delta)^s_p u(x)+2\\int_{B^c_{R\/3}}\\frac{(u(x)-u(y)-u_-(y))^{p-1}-(u(x)-u(y))^{p-1}}{|x-y|^{N+ps}}\\,dy \\\\\n&\\ge -K+2\\int_{\\{u<0\\}}\\frac{u(x)^{p-1}-(u(x)-u(y))^{p-1}}{|x-y|^{N+ps}}\\,dy \\\\\n&\\ge -K+C\\int_{\\{u<0\\}}\\frac{u(x)^{p-1}-(u(x)-u(y))^{p-1}}{|y|^{N+ps}}\\,dy,\n\\end{align*}\nwhere in the end we have used that $|x-y|\\ge 2\/3|y|$. By \\eqref{in7}, for any $\\theta>0$ there exists $C_\\theta>0$ such that weakly in $B_{R\/3}$\n\\begin{align*}\n(-\\Delta)^s_p u_+(x) &\\ge -K-\\theta\\big(\\sup_{B_R}u\\big)^{p-1}\\int_{B^c_R}\\frac{1}{|y|^{N+ps}}\\,dy-\\frac{C_\\theta}{R^{ps}}{\\rm Tail}(u_-;R)^{p-1} \\\\\n&\\ge -K-\\frac{C\\theta}{R^{ps}}\\big(\\sup_{B_R}u\\big)^{p-1}-\\frac{C_\\theta}{R^{ps}}{\\rm Tail}(u_-;R)^{p-1} =: -\\tilde K.\n\\end{align*}\nNow, by applying Theorem \\ref{harnack} to $u_+$ we have for any $\\varepsilon>0$ and $\\theta<(\\varepsilon\/\\bar C)^{p-1}$,\n\\begin{align*}\n\\inf_{B_{R\/4}}u &\\ge \\sigma\\Big(\\Xint-_{B_R\\setminus B_{R\/2}}u^{p-1}\\,dx\\Big)^\\frac{1}{p-1}-\\bar C(\\tilde KR^{ps})^\\frac{1}{p-1} \\\\\n&\\ge \\sigma\\Big(\\Xint-_{B_R\\setminus B_{R\/2}}u^{p-1}\\,dx\\Big)^\\frac{1}{p-1}-\\tilde C(KR^{ps})^\\frac{1}{p-1}-C_\\varepsilon{\\rm Tail}(u_-;R)-\\varepsilon\\sup_{B_R}u\n\\end{align*}\nfor some universal constant $\\tilde C>0$ and a convenient $C_\\varepsilon>0$ depending also on $\\varepsilon$.\n\\vskip2pt\n\\noindent\nNow we turn to the case $p\\in(1,2)$. The argument in this case is in fact easier, as by \\eqref{in2} we have\n\\[\\int_{\\{u<0\\}}\\frac{u(x)^{p-1}-(u(x)-u(y))^{p-1}}{|y|^{N+ps}}\\,dy\\le\\frac{1}{R^{ps}}{\\rm Tail}(u_-;R)^{p-1}\\quad \\text{for a.e.\\ $x\\in B_{R\/3}$},\\]\nthen we argue as above using \\eqref{in3} instead of \\eqref{in1} when required.\n\\end{proof}\n\n\\noindent\nClearly, symmetric versions of Theorem \\ref{harnack} and Lemma \\ref{harnackloc} also hold. Now we use the above results to produce an estimate of the oscillation of a bounded function $u$ such that $(-\\Delta)^s_p u$ is locally bounded. We set for all $R>0$, $x_0\\in {\\mathbb R}^N$\n\\[Q(u; x_0, R) = \\|u\\|_{L^\\infty(B_R(x_0))}+{\\rm Tail}(u; x_0, R),\\quad Q(u; R)=Q(u; 0, R).\\]\nOur result is as follows:\n\n\\begin{theorem}\\label{osc}\nThere exist universal $\\alpha\\in(0,1)$, $C>0$ with the following property: if $u\\in \\widetilde{W}^{s,p}(B_{R_0})\\cap L^\\infty(B_{R_0})$ satisfies $|(-\\Delta)^s_p u|\\le K$ weakly in $B_{R_0}$ for some $R_0>0$, then for all $r\\in(0,R_0)$\n\\[\\underset{B_r}{\\rm osc}\\,u\\le C\\big[(KR_0^{ps})^\\frac{1}{p-1}+Q(u;R_0)\\big]\\frac{r^\\alpha}{R_0^\\alpha}.\\]\n\\end{theorem}\n\\begin{proof}\nFirst we consider the case $p\\ge 2$. For all integer $j\\ge 0$ we set $R_j=R_0\/4^j$, $B_j=B_{R_j}$, and $\\frac{1}{2} B_j=B_{R_j\/2}$. We put forward the following\n\\vskip2pt\n\\noindent\n{\\em Claim.} There exist a universal $\\alpha\\in(0,1)$ and a real $\\lambda>0$ (depending on all the data), a non-decreasing sequence $(m_j)$, and a non-increasing sequence $(M_j)$, such that for all $j\\ge 0$\n\\[m_j\\le\\inf_{B_j}u\\le\\sup_{B_j}u\\le M_j, \\quad M_j-m_j=\\lambda R_j^\\alpha.\\]\nWe argue by induction on $j$. Step zero: we set $M_0=\\sup_{B_0}u$ and $m_0=M_0-\\lambda R_0^\\alpha$, where $\\lambda>0$ satisfies\n\\begin{equation}\\label{io1}\n\\lambda\\ge \\frac{2\\|u\\|_{L^\\infty(B_0)}}{R_0^\\alpha},\n\\end{equation}\nwhich clearly implies\n\\[\\inf_{B_0}u\\ge m_0.\\]\nInductive step: assume that sequences $(m_j)$, $(M_j)$ are constructed up to the index $j$. Then\n\\begin{equation}\\label{iox}\n\\begin{split}\nM_j-m_j &= \\Xint-_{B_j\\setminus\\frac{1}{2} B_j}(M_j-u)\\,dx+\\Xint-_{B_j\\setminus\\frac{1}{2}B_j}(u-m_j)\\,dx \\\\\n&\\le \\Big(\\Xint-_{B_j\\setminus\\frac{1}{2}B_j}(M_j-u)^{p-1}\\,dx\\Big)^\\frac{1}{p-1}+\\Big(\\Xint-_{B_j\\setminus\\frac{1}{2} B_j}(u-m_j)^{p-1}\\,dx\\Big)^\\frac{1}{p-1}.\n\\end{split}\n\\end{equation}\nLet $\\sigma\\in(0,1)$, $\\tilde C>0$ be as in Lemma \\ref{harnackloc}, and multiply the previous inequality by $\\sigma$ to obtain, via \\eqref{hloctesi}, \n\\begin{equation}\n\\label{ioxx}\n\\begin{split}\n\\sigma(M_j-m_j) &\\le \\inf_{B_{j+1}}(M_j-u)+\\inf_{B_{j+1}}(u-m_j)+2\\tilde C(KR_0^{ps})^\\frac{1}{p-1} \\\\\n&\\quad + C_\\varepsilon\\big[{\\rm Tail}((M_j-u)_-;R_j)+{\\rm Tail}((u-m_j)_-;R_j)\\big]\\\\\n&\\quad +\\varepsilon\\Big[\\sup_{B_j}(M_j-u)+\\sup_{B_j}(u-m_j)\\Big].\n\\end{split}\n\\end{equation}\nSetting universally $\\varepsilon=\\sigma\/4$, $C=\\max\\{2\\tilde C,C_\\varepsilon\\}$ and rearranging, we have\n\\begin{equation}\\label{io2}\n\\begin{split}\n\\underset{B_{j+1}}{\\rm osc}\\,u &\\le \\Big(1-\\frac{\\sigma}{2}\\Big)(M_j-m_j)\\\\\n&\\quad +C\\big[(KR_0^{ps})^\\frac{1}{p-1}+{\\rm Tail}((M_j-u)_-;R_j)+{\\rm Tail}((u-m_j)_-;R_j)\\big].\n\\end{split}\n\\end{equation}\nNow we provide an estimate of both non-local tails, firstly noting that\n\\begin{equation}\\label{io3}\n\\begin{split}\n{\\rm Tail}((u-m_j)_-;R_j)^{p-1}&=R_j^{ps}\\sum_{k=0}^{j-1}\\int_{B_k\\setminus B_{k+1}}\\frac{(u(y)-m_j)_-^{p-1}}{|y|^{N+ps}}\\,dy\\\\\n&\\quad +R_j^{ps}\\int_{B^c_0}\\frac{(u(y)-m_j)_-^{p-1}}{|y|^{N+ps}}\\,dy.\n\\end{split}\n\\end{equation}\nWe consider the first term: by the inductive hypothesis, for all $0\\le k\\le j-1$ we have in $B_k\\setminus B_{k+1}$\n\\[(u-m_j)_-\\le m_j-m_k \\le (m_j-M_j)+(M_k-m_k) = \\lambda(R_k^\\alpha-R_j^\\alpha),\\]\nhence\n\\begin{align*}\n\\sum_{k=0}^{j-1}\\int_{B_k\\setminus B_{k+1}}\\frac{(u(y)-m_j)_-^{p-1}}{|y|^{N+ps}}\\,dy &\\le \\lambda^{p-1}R_j^{\\alpha(p-1)}\\sum_{k=0}^{j-1}\\int_{B_k\\setminus B_{k+1}}\\frac{(4^{\\alpha(j-k)}-1)^{p-1}}{|y|^{N+ps}}\\,dy \\\\\n&\\le C\\lambda^{p-1} S(\\alpha)R_j^{\\alpha(p-1)-ps},\n\\end{align*}\nwhere we have set for all $\\alpha\\in(0,1)$\n\\[S(\\alpha)=\\sum_{h=1}^\\infty\\frac{(4^{\\alpha h}-1)^{p-1}}{4^{psh}},\\]\nnoting that $S(\\alpha)\\to 0$ as $\\alpha\\to 0^+$. Regarding the second term, by the inductive hypothesis we have\n\\[m_j\\le\\inf_{B_j}u\\le\\sup_{B_j}u\\le\\|u\\|_{L^\\infty(B_0)},\\]\nhence\n\\[\\int_{B^c_0}\\frac{(u(y)-m_j)_-^{p-1}}{|y|^{N+ps}}\\,dy\\le \\int_{B^c_0}\\frac{(\\|u\\|_{L^\\infty(B_0)}+|u(y)|)^{p-1}}{|y|^{N+ps}}\\,dy \\le \\frac{CQ(u; R_0)^{p-1}}{R_0^{ps}}.\\]\nChoosing $\\alpha2$.\nSo, \\eqref{io4} forces\n\\[\\underset{B_{j+1}}{\\rm osc}\\,u\\le \\lambda R_{j+1}^\\alpha.\\]\nWe may pick $m_{j+1}$, $M_{j+1}$ such that\n\\[m_j\\le m_{j+1}\\le\\inf_{B_{j+1}}u\\le\\sup_{B_{j+1}}u\\le M_{j+1}\\le M_j, \\ M_{j+1}-m_{j+1}=\\lambda R_{j+1}^\\alpha,\\]\nwhich completes the induction and proves the claim.\n\\vskip2pt\n\\noindent\nNow fix $r\\in(0,R_0)$ and find an integer $j\\ge 0$ such that $R_{j+1}\\le r0$ and $\\alpha\\in (0,1)$ with the following property: for all $u\\in \\widetilde{W}^{s,p}(B_{2R_0}(x_0))\\cap L^\\infty(B_{2R_0}(x_0))$ \nsuch that $|(-\\Delta)^s_p u|\\le K$ weakly in $B_{2R_0}(x_0)$,\n\\begin{equation}\n\\label{tesicorloc}\n[u]_{C^\\alpha(B_{R_0}(x_0))}\\le C\\big[(KR_0^{ps})^\\frac{1}{p-1}+Q(u; x_0, 2R_0)\\big]R_0^{-\\alpha}.\n\\end{equation}\n\\end{corollary}\n\n\\begin{proof}\nGiven $x, y$ in $B_{R_0}(x_0)$, let $r=|x-y|$. It suffices to apply Theorem \\ref{osc} to the ball $B_{R_0}(x)\\subseteq B_{2R_0}(x_0)$. Clearly $\\|u\\|_{L^\\infty(B_{R_0}(x))}\\leq \\|u\\|_{L^\\infty(B_{2R_0}(x_0))}$ and \n\\begin{align*}\n&{\\rm Tail}(u; x, R_0)^{p-1}\\\\\n&=R_0^{ps}\\int_{B_{R_0}^c(x)}\\frac{|u(y)|^{p-1}}{|x-y|^{N+ps}}\\, dy\\\\\n&\\leq CR_0^{ps}\\Big[\\int_{B_{2R_0}(x_0)\\setminus B_{R_0}(x)}\\frac{\\|u\\|_{L^\\infty(B_{2R_0}(x_0))}^{p-1}}{|x-y|^{N+ps}}\\, dy+\\int_{B_{2R_0}^c(x_0)}\\frac{|u(y)|^{p-1}}{|x-y|^{N+ps}}\\, dy\\Big]\\\\\n&\\leq C\\|u\\|_{L^\\infty(B_{2R_0}(x_0))}^{p-1}+CR_0^{ps}\\int_{B_{2R_0}^c(x_0)}\\frac{|u(y)|^{p-1}}{|x_0-y|^{N+ps}}\\, dy\n\\end{align*}\nfor a universal $C$, where as usual we used $|x-y|\\geq |x_0-y|\/2$ for $y\\in B^c_{2R_0}(x_0)$, $x\\in B_{R_0}(x_0)$.\nThis implies that\n\\[Q(u; x, R_0)\\leq CQ(u; x_0, 2R_0),\\]\nand thus the desired estimate on the H\\\"older seminorm.\n\\end{proof}\n\n\\subsection{Global H\\\"older regularity}\n\nWe finally prove the stated H\\\"older regularity result up to the boundary.\n\\vskip4pt\n\\noindent\n\\textbf{Proof of Theorem \\ref{main}.} We set $K=\\|f\\|_{L^\\infty(\\Omega)}$. Corollary \\ref{apb} already provides the desired estimate for the $\\sup$-norm, namely\n\\[\\|u\\|_{L^\\infty(\\Omega)}\\le CK^\\frac{1}{p-1},\\]\nso we can focus on the H\\\"older seminorm.\n\\vskip2pt\n\\noindent\nLet $\\alpha$ be the one given in Corollary \\ref{corlocalpha}. We can assume $\\alpha\\in(0,s]$. Through a covering argument, \\eqref{tesicorloc} implies that $u\\in C^{\\alpha}_{\\rm loc}(\\overline\\Omega')$ for all $\\Omega'\\Subset\\Omega$, with a bound of the form\n\\[\\|u\\|_{C^\\alpha(\\overline\\Omega')}\\leq C_{\\Omega'}K^{\\frac{1}{p-1}},\\quad C_{\\Omega'}=C(N, p, s, \\Omega, \\Omega').\\]\nHence it suffices to prove \\eqref{thm57tesi} in the closure of a fixed $\\rho$-neighbourhood of $\\partial\\Omega$. We will suppose that $\\rho>0$ is so small (depending only on $\\Omega$) that Lemma \\ref{geo1} holds, and thus the metric projection \n\\[\\Pi:V\\to \\partial\\Omega, \\quad \\Pi(x)=\\underset{y\\in \\Omega^c}{\\rm Argmin}|x-y|\\]\nis well defined on $V:=\\{x\\in\\overline\\Omega:\\delta(x)\\leq \\rho\\}$. We claim that \n\\begin{equation}\n\\label{claimfin}\n[u]_{C^\\alpha(B_{r\/2}(x))}\\leq C_\\Omega K^{\\frac{1}{p-1}},\\quad \\text{for all $x\\in V$ and $r=\\delta(x)$} \n\\end{equation}\nfor some constant $C_\\Omega=C(N, p, s, \\Omega)$, independent on $x\\in V$. We recall \\eqref{tesicorloc}, which in the present case rephrases (up to a universal constant) as\n\\[[u]_{C^\\alpha(B_{r\/2}(x))}\\le C\\big[(Kr^{ps})^\\frac{1}{p-1}+\\|u\\|_{L^\\infty(B_r(x))}+{\\rm Tail}(u;x,r)\\big]r^{-\\alpha}.\\]\nTo prove \\eqref{claimfin}, it suffices to bound the three terms on the right hand side of the above inequality. The first one it trivially dealt with since $\\alpha\\leq s\\leq ps\/(p-1)$, and thus\n\\[r^{-\\alpha}(Kr^{ps})^\\frac{1}{p-1}\\leq K^{\\frac{1}{p-1}}\\rho^{\\frac{ps}{p-1}-\\alpha}.\\]\nFor the second one we use Theorem \\ref{estid} and $\\alpha\\leq s$ to get\n\\[\\|u\\|_{L^\\infty(B_{r}(x))}\\leq CK^{\\frac{1}{p-1}}(\\delta(x)+r)^s\\leq CK^{\\frac{1}{p-1}}\\rho^{s-\\alpha}r^\\alpha,\\]\nand thus the claimed bound. Similarly for the last term we employ again \\eqref{thm44tesi}, together with \n\\[\\delta(y)\\leq |y-\\Pi(x)|\\leq |y-x|+|x-\\Pi(x)|\\leq |y-x|+r\\leq 2|x-y|,\\quad \\forall y\\in B_r^c(x),\\]\nto get\n\\begin{align*}\n{\\rm Tail}(u;x,r)^{p-1} &\\leq CKr^{ps}\\int_{B_{r}^c(x)}\\frac{\\delta^{s(p-1)}(y)}{|x-y|^{N+ps}}\\, dy\\\\\n&\\leq CKr^{ps}\\int_{B_{r}^c(x)}\\frac{|x-y|^{s(p-1)}}{|x-y|^{N+ps}}\\, dy\\\\\n&\\leq CKr^{s(p-1)}.\n\\end{align*}\nAgain due to $\\alpha\\leq s$ we obtain the claimed bound, and the proof of \\eqref{claimfin} is completed.\nTo prove the theorem, pick $x, y\\in V$ and suppose without loss of generality that $|x-\\Pi(x)|\\geq|y-\\Pi(y)|$. Two cases may occur:\n\\begin{itemize}\n\\item either $2|x-y|< |x-\\Pi(x)|$, in which case we set $r=\\delta(x)$ and apply \\eqref{claimfin} in $B_{r\/2}(x)$, to obtain \n\\[|u(x)-u(y)|\\leq C K^{\\frac{1}{p-1}}|x-y|^\\alpha;\\]\n\\item or $2|x-y|\\geq |x-\\Pi(x)|\\geq |y-\\Pi(y)|$, in which case \\eqref{thm44tesi} ensures\n\\begin{align*}\n|u(x)-u(y)|&\\leq |u(x)|+|u(y)|\\leq C K^{\\frac{1}{p-1}}(\\delta^s(x)+\\delta^s(y))\\\\\n&=C K^{\\frac{1}{p-1}}(|x-\\Pi(x)|^s+|y-\\Pi(y)|^s)\\\\\n&\\leq C K^{\\frac{1}{p-1}}|x-y|^s\\\\\n&\\leq C K^{\\frac{1}{p-1}}\\rho^{s-\\alpha}|x-y|^\\alpha.\n\\end{align*}\n\\end{itemize}\nThus in both cases the $\\alpha$-H\\\"older seminorm is bounded in $V$ and the proof is completed. \\qed\n\n\\begin{remark}\n\\label{remalpha}\nAs the proofs above show, interior regularity (Theorem \\ref{osc}) forces in particular $\\alpha10$ ({\\em top panel}), clusters with $\\log(M_{500}\n [h^{-1}\\,{\\mbox M_\\odot}])>13$ ({\\em central panel}), and AGN with\n $\\log(M_{\\rm BH} [h^{-1}\\,{\\mbox M_\\odot}])>6.3$ ({\\em bottom\n panel}). The redshift-space correlation functions have been\n measured directly from the simulation, while the real-space ones\n have been either measured from the simulation ({\\em red dots}), or\n derived from the {\\small CAMB} power spectrum ({\\em blue\n squares}). The error bars represent the statistical noise as\n prescribed by \\citet{mo1992}. The black solid lines show the\n expected values of $\\xi(s)\/\\xi(r)$ at large scales predicted by\n Eq.~(\\ref{eq:xi0_2}). The dotted blue and dashed red lines are the\n best-fit ratios estimated from the blue and red points,\n respectively.}\n\\label{fig:xi_ratio1}\n\\end{figure}\n\n\n\n\n\\section{Results}\n\\label{sec:results}\n\nIn this section, we present our main results obtained from mock\nsamples of three different tracers -- galaxies, clusters and AGN --\nusing the two approaches described in \\S\\ref{sec:methodology}. For\neach catalogue, we analyse six snapshots corresponding to the\nredshifts $z=\\{0.2, 0.52, 0.72, 1, 1.5, 2\\}$. Moreover, at each\nredshift we consider different sample selections. In total, we analyse\n$270$ mock catalogues, whose main properties, including the number of\nobjects in each sample, are reported in Tables \\ref{tab:table1},\n\\ref{tab:table2} and \\ref{tab:table3}, and in Fig.~\\ref{fig:nz} in\nAppendix~\\ref{app:samples}.\n\n\\begin{figure}\n\\includegraphics[width=0.49\\textwidth]{fsigma8_RSD4.ps}\n\\caption{{\\em Top panel}: the best-fit values of $f(z)\\sigma_8(z)$ for\n galaxies with $\\log(M_{\\rm STAR} [h^{-1}\\,{\\mbox M_\\odot}])>10$\n ({\\em red dots}), clusters with $\\log(M_{500} [h^{-1}\\,{\\mbox\n M_\\odot}])>13$ ({\\em blue squares}), and AGN with $\\log(M_{\\rm\n BH} [h^{-1}\\,{\\mbox M_\\odot}])>6.3$ ({\\em green diamonds}),\n obtained with the method described in \\S\\ref{subsec:resI}. The black\n line shows the function $\\Omega_{\\rm M}(z)^{0.545}\\cdot\\sigma_8(z)$,\n where $\\Omega_{\\rm M}(z)$ and $\\sigma_8(z)$ are the known values of\n the simulation. {\\em Bottom panel}: the percentage systematic errors\n on $f(z)\\sigma_8(z)$, $[(f\\sigma_8)^{\\rm\n measured}$-$(f\\sigma_8)^{\\rm simulation}]\/(f\\sigma_8)^{\\rm\n simulation}\\cdot100$, that is the percentage differences between\n the points and the black line shown in the top panel. The error bars\n have been estimated with Eq.~(\\ref{eq:bianchi}). The values reported\n have been slightly shifted for visual clarity. To guide the eyes,\n the light and grey shaded areas highlight the $5\\%$ and $10\\%$ error\n regions, respectively.}\n\\label{fig:fsigma8}\n\\end{figure}\n\n\n\n\n\\subsection{The linear growth rate from the clustering monopole}\n\\label{subsec:resI}\n\nWe start analysing the spherically averaged two-point correlation\nfunction -- the clustering monopole -- at large linear scales. The aim\nof this exercise is to investigate the accuracy of the RSD model\nin the simplest case possible, that is in the so-called Kaiser limit.\n\nFig.~\\ref{fig:xi_ratio1} shows the ratio between the redshift-space\nand real-space two-point correlation functions,\n$\\xi(s)\/\\xi(r)$. Specifically, we show here the results obtained at\n$z=0.2$, for galaxies with $\\log(M_{\\rm STAR} [h^{-1}\\,{\\mbox\n M_\\odot}])>10$, clusters with $\\log(M_{500} [h^{-1}\\,{\\mbox\n M_\\odot}])>13$, and AGN with $\\log(M_{\\rm BH} [h^{-1}\\,{\\mbox\n M_\\odot}])>6.3$, as reported by the labels. These correspond to\nthe upper left samples reported in Tables \\ref{tab:table1},\n\\ref{tab:table2} and \\ref{tab:table3}. Results obtained from the other\nmock samples are similar, but more scattered due to the lower\ndensities. The redshift-space correlation functions have been\nmeasured directly from the simulation, through the procedure described\nin \\S\\ref{sec:methodology}. The real-space correlation functions have\nbeen estimated in two different ways. Either they are measured\ndirectly from the simulation, or they are derived from the linear\n{\\small CAMB} power spectrum, and assuming a linear bias factor. The\ndifferences at small scales, $r\\lesssim5$ $h^{-1}\\,\\mbox{Mpc}$\\,, between the ratios\ncomputed with the measured (blue squares) and {\\small CAMB} (red dots)\nreal-space correlation functions are due to non-linear effects. We\nverified that using the non-linear {\\small CAMB} power spectrum, via\nthe {\\small HALOFIT} routine \\citep{smith2003}, does not fully remove\nthe discrepancy. Nevertheless, as we model here the large scale\nclustering, this has no effects on our results. On the other hand, the\nsmall discrepancies at scales $r\\gtrsim40$ $h^{-1}\\,\\mbox{Mpc}$\\, can introduce\nsystematics, that however result smaller than the estimated\nuncertainties. The error bars in Fig.~\\ref{fig:xi_ratio1} show the\nstatistical Poisson noise \\citep{mo1992}. Scales larger than $60$\n$h^{-1}\\,\\mbox{Mpc}$\\,, not shown in the plot, are too noisy to affect the fit. More\nspecifically, we verified that a convenient scale range to get robust\nresults is $1010$, at $z=0.2$\n (black contours). The dot-dashed green and solid red contours show\n the best-fit model given by Eq.~(\\ref{eq:ximodel}), with the\n real-space correlation function $\\xi(r)$ measured from the\n simulation, and estimated from the {\\small CAMB} power spectrum,\n respectively. The blue dashed contours show the linear best-fit\n model given by Eq.~(\\ref{eq:ximodellin}) with the {\\small CAMB}\n real-space correlation function.}\n \\label{fig:iso_gal}\n\\end{figure}\n\n\\begin{figure}\n \\includegraphics[width=0.48\\textwidth]{fsigma8_RSD5_gal.ps}\n \\includegraphics[width=0.48\\textwidth]{best_range_gal.ps}\n \\caption{{\\em Top panel of the upper window}: the best-fit values of\n $f(z)\\sigma_8(z)$ for galaxies with $\\log(M_{\\rm STAR}\n [h^{-1}\\,{\\mbox M_\\odot}])>10$. The open and solid blue squares\n show the values obtained by fitting the data with the models given\n by Eqs.~(\\ref{eq:ximodellin}) and (\\ref{eq:ximodel}),\n respectively, in the scale ranges\n $310$, clusters with\n$\\log(M_{500} [h^{-1}\\,{\\mbox M_\\odot}])>13$, and AGN with\n$\\log(M_{\\rm BH} [h^{-1}\\,{\\mbox M_\\odot}])>6.3$. The black line shows\nthe function $\\Omega_{\\rm M}(z)^{0.545}\\cdot\\sigma_8(z)$, where\n$\\Omega_{\\rm M}(z)$ and $\\sigma_8(z)$ are the values of the {\\em\n Magneticum} simulation. In the lower panel we show the percentage\nsystematic errors on $f\\sigma_8$, defined as $[(f\\sigma_8)^{\\rm\n measured}$-$(f\\sigma_8)^{\\rm simulation}]\/(f\\sigma_8)^{\\rm\n simulation}\\cdot100$. The error bars have been estimated by\npropagating on $f\\sigma_8$ the $\\beta$ errors provided by the scaling\nformula presented in \\citet{bianchi2012}, that gives the statistical\nerrors as a function of bias, $b$, volume, $V$, and density, $n$:\n\\begin{equation}\n\\frac{\\delta\\beta}{\\beta}\\simeq\nCb^{0.7}V^{-0.5}\\exp\\left(\\frac{n_0}{b^2n}\\right) \\; ,\n\\label{eq:bianchi} \n\\end{equation}\nwhere $n_0=1.7\\cdot10^{-4}\\,h^{3}\\,\\mbox{Mpc}^{-3}$ and\n$C=4.9\\cdot10^2 \\,h^{-1.5}\\,\\mbox{Mpc}^{1.5}$. In this case, these\nerror bars have to be considered just as lower limits, as the scaling\nformula has been calibrated based on fits of the full two-dimensional\nanisotropic correlation function $\\xi(s_\\perp,s_\\parallel)$\\,. Moreover, they have been\ncomputed in different scale ranges and with Friends-of-Friends DM\nhaloes, differently from this analysis, where we consider tracers\nhosted in DM sub-haloes. Using the full covariance matrix in this\nstatistical analysis does not change our conclusions, as shown in\nAppendix \\ref{app:errors}.\n\n\\begin{figure}\n \\includegraphics[width=0.48\\textwidth]{err_fsigma8_RSD5_gal_sel.ps}\n \\caption{The percentage systematic errors on $f(z)\\sigma_8(z)$ for\n galaxies, $[(f\\sigma_8)^{\\rm measured}$-$(f\\sigma_8)^{\\rm\n simulation}]\/(f\\sigma_8)^{\\rm simulation}\\cdot100$, as a\n function of redshift and sample selections, as indicated by the\n labels. $S1-S5$ refer to the five selection thresholds reported in\n Tables \\ref{tab:table1}, \\ref{tab:table2} and \\ref{tab:table3},\n for the different tracers and properties used for the selection.}\n \\label{fig:errfs8_gal}\n\\end{figure}\n\nAs demonstrated by Figs.~\\ref{fig:xi_ratio1} and \\ref{fig:fsigma8}, it\nis indeed possible to get almost unbiased constraints on $f\\sigma_8$\nfrom the monopole of the two-point correlation function, at all\nredshifts considered, and for both galaxies, clusters and AGN,\nprovided that the linear bias is estimated at sufficiently large\nscales. The advantage of this method is the minimal number of free\nparameters necessary for the modelisation. However, the linear bias\nhas to be assumed, or measured from other probes.\n\n\n\n\n\\subsection{The linear growth rate from the anisotropic 2D clustering}\n\\label{subsec:resII}\n\nTo extract the full information from the redshift-space two-point\ncorrelation function, the anisotropic correlation $\\xi(s_\\perp,s_\\parallel)$\\, or,\nalternatively, all the relevant multipoles have to be modelled. As\ndiscussed in \\S\\ref{sec:methodology}, we consider the first\napproach. Differently from the analysis of \\S\\ref{subsec:resI}, here\nwe can jointly constrain the two terms $f\\sigma_8$ and $b\\sigma_8$. In\nthis section we investigate the accuracy of the constraints on\n$f\\sigma_8$ provided by the dispersion model, as a function of sample\nselection, with $b\\sigma_8$ and $\\sigma_{12}$ as free parameters.\n\n\n\n\n\\subsubsection{Galaxies}\n\\label{subsubsec:galaxies}\n\nWe start presenting our results for the galaxy mock samples. The black\nlines of Fig.~\\ref{fig:iso_gal} show the iso-correlation contours of\nthe redshift-space two-point correlation function, $\\xi(s_\\perp,s_\\parallel)$\\,, for galaxies\nwith $\\log(M_{\\rm STAR} [h^{-1}\\,{\\mbox M_\\odot}])>10$, at\n$z=0.2$. The other lines are the best-fit models obtained with three\ndifferent methods. The dot-dashed green lines are obtained by using\nthe full non-linear dispersion model given by Eq.~(\\ref{eq:ximodel}),\nwith the real-space correlation function $\\xi(r)$ measured directly\nfrom the simulation. The blue dashed and red solid contours are\nobtained with $\\xi(r)$ estimated from the {\\small CAMB} power\nspectrum, and by fitting the linear model given by\nEq.~(\\ref{eq:ximodellin}) and the non-linear one by\nEq.~(\\ref{eq:ximodel}), respectively. The fitting is done in the scale\nranges $313$, at $z=0.2$ (black\n contours). The other contours are as in\n Fig.~\\ref{fig:iso_gal}.}\n \\label{fig:iso_cl}\n\\end{figure}\n\n\\begin{figure}\n \\includegraphics[width=0.48\\textwidth]{fsigma8_RSD5_cl.ps}\n \\includegraphics[width=0.48\\textwidth]{best_range_cl.ps}\n \\caption{The best-fit values of $f(z)\\sigma_8(z)$ for clusters with\n $\\log(M_{500} [h^{-1}\\,{\\mbox M_\\odot}])>13$. All the symbols are\n as in Fig.~\\ref{fig:fs8_gal}.}\n \\label{fig:fs8_cl}\n\\end{figure}\n\n\\begin{figure} \n \\includegraphics[width=0.48\\textwidth]{err_fsigma8_RSD5_cl_sel.ps}\n \\caption{The percentage systematic errors on $f(z)\\sigma_8(z)$ for\n clusters, as a function of redshift and sample selections. All\n symbols are as in Fig.~\\ref{fig:errfs8_gal}.}\n \\label{fig:errfs8_cl}\n\\end{figure}\n\nThe best-fit values of $f\\sigma_8$ as a function of redshift are shown\nin the top panel of the upper window of Fig.~\\ref{fig:fs8_gal}. The\nopen and solid blue squares show the values obtained by fitting the\ndata with the models given by Eqs.~(\\ref{eq:ximodellin}) and\n(\\ref{eq:ximodel}), respectively, that is they correspond to the blue\nand red contours of Fig.~\\ref{fig:iso_gal} (at $z=0.2$). The bottom\npanel of the upper window shows the percentage systematic errors on\n$f\\sigma_8$, that is $[(f\\sigma_8)^{\\rm measured}$-$(f\\sigma_8)^{\\rm\n simulation}]\/(f\\sigma_8)^{\\rm simulation}\\cdot100$.\n\nAs it can be seen, the best-fit values of $f\\sigma_8$ result strongly\nbiased with respect to the true ones, with a systematic error that\ndepends on the redshift. At $z=0.2$, we get an underestimation of\n$\\sim10$ ($20$) $\\%$ with the non-linear (linear) dispersion model, at\n$z=1$ the error reduces to $\\sim5$ ($8$)$\\%$, while at $z=2$ we get an\noverestimation of $\\sim20\\%$, with both linear and non-linear\nmodelling. This result is fairly in agreement with what found by\n\\citet{bianchi2012} at $z=1$, although the two analyses are not\ndirectly comparable, as \\citet{bianchi2012} considered a sample of\nFriends-of-Friends DM haloes, slightly affected by fingers-of-God. As\nexpected, the convolution given by Eq.~(\\ref{eq:ximodel}) reduces the\nsystematic error on $f\\sigma_8$, especially at low redshifts. However,\nit does not totally remove the discrepancies, in agreement with\nprevious findings \\citep[e.g.][and references\n therein]{mohammad2016}. For comparison, the grey dots of\nFig.~\\ref{fig:fs8_gal} show a set of recent observational measurements\nof $f\\sigma_8$ from large galaxy surveys: 6dFGS at $z=0.067$\n\\citep{beutler2012}; SDSS(DR7) Luminous Red Galaxies at $z=0.25, 0.37$\n\\citep{samushia2012} from scales lower than $60$ and $200$ $h^{-1}\\,\\mbox{Mpc}$\\,; BOSS\nat $z=0.3, 0.57, 0.6$ \\citep{tojeiro2012, reid2012}; WiggleZ at\n$z=0.44, 0.6, 0.73$ \\citep{blake2012}; VIPERS at $z=0.8$\n\\citep{delatorre2013b}. These results have been obtained using\ndifferent RSD models, most of them more accurate than the dispersion\nmodel considered in this work. Nevertheless, many of these\nmeasurements underestimate $f\\sigma_8$ with respect to GR+$\\Lambda$CDM\npredictions \\citep[see e.g.][]{macaulay2013}. In line with our\nfindings, this could be explained, at least partially, by model\nuncertainties still present in the more sophisticated RSD models\nconsidered.\n\nThe direct way to reduce these systematics is to improve the\nmodelisation of RSD at non-linear scales \\citep[see\n e.g.][]{scoccimarro2004, taruya2010, seljak2011, wang2014,\n delatorre2013b}. We explore here a different approach, investigating\nthe dependency of the systematic error on the comoving scales\nconsidered in the analysis. In the forthcoming plots, we show the\nresults obtained by repeating our fitting procedure for different\nvalues of the minimum perpendicular separation and the maximum\nparallel separation used in the fit, that is $r_\\perp^{\\rm min}$ and\n$r_\\parallel^{\\rm max}$. By increasing the value of $r_\\perp^{\\rm\n min}$ we can cut the region more affected by fingers-of-God\nanisotropies, while by reducing $r_\\parallel^{\\rm max}$ we avoid the\nregion more affected by shot noise. As we verified, changing also\n$r_\\perp^{\\rm max}$ and $r_\\parallel^{\\rm min}$, or adopting different\nscale selection criteria, do not affect significantly the results.\nThe aim here is to search for optimal regions in this plane to\npossibly get unbiased constraints. As non-linear dynamics impact on\ndifferent scales for different redshifts and biases of the tracers, we\nexpect that the best values of $r_\\perp^{\\rm min}$ and\n$r_\\parallel^{\\rm max}$, that is the ones that minimise systematics,\nwill be different for different sample selections.\n\nThe results of this analysis are shown in Fig.~\\ref{fig:fs8_gal} with\nopen and solid red dots, obtained by fitting the data with the model\ngiven by Eqs.~(\\ref{eq:ximodellin}) and (\\ref{eq:ximodel}),\nrespectively, that is by considering either the linear or the\nnon-linear RSD model. We explore the ranges $56.3$, at $z=0.2$ (black\n contours). The other contours show the best-fit models, as\n in Fig.~\\ref{fig:iso_gal}.}\n \\label{fig:iso_agn}\n\\end{figure}\n\n\\begin{figure}\n \\includegraphics[width=0.48\\textwidth]{fsigma8_RSD5_agn.ps}\n \\includegraphics[width=0.48\\textwidth]{best_range_agn.ps}\n \\caption{The best-fit values of $f(z)\\sigma_8(z)$ for AGN with\n $\\log(M_{\\rm BH} [h^{-1}\\,{\\mbox M_\\odot}])>6.3$. All the symbols\n are as in Fig.~\\ref{fig:fs8_gal}.}\n \\label{fig:fs8_agn}\n\\end{figure}\n\n\\begin{figure} \n \\includegraphics[width=0.48\\textwidth]{err_fsigma8_RSD5_agn_sel.ps}\n \\caption{The percentage systematic errors on $f(z)\\sigma_8(z)$ for\n AGN, as a function of redshift and sample selections. All symbols\n are as in Fig.~\\ref{fig:errfs8_gal}.}\n \\label{fig:errfs8_agn}\n\\end{figure}\n\nFor completeness, we then apply our method to several subsamples with\ndifferent selections. Specifically, we consider galaxy catalogues\nselected in stellar mass, $M_{\\rm STAR}$, g-band absolute magnitude,\n$M_g$, and star formation rate, SFR. The selection thresholds\nconsidered and the number of galaxies in each sample are reported in\nTable \\ref{tab:table1}. The percentage systematic errors on\n$f\\sigma_8$ are reported in Fig.~\\ref{fig:errfs8_gal}, as a function\nof redshift and sample selections, as indicated by the labels. The\nresult is quite remarkable: it is indeed possible to get almost\nunbiased constraints on $f\\sigma_8$ with the dispersion model,\nindependent of sample selections, provided that $r_\\perp^{\\rm min}$\nand $r_\\parallel^{\\rm max}$ are chosen conveniently. For the largest\ngalaxy sample shown in Fig.~\\ref{fig:fsigma8}, the preferred value of\n$r_\\perp^{\\rm min}$ is $\\sim 15$ $h^{-1}\\,\\mbox{Mpc}$\\, at low and high redshifts, and\n$\\sim 5$ $h^{-1}\\,\\mbox{Mpc}$\\, around $z=1$, while $r_\\parallel^{\\rm max}$ is $\\sim\n15$ $h^{-1}\\,\\mbox{Mpc}$\\, at low redshifts, and increases up to $\\sim 35$ $h^{-1}\\,\\mbox{Mpc}$\\, at\n$z=2$. Again, we do not find any clear trend of $r_\\perp^{\\rm min}$\nand $r_\\parallel^{\\rm max}$ as a function of redshift or sample bias.\n\nOverall, all of these results show that our modelling of RSD is mostly\nsensitive to the lower fitting cut-off, depending on how the tracers\nrelate to the underlying mass. Indeed, the effect of the different\nselections considered here is just to select subsamples of haloes of\ndifferent masses (hence bias) hosting the observed galaxies. As noted\nabove, this has to be considered in multi-tracer analyses, where a\nsingle sample selection might introduce systematics.\n\n\n\n\n\\subsubsection{Galaxy groups and clusters}\n\\label{subsubsec:clusters}\n\nIn this section, we perform the same analysis presented in\n\\S\\ref{subsubsec:galaxies} on the mock samples of galaxy groups and\nclusters. To simplify the discussion, in the following we will use the\nterm {\\em clusters} to refer to all of these objects, though the less\nmassive ones should be considered as galaxy groups, or just haloes,\nfrom an observational perspective (see Table \\ref{tab:table2}).\n\nFig.~\\ref{fig:iso_cl} shows the iso-correlation contours of the\nredshift-space two-point correlation function of clusters with\n$\\log(M_{500} [h^{-1}\\,{\\mbox M_\\odot}])>13$, at $z=0.2$. The other\ncontours are the best-fit models, as in Fig.~\\ref{fig:iso_gal}. As it\ncan be seen, the fingers-of-God anisotropies are almost absent,\ndifferently from the galaxy correlation function shown in\nFig.~\\ref{fig:iso_gal}, due to the lower values of non-linear motions\nof clusters at small scales. This makes the convolution of\nEq.~(\\ref{eq:ximodel}) negligible, as it can be seen comparing the\ndashed blue and solid red lines. Interestingly, when the real-space\ncorrelation function $\\xi(r)$ is measured from the mocks (green\ncontours), the dispersion model fails to match the small-scale\nclustering shape. This is primarily caused by the paucity of the\ncluster sample. Due to the low number of cluster pairs at small\nseparations, the small-scale clustering cannot be estimated\naccurately, thus introducing systematics in the model. On the\ncontrary, when $\\xi(r)$ is estimated from {\\small CAMB}, the\nsmall-scale clustering shape can be accurately described. For visual\nclarity, we have shown here only the iso-correlation contours down to\n$\\xi(s_\\perp,s_\\parallel)$\\,$=0.2$, as for lower values they appear too scattered.\n\nFig.~\\ref{fig:fs8_cl} shows the best-fit values of $f\\sigma_8$ as a\nfunction of redshift. In contrast with what we found with galaxies,\nthe constraints on the linear growth rate obtained in the scale ranges\n$30$. \n\nWe can define a stochastic process\n$X=(X_t)_{t \\geq 0}$ on $M$ in the following way: Given an initial state $i\\in S$ and an initial\nvalue $\\xi \\in M$, $X_t$ follows the trajectory generated by the vector field $u_i$ and initial\ncondition $\\xi$ for an \nexponentially distributed random time with parameter $\\lambda_i>0$. Then a new state is selected at random from\n$S\\setminus\\{i\\}$, and,\nfor another exponentially distributed random time, $X_t$ follows the new vector field corresponding to that state. \nIterating this construction we obtain a piecewise smooth\ntrajectory $(X_t)_{t\\ge 0}$ defined for all positive times and driven by one of the vector fields from $D$\nbetween any two switchings.\nWe assume that (i) all the inter-switching times are exponentially distributed and\nindependent conditioned on the sequence of driving vector\nfields, (ii) the parameter $\\lambda_j$ of the exponential time between any two switches depends only on the current\nstate $j$, \n and (iii) the probabilities of\nswitchings between any two states are positive.\n\nIt is convenient to keep track of the driving vector fields at all times. We\ndefine $A_t\\in S$ as the index of the driving vector field at time $t$, also\nreferred to as the regime or state at time $t$. It is a Markov process with continuous time and finitely\nmany states. Its trajectories are right-continuous and piecewise constant. \n\n\nAlthough $X$ alone is not a Markov process, the joint process $(X,A)$ is Markov. We denote elements of the\nassociated Markov family, i.e., the distribution on paths emitted at $(\\xi,i)\\in M \\times S$ and generated by\nthe iterative random procedure above, by $\\mathsf{P}_{\\xi,i}$, and the corresponding transition probability measures by\n$\\mathsf{P}_{\\xi,i}^{t}$, $t \\geq 0$. The transition probability measures are\ndefined on the product $\\sigma$-algebra $\\mathcal{B}(M) \\otimes \\mathcal{P}(S)$, where $\\mathcal{B}(M)$ is the\nBorel $\\sigma$-algebra on $M$ and $\\mathcal{P}(S)$ is the power set of $S$. We write $\\mathsf{E}_{\\xi,i}$ for expectation with\nrespect to $\\mathsf{P}_{\\xi,i}$.\n\n\n\n\n\nLet us recall that if the initial distribution of the Markov process $(X,A)$ is $\\mu$, then the distribution of the\nprocess at time $t$ is given by the measure $\\mu \\mathsf{P}^t$ on $M\\times S$ defined by\n\\begin{equation}\n \\mu \\mathsf{P}^t(E\\times\\{j\\}) = \\sum_{i=1}^{k} \\int_{M} \\mathsf{P}_{\\xi,i}^t(E\\times\\{j\\})\\, \\mu(d\\xi\\times\\{i\\}).\n\\label{eq:convolution} \n\\end{equation}\n\n\nA probability measure $\\mu$ on $M \\times S$ is called\ninvariant for $(\\mathsf{P}^t)$ if \n$\\mu=\\mu \\mathsf{P}^t$ for all $t \\geq 0$. \n\n\nThe main goal of this paper is to give conditions on $D$ that would guarantee absolute\ncontinuity and uniqueness of an invariant measure of the Markov semigroup $(\\mathsf{P}^t)=(\\mathsf{P}^t)_{t\\ge 0}$. The fairly general\nconditions that we suggest are formulated in geometric terms, and we proceed to introduce the\nnecessary\ndefinitions and notation.\n\n\n\\subsection{Auxiliary definitions and notation}\\label{sec:diff_geometry}\n\nLet $V(M)$ denote the set of real smooth vector fields on the manifold $M$, and let $C^{\\infty}(M)$ denote\nthe set of real-valued smooth functions on $M$. As explained above, we assume that $D$ is\ncontained in $V(M)$. Any element of $V(M)$ corresponds uniquely to a derivation on $C^{\\infty}(M)$, that is to\na linear operator $\\delta$ on $C^{\\infty}(M)$ satisfying the Leibniz rule\n\\begin{equation*}\n\\delta(f \\cdot g) = \\delta(f) \\cdot g + f \\cdot \\delta(g).\n\\end{equation*} \nThe Lie bracket of two vector fields $u$ and $v$ in $V(M)$ is defined as the vector field \n\\begin{equation*}\n[u,v](f):=u(v(f)) - v(u(f))\n\\end{equation*} \nfor test functions $f$ in $C^{\\infty}(M)$. The set $V(M)$ equipped with the bilinear operator $[.,.]$ becomes\na Lie algebra over the reals. A subset of $V(M)$ is called involutive if it is closed under taking the Lie bracket. An\ninvolutive subspace of $V(M)$ is called a subalgebra of~$V(M)$. \n\n\nThe smallest subalgebra of $V(M)$ that contains $D$ is denoted $\\mathcal{I}(D)$. The\nderived algebra $\\mathcal{I}'(D)$ is the smallest algebra containing \nLie brackets of vector fields in $\\mathcal{I}(D)$. We have $\\mathcal{I}'(D)\\subset \\mathcal{I}(D)$, but\n$\\mathcal{I}'(D)$ might not contain any elements of $D$ and may therefore be strictly contained in $\\mathcal{I}(D)$.\nFurther, we define $\\mathcal{I}_{0}(D)$ as the set of vector fields of the form\n\\begin{equation*}\nv + \\sum_{i=1}^{k}{\\lambda_{i}u_{i}},\n\\end{equation*}\nwhere $v \\in \\mathcal{I}'(D)$, $u_1,\\ldots,u_k \\in D$ and $\\sum_{i=1}^{k}{\\lambda_{i}}=0$. Finally, we set\n\\begin{equation*}\n\\mathcal{I}(D)(\\xi):=\\{u(\\xi):\\ u \\in \\mathcal{I}(D)\\}\n\\end{equation*}\nand\n\\begin{equation*}\n\\mathcal{I}_{0}(D)(\\xi):=\\{u(\\xi):\\ u \\in \\mathcal{I}_{0}(D)\\}\n\\end{equation*}\nfor any $\\xi \\in M$. The sets $\\mathcal{I}(D)(\\xi)$ and $\\mathcal{I}_{0}(D)(\\xi)$ are finite-dimensional\nvector spaces. \n\n\\medskip\n\nOur main results will be based on the following assumptions that can naturally be called hypoellipticity conditions\nin analogy with H\\\"ormander's theory.\nWe say that a point $\\xi\\in M$ satisfies Condition~A if $\\dim \\mathcal{I}_{0}(D)(\\xi)=n$. \nWe say that a point $\\xi\\in M$\nsatisfies Condition~B if $\\dim \\mathcal{I}(D)(\\xi)=n$. \n\nThe set of points satisfying Condition~A is open and so is the set of\npoints satisfying Condition~B.\n\\medskip\n\n\nFor our absolute continuity results we will need a reference measure on $M$ that will play the role of Lebesgue\nmeasure. As a smooth manifold, $M$ can be endowed with a Riemannian metric. \nThe metric tensor can be used to define measures on coordinate patches of $M$. \nOne can use then a partition of \nunity in a standard way (see, e.g., \\cite[Section 7]{Taylor:MR2245472})\nto construct a Borel measure on $M$ whose\npushforward to $\\mathbb{R}^{n}$ under any chart map is equivalent to Lebesgue measure. We call the measure on $M$\nobtained through this construction Lebesgue measure, denote it by $\\lambda^{M}$, and use\nit as the main reference measure, often omitting ``with respect to Lebesgue measure'' when writing about absolute\ncontinuity. \nThe product of the Lebesgue measure on $M$ and counting measure on $S$ will be called\nthe Lebesgue measure on\n$M\\times S$. We denote the Lebesgue\nmeasure on\n$\\mathbb{R}^{m}$ by~$\\lambda^{m}$. \n\n\\medskip\n\nIt remains to introduce the flows generated by vector fields in $D$ and the concept of reachability.\n\nFor $i\\in S$, we denote the flow function of the vector field $u_i$ by $\\Phi_i$. Due to forward completeness of $u_i$, the\nflow function is uniquely defined for all $t>0$ and $\\eta\\in M$ by\n\\begin{align*}\n \\frac{d}{dt}\\Phi_i(t,\\eta)&=u_i(\\Phi_i(t,\\eta)),\\\\\n \\Phi_i(0,\\eta)&=\\eta.\n\\end{align*}\n\nFor $m\\in\\mathbb{N}$, we will consider vectors \n${\\bf t}=(t_1,\\ldots,t_m)$ of waiting times between subsequent switches and vectors ${\\bf i}=(i_1,\\ldots,i_m)$ of\ndriving states during these waiting intervals. We will restrict ourselves to positive waiting times, but it can also be useful \n(see \\cite{Sussmann-Jurdjevic:MR0338882} and \\cite{Jurdjevic:MR1425878}) to admit flows backwards in\ntime. \n\nWe write $\\mathbb{R}_{+}$ to denote the positive real line $(0;\\infty)$. \n\nFor ${\\bf t}=(t_1,\\ldots,t_m)\\in \\mathbb{R}_{+}^{m}$ and ${\\bf i}=(i_1,\\ldots,i_m)\\in S^{m}$, we define \n\\begin{equation*}\n\\Phi_{{\\bf i}}({\\bf t},\\xi):=\\Phi_{i_m}(t_m,\\Phi_{i_{m-1}}(t_{m-1},\\ldots\\Phi_{i_1}(t_1,\\xi))\\ldots)\n\\end{equation*}\nas the cumulative flow along the trajectories of $u_{i_1},\\ldots,u_{i_m}$ with starting point $\\xi\\in M$.\n\nThe transition probabilities $\\mathsf{P}_{\\xi,i}^{t}$ can be expressed in terms of cumulative flows. We do not specify\nthese straightforward relations in order to avoid heavy notation. \n\n\nA point $\\eta\\in M$ is called $D$-reachable from a point $\\xi\\in M$ if there exist a time vector ${\\bf\nt}$ with\npositive components and a vector~${\\bf i}$ of driving states such that\n\\begin{equation*}\n\\eta = \\Phi_{{\\bf i}}({\\bf t},\\xi).\n\\end{equation*}\nIf the components of ${\\bf t}$ sum up to $t$, we say that $\\eta$ is $D$-reachable from $\\xi$ at time~$t$. \n\nFor $\\xi \\in M$ and $t>0$, let $L_{t}(\\xi)$ denote the set of $D$-reachable points from $\\xi$ at time~$t$, and\nlet $L(\\xi)=\\bigcup_{t>0} L_{t}(\\xi)$ denote the set of $D$-reachable points from $\\xi$. The points in\nthe closure $\\overline{L(\\xi)}$ can be called $D$-approachable from $\\xi$.\nLet $L=\\bigcap_{\\xi \\in M} \\overline{L(\\xi)}$ denote the set of points that are $D$-approachable from all\nother\npoints.\n\n\n\n \n\n\n\n\\subsection{Main results} \\label{sec:absolute-continuity-invariant}\n\nThe following is the main theorem of this paper.\n \n\n\\begin{theorem} \\label{thm:uniqueness}\nSuppose Hypoellipticity Condition~B is satisfied at some $\\xi \\in L$.\nIf $(\\mathsf{P}^t)$ has an invariant measure, then it is unique and absolutely continuous with respect\nto the Lebesgue measure on $M\\times S$.\n\\end{theorem} \n\n\\begin{remark}\\rm\nOf course, Theorem~\\ref{thm:uniqueness} remains true if we replace $L$ by any of its subsets. For example, if\none of the vector fields in $D$ has a minimal global\nattractor, then it is sufficient to check hypoellipticity for some point of the attractor.\n\\end{remark}\n\n\nUniqueness of invariant distributions is tightly connected to the regularity of the Markov semigroup. Various\naspects of regularity in connection with ergodicity have been studied in the literature: the existence of minorizing\nkernels, the strong Feller property, etc. \nThe main task in the proof of Theorem~\\ref{thm:uniqueness} is to establish regularity for\ntransition probabilities under Hypoellipticity Condition~B. However, we begin with a much stronger regularity property that \ncan be established under the stronger Hypoellipticity Condition~A.\n\n\n\n\n\n\\begin{theorem}\\label{thm:AC-component-1}\nIf Condition~A is satisfied at a point $\\xi\\in M$, then for any $i \\in\nS$ and any $t > 0$, the transition kernel $\\mathsf{P}_{\\xi,i}^{t}$ has a nonzero absolutely\ncontinuous component with respect to Lebesgue measure on $M \\times S$. \n\\end{theorem} \n \n\nUnder the weaker Condition~B it may happen that none of the transition \nprobability measures $\\mathsf{P}_{\\xi,i}^t$, $t > 0$, has a nonzero absolutely continuous component. For example, let $M$~be\nthe\n$n$-dimensional torus $\\mathbb{T}^n=\\mathbb{R}^n\/\\mathbb{Z}^n$, and let \n$D=\\{u_1,\\ldots,u_n\\}$ be the standard basis in $\\mathbb{R}^{n}$. Fix an arbitrary time $t > 0$. The set of\npoints $D$-reachable from the origin at time $t$ is the image of \n\\begin{equation*}\n\\biggl\\{(s_1,\\ldots,s_n) \\in [0;\\infty)^{n}: \\quad \\sum_{j=1}^{n} s_j =t \\biggr\\}\n\\end{equation*}\nunder the covering map $\\mathbb{R}^n\\to\\mathbb{T}^n$,\nand has Lebesgue measure zero, so $\\mathsf{P}_{\\xi,i}^t$ is a purely singular measure.\n \nNevertheless, Condition~B guarantees that time averages of transition probabilities have\nnontrivial absolutely continuous components. Specifically, we will establish this for\nthe resolvent probability kernel $\\mathsf{Q}_{\\xi,i}$ defined by \n\\begin{equation}\n\\label{eq:resolvent}\n\\mathsf{Q}_{\\xi,i}(E\\times\\{j\\}):= \\int_{\\mathbb{R}_{+}} e^{- t}\\,\\mathsf{P}_{\\xi,i}^t(E\\times\\{j\\}) dt.\n\\end{equation}\nThe resolvent kernels are useful in the study of invariant distributions due to the following straightforward result.\n\\begin{lemma}\\label{lem:Q-invariance}\nIf a measure $\\mu$ is $(\\mathsf{P}^t)$-invariant it is also $(\\mathsf{Q})$-invariant, i.e., \n$\\mu=\\mu \\mathsf{Q}$, where the convolution $\\mu\\mathsf{Q}$ is defined analogously to~\\eqref{eq:convolution}. \n\\end{lemma}\n\n\\begin{theorem}\\label{thm:AC-component-2}\nIf Condition B is satisfied at some point $\\xi\\in M$, then for any $i\\in S$,\nthe measure $\\mathsf{Q}_{\\xi,i}$ defined by~\\eqref{eq:resolvent}\nhas a nonzero absolutely continuous component with respect to Lebesgue measure on $M \\times S$. \n\\end{theorem}\n\n\n\n\n\n\\bigskip\n\n\n\n\nAlthough the convergence of transition probabilities to the invariant measures is out of\nthe scope of the present paper and will be addressed elsewhere, our analysis suggests that under the assumptions of\nTheorem~\\ref{thm:uniqueness} and Condition~A at $\\xi$ this convergence holds true while Condition~B at $\\xi$ implies\nonly Ces\\`aro convergence.\n\n\n\n\nAt the heart of our proofs of Theorems~\\ref{thm:AC-component-1} and~\\ref{thm:AC-component-2} are classical results\nfrom geometric control theory that can be found in~\\cite{Jurdjevic:MR1425878}. The statements we present are derived\nfrom Theorems~$3.1$,~$3.2$, and~$3.3$ in~\\cite{Jurdjevic:MR1425878}. Analogous results for the special case of analytic\nvector fields on a real analytic manifold are first stated in~\\cite[Theorems~$3.1$ and\n$3.2$]{Sussmann-Jurdjevic:MR0338882}. In their paper, Sussmann and Jurdjevic were able to build on prior\nwork~\\cite{Chow:MR0001880} by Chow who considered symmetric families of analytic vector fields.\nKrener generalized these results to $C^{\\infty}$-vector fields in~\\cite{Krener:MR0383206}. \n\n\nRecall that a regular point of a function $f:\\mathbb{R}^{m} \\to M$ is a point ${\\bf t} \\in \\mathbb{R}^{m}$ such that the differential $Df({\\bf t})$ has full rank. \nIf $Df({\\bf t})$ has deficient rank, ${\\bf t}$ is called a critical point of~$f$.\n\n\n\\begin{theorem} \\label{thm:Jurdjevic-1} Assume that Condition~A holds at\nsome $\\xi\\in M$.\nThen:\n\\begin{enumerate}\n \\item For any $i,j\\in S$,\nthere are an integer $m>n$\nand a vector ${\\bf i} \\in S^{m+1}$ with $i_1=i$ and $i_{m+1}=j$ such that for any $t>0$ the mapping $f_\\mathbf{i}:\\mathbb{R}_{+}^{m} \\to M$\ndefined by\n\\begin{equation}\n\\label{eq:flow_for_transition_probability}\nf_\\mathbf{i}(t_1,\\ldots,t_{m})=\\Phi_{{\\bf i}}\\biggl(t_1,\\ldots,t_{m},t-\\sum_{l=1}^{m} t_l, \\xi \\biggr)\n\\end{equation}\nhas a nonempty open set of regular points in the simplex \n\\begin{equation*}\n\\Delta_{t,m}:=\\biggl\\{(t_1,\\ldots,t_m) \\in \\mathbb{R}_{+}^{m}:\\ \\sum_{l=1}^{m} t_l < t \\biggr\\}.\n\\end{equation*}\n\\item The interior of $L(\\xi)$ is nonempty and dense in $L(\\xi)$.\n\\end{enumerate}\n\\end{theorem} \n \n\\begin{theorem} \\label{thm:Jurdjevic-2}\nAssume that Condition~B holds at some $\\xi\\in M$. \nThen:\n\\begin{enumerate}\n \\item For any $i,j\\in S$,\nthere are an integer $m>n$\nand a vector ${\\bf i} \\in S^{m+1}$ with $i_1=i$ and $i_{m+1}=j$ such that for any $t>0$ the mapping $F_\\mathbf{i}:\\mathbb{R}_{+}^{m+1} \\to M$\ndefined by\n\\begin{equation*}\nF_\\mathbf{i}(t_1,\\ldots,t_{m+1})=\\Phi_{{\\bf i}}(t_1,\\ldots,t_{m+1},\\xi)\n\\end{equation*}\nhas a nonempty open set of regular points in $\\Delta_{t,m+1}$.\n\\item The interior of $L(\\xi)$ is nonempty and dense in $L(\\xi)$.\n\\end{enumerate}\n\\end{theorem}\n\nCondition~A is stronger than Condition~B, so it is not surprising that the conclusion of\nTheorem~\\ref{thm:Jurdjevic-1} implies the conclusion of Theorem~\\ref{thm:Jurdjevic-2}.\n\nThe first statement of Theorem~\\ref{thm:Jurdjevic-2} corresponds to Theorem~$3.1$ in~\\cite{Jurdjevic:MR1425878}, which reads as follows: Under the assumptions of Theorem~\\ref{thm:Jurdjevic-2}, any neighborhood $U$ of $\\xi$ contains points that are normally accessible from $\\xi$ at arbitrarily small times. A point $\\eta$ in $M$ is called normally accessible from $\\xi$ at time $t>0$ if there exist vectors $\\mathbf{i} \\in S^{m+1}$ and $(\\hat{t}_1,\\ldots,\\hat{t}_{m+1}) \\in \\Delta_{t,m+1}$ such that $F_\\mathbf{i}(\\hat{t}_1,\\ldots,\\hat{t}_{m+1}) = \\eta$ and the differential $DF_\\mathbf{i}(\\hat{t}_1,\\ldots,\\hat{t}_{m+1})$ has full rank. It's worth pointing out, though, that in~\\cite{Jurdjevic:MR1425878} only one sequence~$\\mathbf{i}$ resulting in $F$ with a regular point is constructed. But since the flow generated by any vector field is a family of diffeomorphisms, and since the set of points satisying Condition~B is open, one can append any indices in front or at the back of that sequence without destroying the desired properties, and thus recover this part of Theorem~\\ref{thm:Jurdjevic-2} as we state it. \n\nThe fact that the interior of $L(\\xi)$ is nonempty and dense in $L(\\xi)$ follows from Theorem~$3.2.a$ in~\\cite{Jurdjevic:MR1425878}.\n\nTheorem~\\ref{thm:Jurdjevic-1} essentially follows from applying Theorem~$3.1$ (\\cite{Jurdjevic:MR1425878}) to $\\mathbb{R} \\times M$ and\nvector fields ${\\bf 1} \\oplus u_i, i\\in S$, where\n\\begin{equation*}\n({\\bf 1} \\oplus u)(r,\\xi):= (1,u(\\xi)),\\quad (r,\\xi)\\in \\mathbb{R} \\times M,\n\\end{equation*}\nand ${\\bf 1}$ is the unit vector field on $\\mathbb{R}$ corresponding to\nthe derivation $\\partial\/\\partial r$ and identically equal to~$1$ in the natural coordinates on $\\mathbb{R}$. \n\n\n\n\n\\section{Proof of Theorem~\\ref{thm:AC-component-1}} \\label{sec:AC-component-1-proof} \n\nWe need to prove that for any $t>0$ and $i \\in S$, the measure $\\mathsf{P}^t_{\\xi,i}$ is not singular.\n\n\\medskip\n\nFor any finite sequence ${\\mathbf{i}}$ of indices in $S$ with initial index $i$ (we will call these sequences \nadmissible), let $C_{{\\mathbf{i}}}$ be the event that the driving vector fields up to time $t$ appear in the order determined by ${\\mathbf{i}}$. Since\n$\\mathsf{P}_{\\xi,i}(C_{\\mathbf{i}})>0$ for any admissible $\\mathbf{i}$\nit suffices to find an admissible sequence $\\mathbf{i}$ such that \n$\\mathsf{P}_{\\xi,i}^t(\\cdot| C_{\\mathbf{i}})$ is not singular.\nWe claim that this holds true for the the sequence $\\mathbf{i}$ provided by \nTheorem~\\ref{thm:Jurdjevic-1}. According to\nTheorem~\\ref{thm:Jurdjevic-1}, there is an admissible sequence $\\mathbf{i}=(i_1,i_2,\\ldots,i_{m+1})$ with $i_1 = i$ \nsuch that the function\n$f_\\mathbf{i}$\nhas a regular point in $\\Delta_{t,m}$. Since the set of regular points of a differentiable function is open in its domain, the function $f_\\mathbf{i}$ is regular in a nonempty open set $B \\subset \\Delta_{t,m}$. \n\nLet $T_1,T_2,\\ldots,T_{m+1}$ be independent and exponentially distributed random variables such that $T_j$ has\nparameter\n$\\lambda_{i_j}$ for $1\\leq j \\leq m+1$. \n\nOn $C_\\mathbf{i}$ we have $A_t=i_{m+1}$, and the distribution of $X_t$ under $\\mathsf{P}_{\\xi,i}(\\cdot | C_{\\mathbf{i}})$ coincides\nwith the distribution of $f_\\mathbf{i}(T_1,\\ldots,T_m)$ conditioned on the event\n\\begin{equation}\n\\label{eq:event_R}\nR=\\biggl\\{\\sum_{j=1}^{m} T_j < t \\le \\sum_{j=1}^{m+1} T_j \\biggr\\}.\n\\end{equation}\nThe distribution of the random vector $(T_1,\\ldots,T_m)$ conditioned on $R$, \nis equivalent to the uniform distribution on the simplex \n\\begin{equation*}\n\\Delta_{t,m} := \\biggl\\{(t_1,\\ldots,t_m) \\in \\mathbb{R}_{+}^{m}: \\quad \\sum_{j=1}^{m}t_j < t \\biggr\\}.\n\\end{equation*}\nNow the theorem directly follows from the following result:\n\\begin{lemma}\\label{lem:pushforward_full_rank} Let $n,m\\in\\mathbb{N}$, $n\\le m$.\nSuppose that $B$ and $\\Delta$ are nonempty open sets in~$\\mathbb{R}^m$, $B$ is dense in $\\Delta$,\nand $M$ is an $n$-dimensional\nsmooth manifold.\nIf $f:\\Delta\\to M$ is \ndifferentiable on $\\Delta$ and all points in $B$ are regular for $f$, then for any\nabsolutely continuous\nprobability measure $\\mu$ on $\\Delta$ satisfying $\\mu(B)>0$, its\npushforward\n$\\mu f^{-1}$ is not singular with respect to~$\\lambda^M$.\n\\end{lemma}\n \nWe will prove this lemma only for $M=\\mathbb{R}^n$. Modifying the proof for the general case using\ncoordinate patches on $M$ amounts only to notational differences. \n\nWe will use the following statement (see, e.g., Proposition~4.4\nin~\\cite{Davydov-et-al:MR1604537}):\n\\begin{lemma}\\label{lem:pushforward_for_nonzero_det}\n Let $f:B \\to\\mathbb{R}^m$ be a Borel function a.e.-differentiable on an open set $B \\subset\\mathbb{R}^m$\nand satisfying $\\lambda^m\\{\\mathbf{t}\\in B:\\, \\det Df(\\mathbf{t})=0\\}=0$. If $\\mu\\ll \\lambda^m$, then\n$\\mu f^{-1}\\ll\\lambda^m$, and\n\\[\n \\frac{d(\\mu f^{-1})}{d\\lambda^m}(\\mathbf{s})=\\sum_{\\mathbf{t}\\in B: f(\\mathbf{t})=\\mathbf{s}} |\\det Df(\\mathbf{t})|^{-1}\\frac{d\\mu}{d\\lambda^m}(\\mathbf{t})\n\\]\n\n\\end{lemma}\n\n\n\n\n\\bpf[Proof of Lemma~\\ref{lem:pushforward_full_rank}]We must prove that for any $E \\subset \\mathbb{R}^n$ with $\\lambda^n(E)=0$, \n\\begin{equation}\n\\label{eq:not_singular}\n\\mu\\{\\mathbf{t}\\in\\Delta:\\, f(\\mathbf{t})\\in E\\}<1. \n\\end{equation}\nFor any $\\mathbf{t} \\in B$, we can find some $n$\ncolumns of $Df(\\mathbf{t})$ (without loss of generality, first~$n$ columns) that are linearly\nindependent.\nFor $\\rho:B \\to \\mathbb{R}^n \\times \\mathbb{R}^{m-n}$ defined by\n\\[\n\\rho:\\mathbf{t}=(t_1,\\ldots,t_m) \\mapsto (f(\\mathbf{t}), t_{n+1},\\ldots,t_m),\n\\]\nand any $\\mathbf{t}\\in B$, we have\n\\begin{equation*}\n\\det D\\rho(\\mathbf{t} ) \\neq 0. \n\\end{equation*}\n Since $E \\times \\mathbb{R}^{m-n}$ is a set of measure zero, Lemma~\\ref{lem:pushforward_for_nonzero_det} implies\nthat \n\\begin{equation*}\n0 = \\lambda^{m}\\{\\mathbf{t} \\in B:\\, \\rho(\\mathbf{t}) \\in E \\times \\mathbb{R}^{m-n}\\} = \\lambda^{m}\\{\\mathbf{t} \\in B:\\, f(\\mathbf{t}) \\in E\\}.\n\\end{equation*}\nTherefore, $\\mu\\{\\mathbf{t} \\in B:\\, f(\\mathbf{t}) \\in E\\}=0$, and\n\\[\n \\mu\\{\\mathbf{t}\\in\\Delta:\\, f(\\mathbf{t})\\in E\\}= \\mu\\{\\mathbf{t}\\in \\Delta \\setminus B:\\, f(\\mathbf{t})\\in E\\}\\le 1 -\\mu(B)<1. \n\\]\nso \\eqref{eq:not_singular} is established and the proof is complete.\n{{\\hfill $\\Box$ \\smallskip}}\n \n\n\n\\section{Proof of Theorem~\\ref{thm:AC-component-2}} \\label{sec:AC-component-2-proof} \n\n\n \n We need to show that $\\mathsf{Q}_{\\xi,i}$ is not a singular measure. The proof is based on\nTheorem~\\ref{thm:Jurdjevic-2}.\n\nFor the $S$-valued process $A$ we denote by $I_t(A)$ the sequence of states visited by $A$ between $0$\nand $t$.\nFor any $m\\in\\mathbb{N}$ and any sequence $\\mathbf{i}\\in S^m$, we can introduce an auxiliary measure $\\mathsf{Q}_{\\xi,i,\\mathbf{i}}$ on $M$ by \n\\[\n\\mathsf{Q}_{\\xi,i,\\mathbf{i}}(B)=\\int_{R_{+}} e^{-t}\\,\\mathsf{P}_{\\xi,i}\\{X_t\\in B\\ \\text{and}\\ I_t(A)=\\mathbf{i}\\}\\, dt,\\quad B\\in\\mathcal{B}(M).\n\\]\nSince \n\\begin{equation}\n\\label{eq:Qq-via-I_t}\n\\mathsf{Q}_{\\xi,i}(B \\times \\{j\\})=\\sum_{m}\\sum_{\\mathbf{i}=(i,i_2,\\ldots,i_{m-1},j)\\in S^m}\\mathsf{Q}_{\\xi,i,\\mathbf{i}}(B), \n\\end{equation}\nit is sufficient to find $\\mathbf{i}=(i_1,\\ldots,i_m)$ with $i_1=i$ such that $\\mathsf{Q}_{\\xi,i,\\mathbf{i}}(M)>0$ and\n\\[\n\\overline \\mathsf{Q}_{\\xi,i,\\mathbf{i}}(\\cdot)= \\frac{\\mathsf{Q}_{\\xi,i,\\mathbf{i}}(\\cdot)}{\\mathsf{Q}_{\\xi,i,\\mathbf{i}}(M)} \n\\]\n is a nonsingular probability measure. To apply Lemma~\\ref{lem:pushforward_full_rank}, we need to represent\n$\\overline \\mathsf{Q}_{\\xi,i,\\mathbf{i}}$ as the pushforward of a measure, equivalent to Lebesgue measure, under a smooth map with a \nnonempty set of regular points.\n\n\nSince Condition~B holds at $\\xi$, Theorem~\\ref{thm:Jurdjevic-2} yields an integer $m>n$\nand a sequence $\\mathbf{i}=(i_1,i_2,\\ldots,i_{m+1})$ with $i_1=i$, such that the function $F_\\mathbf{i}:\\mathbb{R}_{+}^{m+1} \\to\nM$ defined by\n\\begin{equation*}\nF_\\mathbf{i}(\\mathbf{t})= \\Phi_{\\mathbf{i}}(\\mathbf{t},\\xi)\n\\end{equation*}\nhas a regular point.\nFor this $\\mathbf{i}$ provided by Theorem~\\ref{thm:Jurdjevic-2}, $\\overline \\mathsf{Q}_{\\xi,i,\\mathbf{i}}$ is \nthe distribution of $\\Phi_{{\\bf i}} \\left(T_1,\\ldots,T_m,T-\\sum_{j=1}^{m} T_j, \\xi \\right)$ conditioned on\n\\begin{equation}\nR= \\biggl\\{\\sum_{j=1}^{m} T_j 0$, using the\ninvariance of $\\mu$, we can write\n\\begin{align}\n\\mu_{ac}+\\mu_{s} &= \\mu=\\mu \\mathsf{P}^t\n\\label{eq:decomposition_in_ac_and_s}\n=\n\\mu_{ac} \\mathsf{P}^t+\\mu_s\\mathsf{P}^t\n=\\sum_{j=1}^{k} \\nu_j + \\mu_s\\mathsf{P}^t,\n\\end{align}\nwhere \n\\begin{equation}\n\\label{eq:nu_j}\n\\nu_{j}(\\cdot)= \\int_M \\mathsf{P}^t_{\\xi,j} (\\cdot)\\mu_{ac}(d\\xi\\times\\{j\\}),\\quad j\\in S.\n\\end{equation}\n\n\nWe claim that the measures $\\nu_j$, $j\\in S$,\nare absolutely continuous. To see this we\ncheck that for any sequence $\\mathbf{i}=(i_1,\\ldots,i_{m+1})$ with $i_1=j$, the measure $\\nu_\\mathbf{i}$ defined by\n\\begin{align}\\notag\n \\nu_{\\mathbf{i}}(E)&= \\int_M \\mathsf{P}_{\\xi,j} (X_t\\in E\\, |C_\\mathbf{i})\\mu_{ac}(d\\xi\\times\\{j\\})\\\\\n&=\\int_M \\mathsf{P}\\biggl(\\Phi_{{\\bf i}}\\biggl(T_1,\\ldots,T_m,t-\\sum_{l=1}^{m} T_l,\\xi \\biggr)\\in\nE\\, \\biggr|\\, R\\biggr)\\mu_{ac}(d\\xi\\times\\{j\\})\\label{eq:one_of_continuous_components_disintegrated}\n\\end{align}\n is absolutely continuous (here we use the notation introduced in Section~\\ref{sec:AC-component-1-proof}). Suppose\n$\\lambda^M(E)=0$. For fixed $T_1,\\ldots,T_m,T_{m+1}$, the map $\\Phi_\\mathbf{i}$ is a diffeomorphism in $\\xi$. Therefore, on\nevent $R$ introduced in~\\eqref{eq:event_R}, we have\n\\[\n \\mu_{ac}\\biggl(\\xi \\times \\{j\\}:\\ \\Phi_{{\\bf i}}\\biggl(T_1,\\ldots,T_m,t-\\sum_{l=1}^{m} T_l,\\xi \\biggr)\\in\nE \\biggr)=0,\n\\]\nand $\\nu_\\mathbf{i}(E)=0$ follows from disintegrating the right side\nof~\\eqref{eq:one_of_continuous_components_disintegrated} and changing the order of integration.\n\nNow, using~\\eqref{eq:decomposition_in_ac_and_s} and the absolute continuity of $\\nu_j$, $j\\in S$, we can write\n\\begin{equation}\n\\label{eq:decomposition_of_ac_part}\n\\mu_{ac}= \n\\sum_{j=1}^{k} \\nu_j + (\\mu_s\\mathsf{P}^t)_{ac}\\ .\n\\end{equation}\nSince $\\mathsf{P}^t_{\\xi,j}(M\\times S)=1$ for all $\\xi$ and $j$, \\eqref{eq:nu_j} implies\n$\\sum_{j=1}^{k} \\nu_j(M\\times S)= \\mu_{ac}(M\\times S)$. Therefore, applying~\\eqref{eq:decomposition_of_ac_part}\nto $M \\times S$, we obtain that the absolutely continuous\ncomponent of the measure $\\mu_s\\mathsf{P}^t$ is zero. In other\nwords, $\\mu_s\\mathsf{P}^t$ is singular, and from~\\eqref{eq:decomposition_in_ac_and_s} and the absolute\ncontinuity of $\\nu_j$, $j\\in S$, we obtain\n\\begin{equation}\n\\label{eq:invariance_of_singular_component}\n\\mu_s = \\mu_s\\mathsf{P}^t.\n\\end{equation} \nIn other words, $\\mu_s$ is invariant for $(\\mathsf{P}^t)$. It follows from~\\eqref{eq:decomposition_in_ac_and_s}\nthat $\\mu_{ac}$ is also invariant. Since $\\mu$ is ergodic, it cannot be represented as a sum of two nontrivial\ninvariant measures. This means that either $\\mu=\\mu_{ac}$ or $\\mu=\\mu_s$.\n{{\\hfill $\\Box$ \\smallskip}}\n\n\n\nWe endow the state space $S$ with the discrete topology and recall that a point $(\\xi,i)\\in M\\times S$ is\ncontained in the support of a measure if and\nonly if the measure of every open neighborhood of $(\\xi,i)$ is positive. \n \n\\begin{theorem}\\label{thm:absolute-continuity-for-ergodic}\nLet $\\mu$ be an ergodic invariant measure for $(\\mathsf{P}^t)$. Assume that the support of $\\mu$ contains a point\n$(\\eta,i)$ such that Condition~B holds at~$\\eta$. \nThen, $\\mu$ is absolutely continuous\nwith\nrespect to Lebesgue measure on $M\\times S$. \n\\end{theorem}\n\n\n\n\nWe will need several auxiliary statements. \n\n\n\\begin{lemma} \\label{lem:support}\nLet $\\nu$ be a finite Borel measure on $M \\times S$ with support $K$. If $U$ is any open set in $M \\times\nS$ whose intersection with $K$ is nonempty, we have \n\\begin{equation*}\n\\nu(U \\cap K)>0.\n\\end{equation*}\n\\end{lemma} \n \n\\bpf\nAssume that $\\nu(U \\cap K)=0$. The complement of the support of $K$ has measure zero. Therefore\n\\begin{equation*}\n\\nu(U) = \\nu(U \\cap K) + \\nu(U \\cap K^{c})=0.\n\\end{equation*}\nThus, $U^{c}$ is a closed subset of $M \\times S$ whose complement has measure zero. From the\ndefinition of the support, we obtain $K \\subset U^{c}$. But then, $U \\cap K$ must be empty, a contradiction. \n{{\\hfill $\\Box$ \\smallskip}}\n\n\n\n\n\\begin{lemma} \\label{lem:open-starting-points}\nFix a time $t>0$, an open subset $E$ of $M$ and an index $j \\in S$. The set $H$ of points $(\\xi,i) \\in M \\times\nS$ satisfying $\\mathsf{P}^t_{\\xi,i}(E\\times\\{j\\})>0$ is open. \n\\end{lemma}\n\n\\bpf We will use the notation introduced in Section~\\ref{sec:AC-component-1-proof}.\n\nLet us assume that $H$ is not open and there is a sequence $((\\xi_p,i))_{p\\in\\mathbb{N}}$ in $H^c$ \nconverging to a point $(\\xi,i)\\in H$. \n\nSince $(\\xi,i)$ is an element of $H$, there exists an admissible sequence \\\\ $\\mathbf{i}=(i,i_2,...,i_m,j)$ for which\n$\\mathsf{P}_{\\xi,i}(X_t\\in E|C_\\mathbf{i})>0$.\n\nOn the other hand, we have $\\mathsf{P}_{\\xi_p,i}(X_t\\in E|C_\\mathbf{i})=0$ for all $p\\in\\mathbb{N}$. Therefore,\nfor each $p$, the set $B(\\xi_p)$\nhas zero Lebesgue measure, where\n\\[\nB(\\eta)=\\biggl\\{\\mathbf{t}\\in\\Delta_{t,m}:\\, \\Phi_{{\\bf i}}\\biggl(t_1,\\ldots,t_m,t-\\sum_{j=1}^{m} t_j,\\eta \\biggr)\\in E\n\\biggr\\},\\quad \\eta\\in M.\n\\]\n\nIf $\\mathbf{t}\\in B(\\xi_p)^c$ for all $p$,\nthen $\\Phi_{\\mathbf{i}}(\\mathbf{t},\\xi)\\in E^c$ since $\\Phi_{\\mathbf{i}}(\\mathbf{t},\\cdot)$ is a diffeomorphism and $E^c$ is a closed set.\nHence,\nthe Lebesgue measure of $B(\\xi)$ is also zero, so \n$\\mathsf{P}_{\\xi,i}(X_t\\in E|C_\\mathbf{i})=0$, which contradicts our assumption and finishes the proof.\n{{\\hfill $\\Box$ \\smallskip}}\n\n\n\n\n\n\n\n\n\n\n \n\n\\bpf[Proof of Theorem~\\ref{thm:absolute-continuity-for-ergodic}] According to Theorem~\\ref{thm:AC-or-S} we need to show\nthat $\\mu$ is not singular. If $\\mu$ is singular,\nit is entirely supported on a zero Lebesgue measure set $G\\subset M\\times S$, so\n$\\mu(G^c)=0$. Since $\\mu$ is $(\\mathsf{P}^t)$-invariant, it is also $\\mathsf{Q}$-invariant. Therefore,\n$\\mu(G^c)=\\mu \\mathsf{Q}(G^c)$, and we see that $\\mu(V)=0$ where\n\\[\n V=\\{(\\xi,j)\\in M \\times S:\\ \\mathsf{Q}_{\\xi,j} (G^{c})>0\\}.\n\\]\nLet $U$ be the set of points $\\xi \\in M$ where Condition~B holds.\nDue to Theorem~\\ref{thm:AC-component-2}, $U\\times S \\subset V$, and\nwe conclude that $\\mu(U\\times S)=0$.\n\n\nRecall that $U$ is an open subset of $M$, and $(U\\times S)\\cap \\mathop{\\rm supp} \\mu\\ne\\emptyset$ by assumption.\nLemma~\\ref{lem:support} implies that $\\mu((U\\times S)\\cap \\mathop{\\rm supp} \\mu)>0$. The contradiction with $\\mu(U\\times\nS)=0$ completes the proof.\n{{\\hfill $\\Box$ \\smallskip}}\n\n\nOf course, if one replaces Condition~B in Theorem~\\ref{thm:absolute-continuity-for-ergodic} with \nthe stronger Condition~A the resulting statement holds automatically, but one can give a proof that does not involve\nthe resolvent $Q$:\n\n\\begin{theorem}\\label{thm:absolute-continuity-for-ergodic-under-condition-A}\nLet $\\mu$ be an ergodic invariant measure for $(\\mathsf{P}^t)$. Assume that the support of $\\mu$ contains a point\n$(\\eta,i)$ such that Condition~A holds at~$\\eta$.\nThen, $\\mu$ is absolutely continuous\nwith\nrespect to the Lebesgue measure on $M\\times S$. \n\\end{theorem}\n\n\n\\bpf According to Theorem~\\ref{thm:AC-or-S} we need to show that $\\mu$ is not singular. If $\\mu$ is singular,\nit is entirely supported on a zero Lebesgue measure set $G\\subset M\\times S$, so\n$\\mu(G^c)=0$. Since $\\mu(G^c)=\\mu \\mathsf{P}^t(G^c)$, we see that $\\mu(V)=0$ where\n\\[\n V=\\{(\\xi,j)\\in M \\times S:\\ \\mathsf{P}^t_{\\xi,j} (G^{c})>0\\}.\n\\]\nLet $U$ be the set of points $\\xi \\in M$ where Condition~A holds.\nDue to Theorem~\\ref{thm:AC-component-1}, $U\\times S \\subset V$, and\n we conclude that $\\mu(U\\times S)=0$.\n\n\nRecall that $U$ is an open subset of $M$, and $(U\\times S)\\cap \\mathop{\\rm supp} \\mu\\ne\\emptyset$ by assumption.\nLemma~\\ref{lem:support} implies that $\\mu((U\\times S)\\cap \\mathop{\\rm supp} \\mu)>0$. The contradiction with $\\mu(U\\times\nS)=0$ completes the proof.\n{{\\hfill $\\Box$ \\smallskip}}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Proof of Theorem~\\ref{thm:uniqueness}} \\label{sec:uniqueness-proof}\n\nFirst, we establish two properties of the set $E= L\\cap U$, where\n$U$ is the open set of points satisfying Condition B.\n\\begin{lemma} \\label{lem:non-empty-interior}\nThe set $E$ has nonempty interior.\n\\end{lemma}\n\n\\bpf By assumption, $\\xi\\in E$ , so $U\\ne\\emptyset$ and $L(\\xi) \\cap U\\ne \\emptyset$ by continuity of the vector fields in $D$. Since $\\xi\\in U$, Theorem~\\ref{thm:Jurdjevic-2} implies that $L(\\xi)$\nhas nonempty interior that is dense in $L(\\xi)$. Therefore, the set\n\\begin{equation*}\nV=L(\\xi)^{\\circ} \\cap U\n\\end{equation*}\nis nonempty and open. Clearly, $V\\subset U$, and it remains to prove that $L(\\xi)^{\\circ}\\subset L$.\nIt is sufficient to show that $W\\cap L(\\eta)\\ne\\emptyset$ for any nonempty open subset $W$ of $L(\\xi)$ and any $\\eta \\in M$.\n\nLet us take any $\\zeta\\in W$. Since $\\zeta$ is $D$-reachable from $\\xi$, there exist ${\\bf i}$ and~${\\bf t}$ such that \n\\begin{equation*}\n\\zeta = \\Phi_{{\\bf i}}({\\bf t},\\xi).\n\\end{equation*}\nThe mapping $h$ defined by $h(x)=\\Phi_{{\\bf i}}({\\bf t},x), x\\in M,$ is a diffeomorphism on $M$. Therefore, $h^{-1}(W)$ is an open neighborhood of $\\xi$. \nSince $\\xi \\in E\\subset \\overline {L(\\eta)}$, there is a point $\\rho \\in h^{-1}(W)$, $D$-reachable from $\\eta$. Consequently,\n$h(\\rho)\\in W$ is also $D$-reachable from $\\eta$.\n{{\\hfill $\\Box$ \\smallskip}}\n\nAs an immediate corollary of Lemma~\\ref{lem:non-empty-interior}, the set $L$ has nonempty interior.\n\\begin{lemma} \\label{lem:positive-measure}\nSuppose $\\mu$ is an invariant measure for $(\\mathsf{P}^t)$. If $G$ is a nonempty open subset of~$L$ and $j\\in\nS$, then $\\mu(G\\times\\{j\\})>0$.\n\\end{lemma} \n \n\\bpf Let us assume that $\\mu(G\\times\\{j\\})=0$. Since\n$\\mu$ is $(\\mathsf{P}^t)$-invariant, it is also $\\mathsf{Q}$-invariant, and we have\n\\begin{equation*}\n0 = \\mu(G\\times\\{j\\}) = \\sum_{i=1}^{k} \\int_M \\mathsf{Q}_{\\eta,i}(G\\times\\{j\\}) \\mu(d\\eta\\times\\{i\\}).\n\\end{equation*}\nFor all $i\\in S$ and $\\mu(\\cdot \\times\\{i\\})$-almost every $\\eta\\in M$, we thus obtain\n\\begin{equation}\n\\label{eq:Q-0}\n\\mathsf{Q}_{\\eta,i}(G\\times\\{j\\}) = 0.\n\\end{equation}\nLet us choose $\\eta$ such that~\\eqref{eq:Q-0} holds true.\n\n\n\n\n\nBy assumption, we have $G\\subset L\\subset\\overline{L(\\eta)}$. Since $G$ is open, $G \\cap L(\\eta)\\ne\n\\emptyset$. So there exist a sequence $\\mathbf{i}=(i,i_2,\\ldots,i_m,j)$ and an interswitching time vector\n$\\mathbf{t}=(t_1,\\ldots,t_m,t_{m+1})$ such that $\\Phi_{{\\bf i}}({\\bf t}, \\eta)\\in G$. By continuity of $\\Phi_{\\bf i}$ there\nis a neighborhood $W$ of $\\mathbf{t}$ in $\\mathbb{R}^{m+1}_+$\nsuch that $\\Phi_{{\\bf i}}(\\mathbf{s}, \\eta)\\in G$ for all $\\mathbf{s}\\in W$. Denoting $s=s_1+\\ldots+s_{m+1}$ and using the\nrepresentation of $\\mathsf{P}^s_{\\eta,i}(\\cdot|C_{\\mathbf{i}})$ via exponentially distributed times\nthat we used in the proof of Theorem~\\ref{thm:AC-component-1}, we conclude that $\\mathsf{P}^s_{\\eta,i}(G \\times \\{j\\})>0$ for \n$s$ sufficiently close to $t=t_1+\\ldots+t_{m+1}$. Therefore, $\\mathsf{Q}_{\\eta,i}(G \\times \\{j\\})>0$ contradicting \\eqref{eq:Q-0}.\n{{\\hfill $\\Box$ \\smallskip}}\n\n\\bigskip\n\n\\bpf[Proof of Theorem~\\ref{thm:uniqueness}]\nAny invariant measure can be written as a convex combination of ergodic invariant measures. Therefore, it suffices to\nshow absolute continuity and uniqueness of an ergodic invariant measure. \n\nLet us begin by deriving absolute continuity. If $\\mu$ is an\nergodic invariant measure that satisfies the assumptions of Theorem~\\ref{thm:uniqueness} then, \ndue to Theorem~\\ref{thm:absolute-continuity-for-ergodic}, it suffices to show that\n$L \\subset \\mathop{\\rm supp} \\mu$. Let $\\xi \\in L$, and let $U$ be a neighborhood of $\\xi$ in $M$, $j \\in S$. By\nLemma~\\ref{lem:positive-measure}, we have $\\mu(U \\times \\{j\\})>0$, hence $\\xi \\in \\mathop{\\rm supp} \\mu$. \n\nIn order to prove uniqueness of the ergodic invariant measure, let us assume that $\\mu_1$ and $\\mu_2$ are two distinct ergodic invariant probability measures. Birkhoff's ergodic\ntheorem then implies that $\\mu_1$ and $\\mu_2$ are mutually singular. Hence, the set $M \\times S$ can be partitioned into\ntwo disjoint subsets $H_1$ and $H_2$ with\n$\\mu_1(H_2) = \\mu_2(H_1) = 0.$\nThe two sets can be represented as \n\\[H_\\alpha=\\bigcup_{j=1}^{k} M_{\\alpha,j} \\times \\{j\\},\\quad \\alpha=1,2,\\]\nfor some measurable sets $M_{\\alpha,j}$, $j\\in S$, $\\alpha=1,2$. For all $\\alpha$ and $j$,\n\\begin{equation*}\n\\mu_\\alpha( M_{\\alpha,j} \\times \\{j\\})=\\mu_\\alpha( M \\times \\{j\\})>0,\n\\end{equation*}\nsince the left side is a stationary distribution for the Markov chain on $S$ and by our assumptions, transitions\nbetween all states happen with positive probability.\n\nFix a $j$ in $S$. By virtue of Lemma~\\ref{lem:non-empty-interior}, the set $E^{\\circ}$ is nonempty.\nAccording to Lemma~\\ref{lem:positive-measure}, for all $j \\in S$ we have $\\mu_1(E^\\circ\\times\\{j\\})>0$.\n Since $\\mu_1(M_{2,j}\\times\\{j\\}) =0$, we deduce that $\\mu_1(E_1\\times\\{j\\}) > 0$, where $E_1=E^\\circ \\cap\nM_{1,j}$.\n\nThe measure $\\mu_1$ is $(\\mathsf{P}^t)$-invariant, hence it is also $\\mathsf{Q}$-invariant, and we have\n\\begin{equation}\n\\label{eq:invariance-for-M_2_j}\n0=\\mu_1(M_{2,j}\\times\\{j\\}) \\geq \\int_{E_1} \\mathsf{Q}_{\\eta,j}(M_{2,j}\\times \\{j\\}) \\mu_1(d\\eta\\times\\{j\\}).\n\\end{equation}\nSince $\\mu_1(E_1\\times\\{j\\})>0$, it suffices to show that $\\mathsf{Q}_{\\eta,j}(M_{2,j}\\times\\{j\\})>0$ for all $\\eta \\in\nE_1$, to obtain a contradiction with~\\eqref{eq:invariance-for-M_2_j}.\n\nSince $\\eta$ satisfies Condition~B,\nTheorem~\\ref{thm:Jurdjevic-2} guarantees that there exist an integer $m>n$ and a vector ${\\bf i} =\n(j,i_2,\\ldots,i_m,j)$ such\nthat the function $f:\\mathbb{R}^{m+1}_+ \\to M$ defined by\n\\begin{equation*}\n f(\\bf t) =\\Phi_{\\bf i}({\\bf t},\\eta)\n\\label{eq:F_ibf}\n\\end{equation*}\nhas an open set $O$ of regular points such that for all $t>0$,\n\\begin{equation*}\n\\{\\mathbf{t}=(t_1,\\ldots,t_{m+1})\\in O:\\ t_1+\\ldots+t_{m+1}0.\n\\label{eq:0_is_limiting_point}\n\\end{equation}\n\n\nUsing the representation of $\\mathsf{Q}$ via~\\eqref{eq:Qq-via-I_t} and the family of exponentially distributed times\n$T_1,\\ldots,T_{m+1},T$, we obtain that it is sufficient to prove that\n\\begin{equation}\n\\mathsf{P}\\left\\{F(T_1,\\ldots,T_{m+1},T) \\in M_{2,j}|\\ R \\right\\} > 0,\n\\label{eq:sufficient_for_contradiction}\n\\end{equation}\nwhere $R$ was introduced in~\\eqref{eq:event_R_2}.\n\n\n Since $E^\\circ$ is an open set containing\n$\\eta$, and $F(V)$ is an open set such that $\\eta\\in \\overline{F(V)}$ (due \nto~\\eqref{eq:0_is_limiting_point} and continuity of $F$ at $0$), we obtain that\n$G= E^{\\circ} \\cap F(V)$ is also a nonempty open set. \n\n\n\nLet us choose a vector ${\\bf r}\\in V$ such that $F({\\bf r}) \\in E^{\\circ}$. Since $\\mathbf{r}$ is a regular point for\n$F$, we see that \nfor an arbitrary choice of local smooth coordinates around $\\mathbf{r}$,\nthere are $n$ independent columns of the matrix $DF(\\mathbf{s})$ for $\\mathbf{s}$\nin a small neighborhood of $\\mathbf{r}$. Without loss of generality we\ncan assume that these are the first $n$ columns. \nThen the map $\\rho:\\mathbb{R}^{m+2}\\to M\\times\\mathbb{R}^{m+2-n}$ defined by\n\\[\n \\rho(s_1,\\ldots,s_{m+1},s)=(F(s_1,\\ldots,s_{m+1},s),s_{n+1},\\ldots ,s_{m+1},s)\n\\]\nhas nonzero Jacobian in that neighborhood. So we can choose an open set $W_V$ containing~$\\mathbf{r}$ so that $\\rho$ is a\ndiffeomorphism between $W_V$ and \n$W_G\\times W_{n-m-2}$, where $W_G\\subset G$ and $W_{m+2-n}\\subset \\mathbb{R}_{+}^{m+2-n}$ are some open sets.\n\n\n\nThe set $W_G$ is an open subset of $L$. It is also not empty since it contains~$F({\\bf r})$.\nLemma~\\ref{lem:positive-measure} implies that $\\mu_2( W_G\\times\\{j\\})>0$. Since\n$\\mu_2(M_{2,j}^c\\times\\{j\\})=0$, we conclude that\n$\\mu_2(J\\times\\{j\\})>0$ where $J=M_{2,j} \\cap W_G.$ Since $\\mu_2$ is an ergodic measure, it is absolutely continuous, so\n\\begin{equation}\n\\label{eq:J-Lebesgue-positive}\n\\lambda^M(J)>0. \n\\end{equation}\n\n\n\nSince $J\\subset M_{2,j}$, the desired inequality \\eqref{eq:sufficient_for_contradiction} will follow from\n\\begin{equation}\n\\mathsf{P}\\left\\{F(T_1,\\ldots,T_{m+1},T) \\in J|\\ R \\right\\} > 0.\n\\label{eq:sufficient_for_contradiction-2}\n\\end{equation}\n\n\nSince the joint distribution of $T_1,\\ldots,T_m,T_{m+1},T$ is equivalent to the Lebesgue measure on $\\Delta$, \nLemma~\\ref{lem:pushforward_for_nonzero_det} implies that $\\rho(T_1,\\ldots,T_{m+1},T)$ has positive density\nalmost everywhere in $W_G\\times W_{m+2-n}$. Integrating over $W_{m+2-n}$, we see that \\\\ $F(T_1,\\ldots,T_{m+1},T)$ has\npositive density almost everywhere in $W_G$. Now~\\eqref{eq:sufficient_for_contradiction-2} follows from \n\\eqref{eq:J-Lebesgue-positive}.{{\\hfill $\\Box$ \\smallskip}}\n\n\n\n\n\n\n\n\n\n\n\\bigskip \n\n\n\nOf course, Theorem~\\ref{thm:uniqueness} remains true if one replaces Condition~B by the stronger Condition~A.\nHowever, under that condition one can prove this result without referring to the resolvent $\\mathsf{Q}$. Namely, one can\nuse the regularity of transition probabilities established in Theorem~\\ref{thm:AC-component-1} (which is stronger than\nthe regularity established in Theorem~\\ref{thm:AC-component-2}),\nand invoke Theorems~\\ref{thm:Jurdjevic-1} and~\\ref{thm:absolute-continuity-for-ergodic-under-condition-A} instead of\nTheorems~\\ref{thm:Jurdjevic-2} and~\\ref{thm:absolute-continuity-for-ergodic}.\n\n\n\n\n\\section{Examples} \\label{sec:examples}\n\nIn this section, we apply Theorem~\\ref{thm:uniqueness} to two concrete switching systems. In the first example, we have a closer look at the system on the $n$-dimensional torus $\\mathbb{T}^n=\\mathbb{R}^n\/\\mathbb{Z}^n$ that was introduced in Section~\\ref{sec:absolute-continuity-invariant}. In the second example, we switch between two Lorenz vector fields with different parameter sets. For both systems, uniqueness of the invariant measure is derived from Theorem~\\ref{thm:uniqueness}. For the system on $\\mathbb{T}^n$, we point out the invariant measure explicitly.\n\n\\bigskip\n\nLet $M$ be the $n$-dimensional torus $\\mathbb{T}^n$, and let $D=\\{u_1,\\ldots,u_n\\}$ be the standard basis of $\\mathbb{R}^{n}$.\n We assume for simplicity that the parameter $\\lambda$ of the exponential time between any two switches is independent\nof the current state, and that we have a uniform probability of switching between any two states. In\nSection~\\ref{sec:absolute-continuity-invariant} we implicitly argued that Condition~A does not hold for this system: If\nCondition~A was satisfied at some point $\\xi \\in \\mathbb{T}^n$, the transition probability measures $\\mathsf{P}_{\\xi,i}^t$ would not be\nsingular with respect to Lebesgue measure, according to Theorem~\\ref{thm:AC-component-1}. However, as pointed out in\nSection~\\ref{sec:absolute-continuity-invariant}, the measures $\\mathsf{P}_{\\xi,i}^t$ are purely singular. \n\nIt is also instructive to show directly why Condition~A does not hold. As all the vector fields in $D$ are constant, the derived algebra $\\mathcal{I}'(D)$ contains only the zero vector field. Thus, for any $\\xi \\in \\mathbb{T}^n$,\n\\begin{equation*}\n \\mathcal{I}_{0}(D)(\\xi) = \\biggl\\{\\sum_{i=1}^{n} \\lambda_{i} u_{i}:\\ \\sum_{i=1}^{n} \\lambda_{i} = 0\\biggr\\}. \n\\end{equation*}\nDue to the constraint $\\sum_{i=1}^{n} \\lambda_{i} = 0$, the algebra $\\mathcal{I}_{0}(D)(\\xi)$ does not have full\ndimension, so Condition~A is violated at every point in $\\mathbb{T}^n$.\n\nOn the other hand, Condition~B is clearly satisfied at any point $\\xi \\in \\mathbb{T}^n$, as the standard basis of $\\mathbb{R}^n$ applied to $\\xi$ yields a full-dimensional set of vectors in the tangent space. Also note that any point in $\\mathbb{T}^n$ is $D$-reachable from any other point. Therefore, Theorem~\\ref{thm:uniqueness} guarantees that the associated Markov semigroup has a unique invariant measure, provided that such a measure exists. In this elementary example, it is possible to point out the invariant measure explicitly. For Borel sets $E \\subset \\mathbb{T}^n$ and states $i \\in S$, it is given by\n\\begin{equation*}\n \\mu(E \\times \\{i\\}) = \\frac{1}{n} \\cdot \\lambda(E).\n\\end{equation*}\nHere, $\\lambda$ denotes Lebesgue measure on $\\mathbb{T}^n$. \n\n\\bigskip\n\nThe second example provides a situation where (i) the number of vector fields in~$D$ is less than the\ndimension of the manifold $M$, and (ii) each individual vector field in~$D$ gives rise to dynamics with a\nstrange attractor and no absolutely continuous invariant measures, but (iii) the switched system has a unique invariant\nmeasure and it is absolutely continuous.\n\n\nNamely, we consider switching between two Lorenz vector fields with different parameter values. A Lorenz\nvector field is a vector field defined in $\\mathbb{R}^3$, of the form\n\\begin{equation*}\n u(x,y,z) = \\begin{pmatrix}\n \\sigma \\cdot (y-x) \\\\\n rx-y-xz \\\\\n xy-bz\n \\end{pmatrix},\n\\end{equation*}\n where $\\sigma$, $r$ and $b$ are physical parameters. Let the set $D$ contain exactly two Lorenz vector fields $u_1$ and\n$u_2$ such that $u_1$ has Rayleigh number $r=28$ and $u_2$ has a Rayleigh number different from, but close to, $28$. We\nassume for both vector fields that $\\sigma=10$ and that $b=\\tfrac{8}{3}$, which is the classical parameter choice for\nthe Lorenz system. In~\\cite{Tucker:MR1701385}, Tucker shows that the Lorenz system with parameters $\\sigma=10$, $r=28$\nand $b=\\tfrac{8}{3}$, corresponding to vector field $u_1$, admits a robust strange attractor $\\Lambda$ as well as a\nunique SRB-measure supported on $\\Lambda$. Robustness implies that the dynamical structure of the system remains intact\nunder small parameter changes, so the dynamics induced by $u_2$ share these features if $r$ is sufficiently close to\n$28$. Moreover, the SRB-measure on $\\Lambda$ satisfies a dissipative ergodic theorem, see\ne.g.~\\cite[Section~5.1]{Bunimovich-Sinai:MR755521}. It follows that any point $\\xi\\in\\Lambda$ is\n$\\{u_1\\}$-approachable (and thus $D$-approachable) from every point in a set $S_\\xi\\subset\\mathbb{R}^3$ with zero Lebesgue\nmeasure complement.\n\nAssisted by a computer algebra system, we checked that Condition~A is satisfied for this system at any point in $\\mathbb{R}^3$\nthat does not lie on the $z$-axis. Since the $z$-axis is invariant under the flows of both vector fields, we disregard\nit and set $M$ to be $\\mathbb{R}^3$ without points on the $z$-axis. With this provision, every point on the attractor $\\Lambda$\nis $D$-approachable from any point in $M$: \n\nConsider a point $\\xi \\in \\Lambda$ and a point $\\eta \\in M$. By Theorem~\\ref{thm:Jurdjevic-1}, there is a nonempty open\nset of $D$-reachable points from $\\eta$ (recall that Condition~A holds at any point in $M$). And since this open set has\npositive Lebesgue measure, it contains a point belonging to~$S_\\xi$.\nHence, $\\xi$ is $D$-approachable from $\\eta$. As in the first example, uniqueness and absolute continuity of an invariant measure\nfollow from Theorem~\\ref{thm:uniqueness}. \n \n \n\n\\bibliographystyle{alpha}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}