diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzgqth" "b/data_all_eng_slimpj/shuffled/split2/finalzzgqth" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzgqth" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\nHigh-intensity and monochromatic X-ray sources are important tools for research in fundamental to applied science. Nowadays, intense and monochromatic soft X-ray beams are produced by means of Free-Electron Lasers (FELs). FELs are based on magnetic undulators \\cite{Ginzburg,Motz}, which force electrons to an oscillatory motion, resulting in electromagnetic radiation generation. With currently available magnetic undulators, the minimum achievable oscillation period $\\lambda_u$ is of the order of the centimeters, thus limiting the generation of X-ray to a few tens of keV at the highest synchrotron electron energies \\cite{1325024}.\n\nThe availability of harder X-rays or even $\\gamma$-ray source will pave the way to the development of innovative applications. For instance, a $\\gamma$-ray beam can induce nuclear reactions through photo-transmutation \\cite{nuclearph}, i.e., it can be employed for changing the atomic number of nuclei. This technique can be employed for eliminating nuclear waste by trasmuting it into short-lived nuclei for medicine as $^{126}$Sn$(\\gamma,$n$)^{125}$Sn \\cite{Irani2012466} or the $^{100}$Mo$(\\gamma,$n$)^{99}$Mo reaction followed by a $\\beta$-decay used to produce the $^{99m}$Tc isotope \\cite{national2009Medical}. Another possible application lies in the field of photo-induced nuclear fission to induce the fragmentation of heavy nuclei \\cite{PhysRev.56.426,PhysRev.59.57}. Supply of energy from the gamma-quanta causes the excitation of nuclei. Provided that the photon energy matches the transition energy between nuclear states, the reaction acquires a resonant character. As a possible result, heavy nuclei are splitted into two or more fragments. As an example of application, this process can be used in the production of medium mass neutron-rich nuclei.\n\nIn order to produce photon with the suitable energies for the applications, undulators with a shorter ${\\lambda}_u$ than currently available FELs are needed. First efforts in this direction date back to the mid 80s \\cite{PhysRevSTAB.15.070703}. A design for mm-scale pulsed electromagnetic undulator was proposed in Ref. \\cite{Granatstein}, a mm-scale undulating field produced using periodic grooves ground into samarium cobalt blocks was proposed and obtained \\cite{Ramian,Paulson}, and hybrid-bias-permanent-magnet undulators with period lengths in the range of ${\\lambda}_u$ = $700 - 800$ $\\mu$m were fabricated \\cite{Tatchyn1,Tatchyn2,Tatchyn3}.By exploiting the recent progress in the fabrication of MicroElectroMechanical Systems (MEMS), the realization of undulators with ${\\lambda}_u$ in the 10 - 1000 $\\mu$m range has been proposed \\cite{PhysRevSTAB.15.070703}. Drawbacks of the MEMS undulators are the demand for an expensive cooling system and the strict requirements on beam quality.\n\t\nA promising solution for reaching higher photon energies is the usage of a Crystalline Undulator (CU) \\cite{Baryshevsky198061,Baryshevsky2,Baryshevsky201330}. In a CU, the electrons (positrons) are forced to an oscillatory motion by the strong electrostatic field generated by the aligned atoms of a crystal planes or axes. For instance, the Si (110) planes act on charged particles with an electric field of 6 GV\/cm, which is equivalent to a 2 kT magnetic field. Therefore, the trajectories of the particles entering a crystal at small angle with respect to the major crystallographic planes or axes, are confined \\cite{Dansk.Fys.34.14}. The quasi-oscillatory motion of the channeled particles is accompanied by channeling radiation.\n\nA crystal can tolerate a significant amount of stress and torsion while remaining into the elastic regime. Therefore, a complex geometry, such as the periodically bent crystal needed to drive the particles inside manipulated channels, can be achieved. By exploiting currently available techniques, crystals with an undulated geometry with a period ranging from 1 $\\mu$m to 1 cm can be fabricated.\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=1\\columnwidth]{schema.eps}\n\\caption{Schematic representation of the crystalline undulator obtained through the grooving method. The red curved line represents the undulated planes in the center of the crystal. $\\textit{A}$ is the amplitude of the undulated planes, $\\textit{a}$ the groove width and $\\textit{d}$ the groove depth.}\\label{fig:schema}\n\\end{center}\n\\end{figure}\n\n\\section{Fabrication}\n\\begin{figure}\n\\begin{center}\n\\subfloat[3D reconstruction of the sample surface obtained through interferometric profilometry]{\\includegraphics[width=0.8\\columnwidth]{onduinterESRFa.eps}}\n\\newline\n\\subfloat[Angular analysis of (220) planes through X-ray diffraction.]{\\includegraphics[width=0.8\\columnwidth]{onduinterESRFb.eps}}\n\\caption{Morphological and structural measurement of the sample.}\\label{fig:onduinterESRF}\n\\end{center}\n\\end{figure}\nIn order to manufacture a working CU, a crystal with very low density of defects should be used, together with high accuracy in bending the crystallographic planes. Indeed, the capture efficiency of channeling is strongly reduced by the presence of crystalline defects \\cite{PhysRevLett.110.175502}, and the coherence of the emitted X- and $\\gamma$-rays relies on the accuracy in the periodically bending of the crystallographic planes. Various methods have been proposed for the realization of a periodically bent crystal, such as acoustic-wave transmission \\cite{Baryshevsky198061}, periodically-graded $Si_{1-x}Ge_x$ structures \\cite{Breese1997540}, periodic-surface deformations obtainable via superficial grooves \\cite{Afonin2005122,PhysRevSTAB.7.023501}, laser ablation \\cite{Balling20092952}, or film deposition \\cite{Guidi200540,Guidi2007apl}.\n\nThe CU radiation can be distinguished from the channeling radiation only if ${\\lambda}_u$ is not equal to the average channeling oscillation period ${\\lambda}_c$. Traditional CUs work under the condition ${\\lambda}_u<{\\lambda}_c$, though an alternative scheme with ${\\lambda}_u>{\\lambda}_c$ was recently proposed \\cite{PhysRevLett.110.115503}.\n\nSeveral experiments aimed to demonstrating the feasibility of a working CU for the traditional configuration \\cite{PhysRevLett.90.034801,Afonin2005122,biryukovundu,Baranov200632,Baranov2005,Backe201337,1742-6596-438-1-012017} were performed, and some theoretical works were accomplished to foresee the energy generated by CUs \\cite{korol2004,PhysRevLett.98.164801}. Recently, a first evidence of the radiation generated by a sub-GeV electrons interacting with a CU having ${\\lambda}_u<{\\lambda}_c$ was observed in the 7-15 MeV range, using a 600-855 MeV electron beam \\cite{PhysRevLett.112.254801}.\n\nThe grooving method \\cite{PhysRevLett.90.034801} may be a simple and reproducible solution to fabricate a CU that emits radiation in the MeV range working within the traditional ${\\lambda}_u<{\\lambda}_c$ scheme.\n\nIn this paper, we present the first experimental evidence of planar channeling in a CU fabricated via superficial grooves.\n\n\n\nThe superficial grooving method consists of making a series of grooves on the major surfaces of a crystal. In the framework of the LAUE project \\cite{SPIE2013Dante}, such method has been deeply investigated and mastered to produce bent crystals tailored for the realization of optics for hard-X- and $\\gamma$-rays, i.e. Laue lenses \\cite{CamaQM,150Ge}. It was shown that a series of grooves may cause a permanent and reproducible deformation of the whole sample \\cite{indentazioni1,indentazioni2}. Indeed, the plasticization that occurs in the thin superficial layer transfers co-active forces to the crystal bulk, producing an elastic strain field within the crystal itself. Making an alternate pattern of parallel grooves on both surfaces of a crystal, the realization of a millimetric or even a sub-millimetric undulator could be envisaged.\n\nThe CU described in the present paper was realized by making a series of grooves on a 0.2$\\times$45.0$\\times$5.0 mm$^3$ silicon crystal. In particular, an alternate and periodic pattern of $150$ $\\mu$m-wide parallel grooves was plotted on the largest surfaces of the strip, with the distance between consecutive grooves being 1 mm. Grooves were carved by means of a diamond blade through the usage of a DISCO DAD 3220 dicing machine. The groove depth was ($54\\pm2$) $\\mu$m. A schematic representation of the sample is illustrated in Fig.\\ref{fig:schema}.\n\n\n\nThe grooving parameters used for the CU fabrication were employed as input parameters to compute the expected amplitude of the undulated planes. In the framework of the classical theory of elasticity, the Stoney formula can be extended to evaluate the curvature just beneath each groove and, in turn, the undulator amplitude \\cite{indentazioni2}. The undulator amplitude resulted to be $(4.75\\pm0.50)$ nm.\n\n\\section{Interferometric and diffractometric characterization}\n\n\n\nA morphological analysis of the sample surface was attained at the Sensor and Semiconductor Laboratory (SSL) of Ferrara (Italy) through a VEECO NT-1100 white-light interferometer, which allowed the reconstruction of the three-dimensional surfaces with lateral resolution of about 1 $\\mu$m and vertical resolution of 1 nm. The profilometric pattern of the sample surface (shown in Fig. \\ref{fig:onduinterESRF}b), allows the estimation of the average undulator amplitude, which resulted to be ($4.5\\pm1.0$) nm.\n\nThe CU was also tested through X-ray diffraction at the ID15A beamline of the European Synchrotron Radiation Facility (ESRF, Grenoble, France). A highly monochromatic beam was set at 150 keV, the beam size was 50$\\times$50$ \\mu$m$^2$. The sample was characterized by recording the position of the Bragg peak for the (220) lattice plane for 50 consecutive points along the [110] direction. In order to subtract a possible shift due to instrumental tolerances, a flat reference crystal with identical lattice orientation was set behind the undulator, and the (220) Bragg peak of the reference crystal was recorded together with the Bragg peak of the undulator. The shift of the angular position of the CU Bragg peak with respect to the reference as a function of the impact position of the X-ray beam is shown in Fig. \\ref{fig:onduinterESRF}. The measured average amplitude of undulation was ($4.1\\pm0.5$) nm with an average period of ($1.01\\pm0.03$) mm.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.8\\columnwidth]{h8exp.eps}\n\\caption{(a) Measured distribution of outgoing particles after the interaction with the undulator as a function of the incoming angle and the deflection angle with respect to the crystal plane. (b) The Monte Carlo simulation of the same distribution. (c) The Monte Carlo simulation of the same distribution convoluted with the experimental resolution.}\\label{fig:h8exp}\n\\end{center}\n\\end{figure}\n\\section{On-beam characterization}\n\nThe CU was exposed to a 400 GeV\/c proton beam in the H8 beam line at CERN-SPS. It was mounted on a two-axis rotational stage with 2 $\\mu$rad resolution. The beam was tracked before and after the interaction with the CU by using a telescope system of Si strip detectors \\cite{Celano199649,PhysRevLett.101.234801}. The angular resolution of the tracker is $\\sim3.5$ $\\mu$rad \\cite{HasanPhDThesis}. The beam size was ($1.36\\pm0.02$) mm $\\times$ ($0.73\\pm0.01$) mm and the angular divergence ($10.15\\pm0.04$) $\\mu$rad $\\times$ ($8.00\\pm0.03$) $\\mu$rad, as measured with the telescope system.\n\nRemaining in the \\emph{xz} plane, the \\emph{z} axis of the CU was purposely misaligned with respect to the beam direction to avoid the undesired axial-channeling effect. The beam enters the crystal through a (110) oriented surface with an area of 0.2$\\times$45.0 mm$^2$. The (111) planes were exploited as channeling planes.\n\nThe distribution of the deflection angle of the particles as a function of the horizontal incoming angle was measured by rotating the goniometer around the channeling position (see Fig. \\ref{fig:h8exp}). Particles channeled along the whole crystal length acquire a null deflection angle. On the contrary, if a channeled particle undergoes dechanneling, the trajectory remains parallel to the tangent of the local plane curvature where dechanneling has occurred. As a consequence, particles may acquire either a positive or negative deflection angle. By comparing the experimental pattern with the Si (110) angular scan from the literature (see Figs. 4, 6, and 7 of \\cite{PhysRevSTAB.11.063501}), a noticeable difference holds. In fact, the particles channeled in a uniformly bent crystal are deflected only to one among the two possible (positive or negative) directions.\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=0.8\\textwidth]{h8MC.eps}\n\\caption{Simulated distribution of particles during the interaction with the undulator as a function of the incoming angle and the deflection angle with respect to the crystal plane at four different penetration depth, i.e., 250 $\\mu$m (a), 500 $\\mu$m (b), 750 $\\mu$m (c) and 1000 $\\mu$m (d).}\\label{fig:h8MC}\n\\end{center}\n\\end{figure*}\n\n\\section{Monte Carlo simulations}\n\nIn order to get an insight in the experimental result, a Monte Carlo simulation was performed through the DYNECHARM++ toolkit \\cite{Bagli2013124,DYNECHARM_PHI}. The code solved the equation of motion in the non-inertial reference frame orthogonal to the crystal plane via numerical integration under the continuum potential approximation \\cite{Dansk.Fys.34.14}. The electrical characteristics of the crystals were evaluated through the ECHARM software \\cite{PhysRevE.81.026708}, which made use of diffraction data for the computation of the atomic form factors. The CU geometry was approximated through a sinusoidal function, with the amplitude being measured at ESRF. The oscillation period is taken equal to the groove distance, i.e. 1 mm. The influence of such geometry on particle trajectory was evaluated through the application of a position-dependent centrifugal force. Intensity of scattering on nuclei and electrons was averaged over the distribution density in the channel in order to reliably estimate the dechanneling probability, i.e., the probability for a channeled particle to leave the channeling condition.\n\nFigure \\ref{fig:h8MC} shows the distribution of the horizontal deflection angle as a function of the horizontal incoming angle at four different penetration depths. As shown in Fig. \\ref{fig:h8MC}a, at the penetration depth of 250 $\\mu$m, the clear typical markers of a bent crystal are present, namely the channeling spot and the volume reflected particles. In fact, channeled particles can be distinguished due to the $+20$ $\\mu$rad deflection kick they received. On the other side, the particles that impinge the target with a direction tangent to the crystalline curved planes are deflected to the opposite direction, i.e., they undergo volume reflection (VR). In Fig. \\ref{fig:h8MC}b, at the penetration depth of 500 $\\mu$m, the channeled particles are deflected to the opposite angle and acquire a $-20$ $\\mu$rad kick. Particles that underwent the VR effect are reflected by the opposite curvature. Thus, a null average deflection is observed. Particles pertaining to the region at the left of the channeling spot are also reflected. Then, Fig. \\ref{fig:h8MC}c shows a shift of the channeling spot of $-35$ $\\mu$rad at the 750 $\\mu$m depth. Lastly, in Fig. \\ref{fig:h8MC}d, at the 1000 $\\mu$m depth, the general picture evolves towards a double fold distribution with a net overall null average deflection angle. The simulated pattern of the beam interacting with the whole CU is shown in Fig. \\ref{fig:h8exp}b. Figure \\ref{fig:h8exp}c shows the simulation convoluted with the detector angular resolution. The simulation is in good agreement with the experimental results.\n\nWith the same Monte Carlo code, the fraction of planar-channeled particles along the crystal length, i.e., the channeling efficiency, can be calculated. For a 400 GeV\/c proton beam entering the CU parallel to the crystalline planes the efficiency is $(48 \\pm 1) \\%$. Such quantity can also be estimated with an analytical model usually adopted for uniformly-bent crystals \\cite{doi.org\/10.1140\/epjc\/s10052-014-2740-7}. By setting the bending radius equal to the $6.2$ m minimum curvature of the planes in the CU, the efficiency results to be $\\sim$ 60$\\%$ for a zero-divergence beam.\n\nThe chosen CU sample parameters fulfill the condition for an optimal undulator in the case of 15 GeV positrons. At these energies, with this CU, the possibility to use an electron beam instead of a positron beam has to be excluded because the dechanneling length is much smaller than the CU length \\cite{Scandale201370}. Thereby, only positrons can be used for such a test. In the next future, a positron beam in the energy range of 10-20 GeV will be available at FACET-SLAC. As done for the 400 GeV\/c proton beam, the expected efficiency for 15 GeV\/c positrons is higher than $40 \\%$, so that the CU described in this paper could be applied straight in that facility.\n\nA prediction regarding the electromagnetic radiation emitted by a 15 GeV positron beam passing through the CU sample was done through a specific Monte Carlo code \\cite{Baryshevsky201330}. Since the undulator radiation is expected in the $1$ MeV region, the Ter-Mikaelian density effect and the transition radiation were also taken into account. The beam divergence was set to zero. The radiation emission probability was averaged over a large number of particle trajectories and photon emission angles.\n\nThe distribution of the radiation emission probability generated by 15 GeV positrons interacting with the CU sample is shown in Fig. \\ref{fig:Rad}. The emission spectrum has a sharp peak at $\\sim$ 0.8 MeV. Because the channeling radiation (CR) is expected in the 100 MeV region \\cite{BaierKatkov}, the main CR and CU peaks present no overlap. In the energy range from 0.5 to 1.5 MeV the peak of the CU radiation is $\\sim$4 times higher than for CR.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=1\\columnwidth]{Rad2.eps}\n\\caption{Simulated radiation emission probability for 15 GeV positrons interacting with the CU sample presented in the paper (solid line) and with a straight Si crystal having the same length along the beam under channeling condition (dashed line). The simulated spectrum emitted by 15 GeV positrons interacting with an amorphous Si target of the same length (dotted curve) is also reported.}\\label{fig:Rad}\n\\end{center}\n\\end{figure}\n\n\\section{Conclusions}\n\nA crystalline undulator to be worked out under the condition ${\\lambda}_u<{\\lambda}_c$ was manufactured through the grooving method at SSL (Ferrara, It). The sample was extensively characterized both morphologically and structurally. The surfaces were reconstructed through an interferometric measurement at SSL. The crystalline structure was analyzed through monochromatic hard X-rays diffraction at ESRF (Grenoble, Fr). The crystal was tested at the H8-SPS line of CERN. The expected planar channeling of 400 GeV\/c protons between the cristalline planes bent with the grooving method was observed. Monte Carlo simulations were worked out to evaluate the channeling efficiency for the 400 GeV proton beam and to predict the radiation spectrum emitted by 15 GeV positrons beam entering the crystalline undulator. The manufactured crystalline undulator is now ready to be tested with high intensity positrons beam at 10-20 GeV energy.\n\n\\section{Acknowledgements}\nWe are thankful to the CERN-SPS and ESRF coordination board for the possibility to use the H8-SPS external line at CERN and the ID15A line at ESRF. We acknowledge the partial support of INFN under the ICE-RAD experiment and the European CUTE Project. We acknowledge Gerald Klug and Eugen Eurich from Disco Europe (Munich, Germany) for their support in crystal manufacturing, Persiani Andrea and Manfredi Claudio of Perman (Loiano, Italy) for their support with crystal holders manufacturing.\n\n\\bibliographystyle{unsrt}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Spin-3\/2 matrices}\\label{AppMat}\nThe spin-3\/2 matrices in their standard matrix representation read\n\\begin{align}\n \\label{mat1} J_x &= \\begin{pmatrix} 0 & \\frac{\\sqrt{3}}{2} & 0 & 0 \\\\ \\frac{\\sqrt{3}}{2} & 0 & 1 & 0 \\\\ 0 & 1 & 0 & \\frac{\\sqrt{3}}{2} \\\\ 0 & 0 & \\frac{\\sqrt{3}}{2} & 0 \\end{pmatrix},\\\\\n \\label{mat2} J_y &=\\begin{pmatrix} 0 & -\\rmi \\frac{\\sqrt{3}}{2} & 0 & 0 \\\\ \\rmi \\frac{\\sqrt{3}}{2} & 0 & -\\rmi &0 \\\\ 0 & \\rmi & 0 & -\\rmi \\frac{\\sqrt{3}}{2} \\\\ 0 & 0 & \\rmi \\frac{\\sqrt{3}}{2} & 0\\end{pmatrix},\\\\\n \\label{mat3} J_z &= \\begin{pmatrix} \\frac{3}{2} & 0 & 0 & 0 \\\\ 0 & \\frac{1}{2} & 0 & 0 \\\\ 0 & 0 & -\\frac{1}{2} & 0 \\\\ 0 & 0 & 0 & -\\frac{3}{2}\\end{pmatrix}.\n\\end{align}\nThe matrices satisfy $[J_i,J_j]=\\rmi \\vare_{ijk}J_k$ and $\\sum_i J_i^2 = \\frac{15}{4} \\mathbb{1}_4$, and all results obtained in the main text result from these relations. Some insight into the operators that appear in the analysis can be gained from applying the basis change\n\\begin{align}\n \\label{mat4} S = \\begin{pmatrix} 1 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 1 \\\\ 0 & 0 & 1 & 0 \\\\ 0 & 1 & 0 & 0 \\end{pmatrix}\n\\end{align}\nwith $S^{-1}=S ^T=S$. The spin-3\/2 matrices in this frame (denoted with an overbar) read\n\\begin{align}\n \\label{mat5} \\bar{J}_x &= SJ_x S = \\begin{pmatrix} 0 & 0 & 0 &\\frac{\\sqrt{3}}{2} \\\\ 0 & 0 & \\frac{\\sqrt{3}}{2} & 0 \\\\ 0 & \\frac{\\sqrt{3}}{2} & 0 & 1 \\\\ \\frac{\\sqrt{3}}{2} & 0 & 1 & 0 \\end{pmatrix},\\\\\n \\label{mat6} \\bar{J}_y &= SJ_y S = \\begin{pmatrix} 0 & 0 & 0 &-\\frac{\\sqrt{3}}{2}\\rmi \\\\ 0 & 0 & \\frac{\\sqrt{3}}{2}\\rmi & 0 \\\\ 0 & -\\frac{\\sqrt{3}}{2}\\rmi & 0 & \\rmi \\\\ \\frac{\\sqrt{3}}{2}\\rmi & 0 & -\\rmi & 0 \\end{pmatrix},\\\\\n \\label{mat7} \\bar{J}_z &= S J_z S = \\begin{pmatrix} \\frac{3}{2} & 0&0 & 0\\\\ 0& -\\frac{3}{2} &0 &0\\\\ 0& 0& -\\frac{1}{2} &0 \\\\ 0& 0& 0& \\frac{1}{2} \\end{pmatrix}.\n\\end{align}\nDefining the matrices $\\bar{V}_i$ and $\\bar{\\gamma}$ as in the main text with $J_i\\to \\bar{J}_i$ we find\n\\begin{align}\n \\label{mat8} \\bar{V}_1 =\\mathbb{1}_2\\otimes \\sigma_1,\\ \\bar{V}_2 = -\\mathbb{1}_2\\otimes \\sigma_2,\\ \\bar{V}_3=\\mathbb{1}_2\\otimes \\sigma_3,\n\\end{align}\nor \n\\begin{align}\n\\label{mat9} \\bar{V}_i=\\mathbb{1}\\otimes \\sigma_i^*,\n\\end{align}\nThis clearly shows that the representation of the Clifford algebra that specifies the Hamiltonian for $\\alpha=0$ is of the ``first type''. Furthermore, the matrix $\\bar{\\gamma}_{45}$ that enters the time-reversal operator $\\bar{\\mathcal{T}}=\\bar{\\gamma}_{45}\\mathcal{K}$ reads\n\\begin{align}\n \\label{mat10} \\bar{\\gamma}_{45} = \\mathbb{1}_2 \\otimes \\sigma_2,\n\\end{align}\nwhereas we have\n\\begin{align}\n \\label{mat11} \\bar{\\mathcal{W}} = \\bar{\\gamma}_{12} = \\sigma_2\\otimes \\mathbb{1}_2.\n\\end{align}\n\n\n\n\n\n\n\\section{Effective Weyl Hamiltonian and monopole charge}\nWe first construct the effective $2\\times 2$ Weyl Hamiltonian at the nodes $\\textbf{p}_n$, $n=1,\\dots,4$, in the chiral topological semimetal phase with $\\alpha>0$. The two orthogonal zero modes of $H_{\\rm mf}(\\textbf{p}_1)$ read\n\\begin{align}\n \\label{weyl1} |0_1\\rangle = \\frac{1}{\\sqrt{6}}\\begin{pmatrix}\\rmi\\sqrt{3} \\\\ 1-\\rmi \\\\ 1 \\\\ 0 \\end{pmatrix},\\ |0_1'\\rangle=\\mathcal{T}|0_1\\rangle = \\frac{1}{\\sqrt{6}}\\begin{pmatrix} 0 \\\\ \\rmi \\\\ 1-\\rmi \\\\ \\sqrt{3} \\end{pmatrix},\n\\end{align}\nwith similar expressions for $|0_n\\rangle$ and $|0_n'\\rangle$. From this we construct the projected Hamiltonian for momenta $\\textbf{p}=\\textbf{p}_n+\\delta\\textbf{p}$ close to one of the nodes according to\n\\begin{align}\n\\nonumber H_0^{(n)} &= \\begin{pmatrix} \\langle 0_n|H_{\\rm mf}(\\textbf{p}_n+\\delta \\textbf{p})|0_n\\rangle & \\langle 0_n|H_{\\rm mf}(\\textbf{p}_n+\\delta \\textbf{p})|0_n'\\rangle \\\\ \\langle 0_n'|H_{\\rm mf}(\\textbf{p}_n+\\delta \\textbf{p})|0_n\\rangle & \\langle 0_n'|H_{\\rm mf}(\\textbf{p}_n+\\delta \\textbf{p})|0_n'\\rangle \\end{pmatrix}\\\\\n\\label{weyl2} &= \\begin{pmatrix} \\langle 0_n|H(\\delta\\textbf{p})|0_n\\rangle & \\langle 0_n|H(\\delta\\textbf{p})|0_n'\\rangle \\\\ \\langle 0_n'|H(\\delta\\textbf{p})|0_n\\rangle & \\langle 0_n'|H(\\delta\\textbf{p})|0_n'\\rangle \\end{pmatrix}.\n\\end{align}\nNote that the $n$-dependence only results from the $n$-dependence of $|0_n\\rangle$ and $|0'_n\\rangle$ due to the linearity of the Hamiltonian. We arrive at $H_0^{(n)}=v^{(n)}_{ij}\\delta p_i\\sigma_j$ with\n\\begin{align}\n \\nonumber v^{(1)} &=\\frac{1+2\\alpha}{3}\\begin{pmatrix} 1 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & 1 \\end{pmatrix} + \\frac{1-\\alpha}{3} \\begin{pmatrix} 0 & 1 & 1 \\\\ 1 & 0 & 1 \\\\ 1 & 1 & 0 \\end{pmatrix},\\\\\n \\nonumber v^{(2)} &=\\frac{1+2\\alpha}{3}\\begin{pmatrix} 1 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & 1 \\end{pmatrix} + \\frac{1-\\alpha}{3} \\begin{pmatrix} 0 & 1 & -1 \\\\ 1 & 0 & -1 \\\\ -1 & -1 & 0 \\end{pmatrix},\\\\\n \\nonumber v^{(3)} &=\\frac{1+2\\alpha}{3}\\begin{pmatrix} -1 & 0 & 0 \\\\ 0 & -1 & 0 \\\\ 0 & 0 & 1 \\end{pmatrix} + \\frac{1-\\alpha}{3} \\begin{pmatrix} 0 & 1 & 1 \\\\ 1 & 0 & -1 \\\\ -1 & 1 & 0 \\end{pmatrix},\\\\\n \\label{weyl3} v^{(4)} &=\\frac{1+2\\alpha}{3}\\begin{pmatrix} -1 & 0 & 0 \\\\ 0 & -1 & 0 \\\\ 0 & 0 & 1 \\end{pmatrix} + \\frac{1-\\alpha}{3} \\begin{pmatrix} 0 & 1 & -1 \\\\ 1 & 0 & 1 \\\\ 1 & -1 & 0 \\end{pmatrix}.\n\\end{align}\nWe have \n\\begin{align}\n\\mbox{det}(v^{(n)})=\\alpha^2\n\\end{align}\nfor all $n=1,\\dots,4$. The resulting monopole charge at the node $\\textbf{p}_n$ is $q_n = \\sgn[\\mbox{det}(v^{(n)})]=+1$ for $\\alpha>0$.\n\nIn the Weyl semimetal phase with $m\\neq 0$ we consider $H_V(\\textbf{p}_n)$ with $n=\\text{a},\\text{b}$. Since the x- and y-components of the nodal points vanish, we have a diagonal mean-field Hamiltonian at the nodes, namely\n\\begin{align}\n \\label{weyl5} H_V(\\textbf{p}_{\\text{a}}) &= \\frac{2m\\alpha}{1+\\alpha}\\begin{pmatrix} 0 & 0 & 0 & 0 \\\\ 0 & -1 & 0 & 0 \\\\ 0 & 0 & 1 & 0 \\\\ 0 & 0 & 0 & 0 \\end{pmatrix},\\\\ \\label{weyl6} H_V(\\textbf{p}_{\\text{b}}) &= \\frac{2m\\alpha}{1-\\alpha}\\begin{pmatrix} -1 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 1 \\end{pmatrix}.\n\\end{align}\nThe zero modes $|0_{n=\\text{a},\\text{b}}\\rangle, |0'_{n=\\text{a},\\text{b}}\\rangle=\\mathcal{T}|0_{n=\\text{a},\\text{b}}\\rangle$ immediately follow from this and we again define the projected Hamiltonian as in Eq. (\\ref{weyl2}). We find $H_0^{(n)}=\\tilde{v}^{(n)}_{ij}\\delta p_i\\sigma_j$ with\n\\begin{align}\n\\label{weyl7} \\tilde{v}^{(\\text{a})} &=\\frac{1}{2}\\begin{pmatrix} 0 & -(2-\\alpha) & 0 \\\\ -(2-\\alpha) & 0 & 0 \\\\ 0 & 0 & 2(1+\\alpha)\\end{pmatrix},\\\\\n \\label{weyl8} \\tilde{v}^{(\\text{b})} &= \\frac{1}{2}\\begin{pmatrix} 0 & 2+\\alpha & 0 \\\\ -(2+\\alpha) & 0 & 0 \\\\ 0 & 0 & -2(1-\\alpha)\\end{pmatrix}.\n\\end{align}\nWe have\n\\begin{align}\n \\label{weyl9} \\mbox{det}(\\tilde{v}^{(\\text{a})}) &= -\\frac{1}{4}(2-\\alpha)^2(1+\\alpha),\\\\\n \\label{weyl10} \\mbox{det}(\\tilde{v}^{(\\text{b})}) &= -\\frac{1}{4}(2+\\alpha)^2(1-\\alpha),\n\\end{align}\nso that the monopole charges are given by $q_{\\text{a}}=\\sgn[\\mbox{det}(\\tilde{v}^{(\\text{a})})]=-1$ and $q_{\\text{b}}=\\sgn[\\mbox{det}(\\tilde{v}^{(\\text{b})})]=\\sgn(\\alpha-1)$. \n\n\n\n\n\n\n\\section{Details of the renormalization group analysis}\n\n\n\\subsection{Perturbative propagator}\nIn order to determine the perturbative propagator $G_0(Q)$ we need to invert\n\\begin{align}\nG_0^{-1}(Q) = A= \\rmi q_0\\mathbb{1}_4 + q_i (V_i + \\alpha U_i)\n\\end{align}\nwith frequency $q_0$. For arbitrary $\\alpha$ this can be achieved with the help of the Cayley--Hamiltonian theorem which implies that the inverse of the $4\\times 4$ matrix $G_0^{-1}$ is given by\n\\begin{align}\n \\nonumber G_0(Q) ={}& \\frac{1}{\\mbox{det}(A)}\\Biggl[\\frac{1}{6}\\Bigl([\\mbox{tr} A]^3-3 (\\mbox{tr}A)\\mbox{tr}(A^2)+2\\mbox{tr}(A^3)\\Bigr)\\mathbb{1}_4\\\\\n &-\\frac{1}{2}\\Bigl([\\mbox{tr}A]^2-\\mbox{tr}(A^2)\\Bigr)A+(\\mbox{tr}A)A^2-A^3\\Biggr].\n\\end{align}\nThe electric charge enters the RG beta functions only through the combination \n\\begin{align}\ng_1'=g_1+\\frac{e^2}{2}.\n\\end{align}\nThis is related to the fact that $G_0(Q)$ satisfies\n\\begin{align}\n\\label{intGG} \\int_{q_0,\\Omega} G_0(Q)^2 =0,\n\\end{align}\nwhere $\\int_{q_0}$ and $\\int_{\\Omega}$ denote the frequency and angular integration, respectively. Indeed, given Eq. (\\ref{intGG}), the same reasoning as in Eqs. (A101)-(A104) of Ref \\cite{PhysRevB.95.075149} can be applied to show this feature.\n\n\n\n\n\n\n\n\n\n\\subsection{Short-range interactions}\\label{AppShort}\n\nThe RG flow of the couplings $\\bar{g}_i$ is determined by the same procedure as laid out for Luttinger semimetals in Appendix 5 of Ref. \\cite{PhysRevB.95.075149}. We confine our analysis to local point-like interaction terms. To incorporate the most general four-fermion interaction we write the interaction part of the Lagrangian as\n\\begin{align}\n\\label{model6b} L_{\\rm short}=\\sum_{A=1}^{16} \\bar{g}_A (\\psi^\\dagger \\Sigma^A\\psi)^2,\n\\end{align}\nwhere $\\Sigma^A$ constitutes an $\\mathbb{R}$-basis of Hermitean $4\\times 4$ matrices satisfying $\\mbox{tr}(\\Sigma^A\\Sigma^B)=4\\delta^{AB}$. The symmetry properties of $H$ dictate which of the 16 entries of $\\{\\Sigma^A\\}$ are independent under RG. \n\n\n\nIn the rotation invariant case (i.e. for $\\alpha=2$) we have\n\\begin{align}\n \\nonumber L_{\\rm short}^{(\\rm rot)} ={}& \\bar{g}_1(\\psi^\\dagger\\psi)^2 + \\bar{g}_2(\\psi\\gamma_a\\psi)^2 \\\\\n \\label{model7} &+ \\bar{g}_\\mathJ(\\psi^\\dagger \\mathJ_i \\psi)^2+\\bar{g}_W(\\psi^\\dagger W_\\mu \\psi)^2,\n\\end{align}\nwhere $\\mathJ_i$, $\\gamma_a$,\\ $W_\\mu$ are the three, five, and seven components of the irreducible $\\text{SO}(3)$-invariant first-, second-, third-rank tensors made from products of the $J_i$. They read\n\\begin{align}\n \\label{model8} \\mathJ_i &= \\frac{2}{\\sqrt{5}}J_i,\n\\end{align}\nand\n\\begin{align}\n \\label{model9} \\gamma_1 &= \\frac{1}{\\sqrt{3}}(J_x^2-J_y^2),\\ \\gamma_2 = J_z^2-\\frac{5}{4}\\mathbb{1}_4,\\\\\n \\label{model11} \\gamma_3 &= \\frac{1}{\\sqrt{3}}\\{J_x,J_z\\},\\ \\gamma_4 = \\frac{1}{\\sqrt{3}}\\{J_y,J_z\\},\\\\\n \\label{model13} \\gamma_5 &=\\frac{1}{\\sqrt{3}}\\{J_x,J_y\\},\n\\end{align}\nand\n\\begin{align}\n \\label{model14} W_1 &= \\frac{2\\sqrt{5}}{3}\\Bigl(J_{x}^3-\\frac{41}{20}J_{x}\\Bigr),\\\\\n \\label{model15} W_2 &= \\frac{2\\sqrt{5}}{3}\\Bigl(J_{y}^3-\\frac{41}{20}J_{y}\\Bigr),\\\\\n \\label{model16} W_3 &= \\frac{2\\sqrt{5}}{3}\\Bigl(J_{z}^3-\\frac{41}{20}J_{z}\\Bigr),\\\\\n \\label{model17} W_4 &= \\frac{1}{\\sqrt{3}}\\{J_x,(J_y^2-J_z^2)\\},\\\\\n \\label{model18} W_5 &= \\frac{1}{\\sqrt{3}}\\{J_y,(J_z^2-J_x^2)\\},\\\\\n \\label{model19} W_6 &= \\frac{1}{\\sqrt{3}} \\{J_z,(J_x^2-J_y^2)\\},\\\\\n \\label{model20} W_7 &= \\frac{2}{\\sqrt{3}}(J_xJ_yJ_z+J_zJ_yJ_x).\n\\end{align}\nNote that \n\\begin{align}\n\\mathcal{W}=W_7.\n\\end{align}\nThe matrices $\\gamma_a$ are chosen such that $\\gamma_{1,2,3}$ are real and $\\gamma_{4,5}$ are imaginary. For a very detailed discussion of the decomposition of interaction vertices in Eq. (\\ref{model7}), also in the cubic symmetric case, we refer to Ref. \\cite{PhysRevB.95.075149}, where an identical notation was used to study systems described by a $4\\times 4$ quadratic band touching Hamiltonian. Obviously the momentum dependence of $H$ does not affect the form of $L_{\\rm short}$ and so all observations made in the mentioned reference are valid here as well.\n\n\n\n\nFor $\\alpha\\neq 2$ the system is invariant under the rotational cubic group $\\text{O}$ only, and the irreducible tensors under $\\text{SO}(3)$ need to be subdivided into irreducible representations of $O$. We write\n\\begin{align}\n \\label{model21} \\vec{E}=\\begin{pmatrix} \\gamma_1 \\\\ \\gamma_2 \\end{pmatrix},\\ \\vec{T}= \\begin{pmatrix} \\gamma_3 \\\\ \\gamma_4 \\\\ \\gamma_5 \\end{pmatrix},\\ \\vec{W}=\\begin{pmatrix} W_1 \\\\ W_2 \\\\ W_3 \\end{pmatrix},\\ \\vec{W}' = \\begin{pmatrix} W_4 \\\\ W_5 \\\\ W_6\\end{pmatrix}.\n\\end{align}\nThe set $\\{\\mathbb{1},E_a,T_a,\\mathJ_i,W_i,W'_i,W_7\\}$ constitutes an appropriate basis of interactions in the cubic case. However, we are free to replace the elements $\\mathJ_i$ and $W_i$ by\n\\begin{align}\n \\label{model22} \\vec{V} &= \\frac{1}{\\sqrt{5}}(\\vec{\\mathcal{J}}+2\\vec{W}),\\\\\n \\label{model23} \\vec{U} &= \\frac{1}{\\sqrt{5}}(2\\vec{\\mathcal{J}}-\\vec{W}),\n\\end{align}\nwhich are the same matrices $\\vec{V}$ and $\\vec{U}$ as they appear in $H$. The set of matrices\n\\begin{align}\n \\label{model24} \\{\\Sigma^A\\} = \\{ \\mathbb{1},\\ E_a,\\ T_a,\\ V_i,\\ U_i,\\ W'_i, W_7\\}\n\\end{align}\nthen comprises a computationally advantageous orthogonal $\\mathbb{R}$-basis of Hermitean $4\\times 4$ matrices with $\\mbox{tr}(\\Sigma^A\\Sigma^B) = 4\\delta_{Ab}$. The seven elements in Eq. (\\ref{model24}) allow us to construct seven distinct insulating ordering channels, namely\n\\begin{align}\n L_1 &=\\bar{g}_1 (\\psi^\\dagger \\psi)^2,\\\\\n L_2 &=\\bar{g}_2 (\\psi^\\dagger\\vec{E}\\psi)^2,\\\\\n L_3 &=\\bar{g}_3 (\\psi^\\dagger\\vec{T}\\psi)^2,\\\\\n L_4&=\\bar{g}_4 (\\psi^\\dagger\\vec{V}\\psi)^2,\\\\\n L_5 &=\\bar{g}_5 (\\psi^\\dagger\\vec{U}\\psi)^2,\\\\\n L_6 &=\\bar{g}_6 (\\psi^\\dagger\\vec{W}^\\prime\\psi)^2,\\\\\n L_7 &=\\bar{g}_7 (\\psi^\\dagger W_7\\psi)^2.\n\\end{align}\nOnly three of these expressions are linearly independent, and we choose to parametrize the interaction in $L_{\\rm int}$ by $L_{1,2,3}$. The remaining four are related to these by the Fierz identities\n\\begin{align}\n L_4 &= -\\frac{3}{2}L_1-\\frac{3}{2}L_2+\\frac{1}{2}L_3,\\\\\n L_5 &= -\\frac{3}{2}L_1 -\\frac{1}{2}L_3,\\\\\n L_6 &= -\\frac{3}{2}L_1-\\frac{1}{2}L_3,\\\\\n L_7 &= -\\frac{1}{2}L_1 +\\frac{1}{2}L_2-\\frac{1}{2}L_3.\n\\end{align}\nNote that $L_5=L_6$. In the rotationally invariant case we have $\\bar{g}_2=\\bar{g}_3$. Each individual term $L_1,\\dots,L_7$ is invariant under $\\psi \\to \\mathcal{W}\\psi$ since we have the (anti)commutation relations\n\\begin{align}\n [\\mathcal{W},V_i] &= [\\mathcal{W},T_i] = 0,\\\\\n \\{\\mathcal{W},U_i\\} &=\\{\\mathcal{W},E_a\\} = \\{\\mathcal{W},W'_i\\}=0.\n\\end{align} \nConsequently, the invariance of $L_{\\rm short}$ under this transformation is independent of the choice of Fierz basis.\n\n\n\n\nWe may alternatively express $L_{\\rm short}$ in terms of the superconducting channels of the system. The number of such terms is identical to the number of Fierz-inequivalent insulating channels. We have\n\\begin{align}\n L_{\\rm short} = \\bar{g}_{\\rm s} L_{\\rm s} + \\bar{g}_{\\rm d,E}L_{\\rm d,E} + \\bar{g}_{\\rm d,T} L_{\\rm d,T}\n\\end{align}\nwith\n\\begin{align}\n L_{\\rm s} ={}& (\\psi^{\\dagger} \\gamma_{45}\\psi^*)(\\psi^{\\rm T}\\gamma_{45}\\psi)\\\\\n L_{\\rm d,E} &= \\sum_{a=1}^2(\\psi^{\\dagger} \\gamma_a\\gamma_{45}\\psi^*)(\\psi^{\\rm T}\\gamma_{45}\\gamma_a\\psi)\\\\\n L_{\\rm d,T} &=\\sum_{a=3}^5(\\psi^{\\dagger} \\gamma_a\\gamma_{45}\\psi^*)(\\psi^{\\rm T}\\gamma_{45}\\gamma_a\\psi),\n\\end{align} \nwith the linear relation \\cite{PhysRevB.93.205138,PhysRevB.95.075149}\n\\begin{align*}\n L_{\\rm s} &= \\frac{1}{4}(L_1+2L_2+3L_3),\\\\\n L_{\\rm d,E} &= \\frac{1}{4}(L_1-3L_3),\\\\\n L_{\\rm d,T} &=\\frac{1}{4}(L_1-2L_2-L_3).\n\\end{align*}\nIn the rotationally symmetric case ($\\alpha=2$) we have $g_{\\rm d,E}=g_{\\rm d,T}$, again reducing the number of independent couplings to two.\n\n\n\n\\subsection{Susceptibility exponents and admixture $\\kappa(\\alpha)$}\\label{AppSusc}\n\n\nThe susceptibility exponents determine which order parameter condenses at the quantum critical point described by a certain RG fixed point. We define the susceptibility exponent $\\eta=\\eta_M$ of a fermion bilinear $\\psi^\\dagger M \\psi$ or $\\psi^\\dagger M \\psi^*$ through the scaling dimension\n\\begin{align}\n\\label{susc1} [h_M] = z +\\eta_M,\n\\end{align}\nwhere $h=h_M$ is introduced by coupling a term $L_M = h_M (\\psi^\\dagger M \\psi^{(*)})$ to the Lagrangian. For a detailed discussion which fully applies here see App. 6 of Ref. \\cite{PhysRevB.95.075149}. In our case $z=1$ is the trivial dynamic critical exponent due to $e=0$ at the fixed point. Importantly, for each fixed point we have to test every ordering channel individually and determine the one with the largest susceptibility. On the other hand, due to cubic symmetry, we can restrict to the ten distinct cubic channels. For this purpose we couple\n\\begin{align}\n L_{\\mathbb{1}} &= h_{\\mathbb{1}} (\\psi^\\dagger \\psi),\\\\\n L_{E} &= h_{E} (\\psi^\\dagger \\gamma_1\\psi),\\\\\n L_{T} &= h_{T} (\\psi^\\dagger \\gamma_3\\psi),\\\\\n L_{V} &= h_{V} (\\psi^\\dagger V_1\\psi),\\\\\n L_{U} &= h_{U} (\\psi^\\dagger U_1\\psi),\\\\\n L_{W'} &= h_{W'} (\\psi^\\dagger W_4\\psi),\\\\\n L_{\\mathcal{W}} &= h_{\\mathcal{W}} (\\psi^\\dagger W_7\\psi),\\\\\n L_{\\rm s} &= h_{\\rm s} (\\psi^\\dagger \\gamma_{45}\\psi^*),\\\\\n L_{\\rm d,E} &= h_{\\rm d,E} (\\psi^\\dagger \\gamma_1 \\gamma_{45}\\psi^*),\\\\ \nL_{\\rm d,T} &= h_{\\rm d,T} (\\psi^\\dagger \\gamma_3\\gamma_{45}\\psi^*)\n\\end{align}\nto the Lagrangian and determine the corresponding flow equations $\\dot{h}_M=(z+\\eta_M)h_M$ to read off the susceptibilities. \n\nTo give some example we present the susceptibilities for the analytically tractable cases $\\alpha=0$ and $\\alpha=2$. For $\\alpha=0$ we have\n\\begin{align}\n\\nonumber \\eta_{\\mathbb{1}}&=\\eta_{E} =\\eta_{\\mathcal{W}}=0,\\ \\eta_T = \\frac{2}{3}(g_1'-2g_2-5g_3),\\\\\n\\nonumber \\eta_V& =\\frac{2}{3}(g_1'+2g_2-g_3),\\ \\eta_U= \\eta_{W'}=\\frac{2}{3}(g_1'+g_3),\\\\\n\\nonumber \\eta_{\\rm s} &= -(g_1'+2g_2+3g_3),\\ \\eta_{\\rm d,E} =-(g_1'-3g_3),\\\\\n \\label{susc2} \\eta_{\\rm d,T} &= \\frac{1}{3}(-g_1'+2g_2+g_3).\n\\end{align}\nFor the rotation invariant case with $\\alpha=2$ we have\n\\begin{align}\n \\nonumber \\eta_{\\mathbb{1}} &=0,\\ \\eta_E = \\frac{1}{5}(g_1'-4g_2-3g_3),\\\\ \n \\nonumber \\eta_T &= \\frac{1}{5}(g_1'-2g_2-5g_3),\\ \\eta_{W'} = \\frac{43}{105}(g_1'+g_3)\\\\\n \\nonumber \\eta_{\\mathcal{W}} &= \\frac{43}{105}(g_1'-2g_2+3g_3),\\ \\eta_{\\rm s} = -\\frac{2}{3}(g_1'+2g_2+3g_3),\\\\\n \\label{susc3}\\eta_{\\rm d,E} &=-\\frac{1}{3}g_1'+g_3,\\ \\eta_{\\rm d,T} = -\\frac{1}{3}(g_1'-2g_2-g_3).\n\\end{align}\nWe omitted $\\eta_V$ and $\\eta_U$ here for a reason that will be explained in the next paragraph. Further imposing rotation invariance onto the couplings by setting $g_2=g_3$ we obtain\n\\begin{align}\n \\nonumber \\eta_{\\mathbb{1}} &=0,\\ \\eta_{\\gamma_a} = \\eta_E = \\eta_T = \\frac{1}{5}(g_1'-7g_2),\\\\\n \\nonumber \\eta_{\\mathcal{J}} &= \\frac{4}{15}(g_1'+g_2),\\ \\eta_W =\\eta_{W'}=\\eta_{\\mathcal{W}} = \\frac{43}{105}(g_1'+g_2),\\\\\n \\label{susc4} \\eta_{\\rm s} &= -\\frac{2}{3}(g_1'+5g_2),\\ \\eta_{\\rm d}=\\eta_{\\rm d,E}=\\eta_{\\rm d,T} = -\\frac{1}{3}g_1'+g_2.\n\\end{align}\nAs expected, the susceptibilities of different components of same-rank tensors in Eq. (\\ref{model6b}) coincide in the rotation invariant limit.\n\n\nIn most cases, by coupling a term $h_M(\\psi^\\dagger M \\psi)$ to the Lagrangian while setting $h_N=0$ for all other matrices $N\\neq M$, we only generate a running of the coupling $h_M$. However, if there is a cubic transformation that relates $M$ and $N$, this is no longer true. In our case, coupling a term $h_V (\\psi^\\dagger V_i\\psi)$ to the Lagrangian generates a term $h_U (\\psi^\\dagger U_i\\psi)$, and vice versa. Referring once more to Ref. \\cite{PhysRevB.95.075149}, Eqs. (A110)-(A114) for details, we briefly review here how to determine the correct scaling behavior. \n\n\n\n\n\n\nWe write\n\\begin{align}\n \\nonumber h(\\psi^\\dagger V_i\\psi) \\Rightarrow h(\\psi^\\dagger V_i\\psi)+h\\Bigl[\\eta_V (\\psi^\\dagger V_i\\psi) +a (\\psi^\\dagger U_i\\psi)\\Bigr],\\\\\n h(\\psi^\\dagger _iU\\psi) \\Rightarrow h(\\psi^\\dagger U_i\\psi)+ h\\Bigl[b (\\psi^\\dagger V_i\\psi) +\\eta_U (\\psi^\\dagger U_i\\psi)\\Bigr],\n\\end{align}\nwhere ``$\\Rightarrow$'' stands for ``generates a term under RG''. In general, $a$ and $b$ depend on $\\alpha$ and the couplings $g_i$. We find that for all values of $\\alpha$ we have\n\\begin{align}\n a &= -K(g_1'+g_3),\\\\\n b&= -K(g_1'+2g_2-g_3)\n\\end{align}\nwith $K=K(\\alpha)>0$ a positive constant. Some values are:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|}\n\\hline $\\alpha$ & 0 & 0.5 & 1 & 1.5 & 2 & 2.296 & 2.5 & 5 \\\\ \n\\hline $K$ & 0 & 0.12 & 0.25 & 0.106 & 0.057 & 0.0425 & 0.035 & 0.0077 \\\\ \n\\hline \n\\end{tabular} \n\\end{center}\nWe observe that $K$ is small and vanishes for $\\alpha\\to0$ and $\\alpha\\to \\infty$, with a maximum around $\\alpha \\approx 1$. \n\n\nWe introduce $M_i = c V_i + c' U_i$ with real coefficients such that $c^2+c'{}^2=1$. Obviously, only the ratio $\\kappa=c'\/c$ is of relevance. The maximal (and also the minimal) susceptibility $\\eta_M$ will then come from a linear combination that satisfies the self-consistent relation\n\\begin{align}\n \\label{susc5} h(\\psi^\\dagger M_i\\psi) \\Rightarrow h(1+\\eta_{M})(\\psi^\\dagger M_i \\psi)\n\\end{align}\nwith $\\eta_{M} =\\eta_V + \\frac{c'}{c}b$ and\n\\begin{align}\n 0 &\\stackrel{!}{=} c'(\\eta_U-\\eta_V) +\\frac{1}{c}\\Bigl(c^2 a - c'{}^2 b\\Bigr).\n\\end{align}\nTe last equation is solved by\n\\begin{align}\n \\kappa &= \\frac{c'}{c} = \\frac{\\eta_U-\\eta_V}{2b}\\pm\\sqrt{\\frac{(\\eta_U-\\eta_V)^2}{4b^2}+\\frac{a}{b}},\\\\\n \\eta_{M} &=\\frac{\\eta_V+\\eta_U}{2} \\pm \\frac{1}{2}\\sqrt{(\\eta_U-\\eta_V)^2+4ab}.\n\\end{align}\nSince we are after the largest susceptibilities, we are interested in the ``$+$'' solution of $\\eta_{M}$. This corresponds to choosing ``$-$'' in $\\kappa$ at fixed point V, and ``$+$'' in $\\kappa$ at fixed point W. For V we have $|\\kappa| < 0.05$ with the largest value around $\\alpha\\sim 1.25$. Consequently, up to a tiny correction we have\n\\begin{align*}\n M_i \\approx V_i\n\\end{align*}\nat fixed point V. In contrast, at W we find that $\\kappa\\gg1$ (of order 10) for all $\\alpha$, and so we can regard this case as\n\\begin{align*}\n M_i \\approx U_i.\n\\end{align*} \nThe exponent $\\eta_{M}$ at W is larger than $\\eta_{\\mathcal{W}}=3$ for $\\alpha \\leq 0.7$. \n\n\n\\begin{figure}[t!]\n\\centering\n\\begin{minipage}{0.48\\textwidth}\n\\includegraphics[width=8cm]{FigureSM1}\n\\includegraphics[width=8cm]{FigureSM2}\n\\caption{The ratio $\\kappa=c'\/c$ that maximizes $\\eta_M$ for the bilinear $\\psi^\\dagger(cV_i+c'U_i)\\psi$ at the fixed point V (upper panel) and W (lower panel). \\emph{Upper panel.} At V we have $|\\kappa|<5\\%$ and so the order parameter is to a very good approximation given by $\\langle \\psi^\\dagger V_i\\psi\\rangle$. \\emph{Lower panel.} At W the ratio $\\kappa\\gg1$ is large and so the corresponding order parameter is approximately $\\langle \\psi^\\dagger U_i \\psi\\rangle$. However, for $\\alpha\\geq 0.7$, the leading instability at W is the condensation of $\\chi=\\langle \\psi^\\dagger \\mathcal{W}\\psi\\rangle$ since $\\eta_{\\mathcal{W}}=3$ is the larger susceptibility.}\n\\label{Figkappa}\n\\end{minipage}\n\\end{figure}\n\n\n\n\n\n\n\\subsection{Flow equations at high-symmetry points}\n\n\n\n\\noindent The flow equations for $\\alpha=0$ read\n\\begin{align}\n \\label{rg3} \\dot{g}_1 &= -2g_1 -g_2^2-6g_2g_3-5g_3^2,\\\\\n \\label{rg4} \\dot{g}_2 &=-2g_2+g_2^2-2g_2g_3-3g_3^2,\\\\\n \\label{rg5} \\dot{g}_3 &= -2g_3 -\\frac{5}{3}g_2^2-\\frac{14}{3}g_2g_3-3g_3^2.\n\\end{align}\nNote that, apart from the trivial term $-2g_1$ in the first line, $g_1$ and $e$ are absent in these equations. We find the quantum critical points SC and V given by\n\\begin{align}\n \\label{rg6} \\text{SC}:\\ (g_1,g_2,g_3)_\\star &= \\Bigl(-\\frac{7}{16},-\\frac{3}{16},-\\frac{5}{16}\\Bigr),\\\\\n \\label{rg7} \\text{V}:\\ (g_1,g_2,g_3)_\\star &= \\Bigl(\\frac{1}{2},\\frac{3}{2},-\\frac{1}{2}\\Bigr).\n\\end{align} \nThe largest susceptibility exponents at SC and V are $\\eta_{\\rm s}=7\/4=1.75$ and $\\eta_V=8\/3=2.67$, respectively.\n\n\n\n\n\nIn the rotation invariant case ($\\alpha=2$) we have\n\\begin{align}\n\\nonumber \\dot{g}_1 ={}&-2g_1 -\\frac{2}{15}g_1'g_2 -\\frac{1}{5}g_1'g_3 -\\frac{76}{105}g_2^2 \\\\\n \\label{rg8} &- \\frac{164}{35} g_2g_3 -\\frac{79}{35}g_3^2,\\\\\n \\nonumber \\dot{g}_2={}& -2g_2 +\\frac{12}{35}g_1'g_2-\\frac{19}{35}g_1'g_3-\\frac{4}{15}g_2^2\\\\\n \\label{rg9} &-\\frac{58}{35}g_2g_3-\\frac{15}{7}g_3^2,\\\\\n \\nonumber \\dot{g}_3 ={}& -2g_3 -\\frac{38}{105}g_1'g_2+\\frac{17}{105}g_1'g_3 -\\frac{24}{35}g_2^2\\\\\n \\label{rg10} &-\\frac{272}{105}g_2g_3-\\frac{83}{105}g_3^2.\n\\end{align}\nThe quantum critical points are given by \n\\begin{align}\n \\label{rg11} \\text{SC}:\\ (g_1,\\ g_2,\\ g_3)_\\star &= (-0.841,\\ -0.450,\\ -0.450),\\\\\n \\label{rg12} \\text{W}:\\ (g_1,\\ g_2,\\ g_3)_\\star &= (1.22,\\ -1.22,\\ 1.22),\\\\\n \\label{rg13} \\text{V}:\\ (g_1,\\ g_2,\\ g_3)_\\star &= (3.31,\\ 3.07,\\ -1.20).\n\\end{align}\nthe leading susceptibility exponents are given by $\\eta_{\\rm s}=2.062$,\\ $\\eta_{\\mathcal{W}}=3$ and $\\eta_{V_i}= 4.06$, respectively. Note that the fixed point W satisfies $g_1=-g_2=g_3$ and $\\eta_{\\mathcal{W}}=d$. For $g_2=g_3$ the flow equations read\n\\begin{align}\n \\label{rg14} \\dot{g}_1 &= -2g_1 -\\frac{1}{3}g_1'g_2-\\frac{23}{3}g_2^2,\\\\\n \\label{rg15} \\dot{g}_2 &= -2g_2 -\\frac{1}{5}g_1'g_2-\\frac{61}{15}g_2^2.\n\\end{align}\nThis set of equations only supports the superconducting quantum critical point SC. We should think of the fixed points W and V in this limit as approached in an anisotropic system with $\\alpha\\to 2$ from above or from below.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Cubic and tetrahedral symmetry group}\\label{AppCubic}\nIn this appendix we summarize some properties of the rotational cubic point group which are relevant for our analysis. We begin by deriving explicit expressions for the group elements and then discuss the rotational or chiral tetrahedral point group. \n\n\nThe cubic point group $O_h$ consists of the transformations that leave a three-dimensional cube invariant. Clearly, inversion is such a symmetry transformation. All other elements of the group can be expressed as a rotation or a rotation combined with an inversion. The ``rotational'' subgroup $O$, which is the symmetry group of our problem at hand, consists of the true rotations. Whereas $O_h$ consists of 48 elements, there are 24 elements in $O$. \n\nIn the following we consider the three-dimensional representation of the group acting on position or momentum space vectors $\\textbf{x}=(x_1,x_2,x_3)^T$ or $\\textbf{p}=(p_1,p_2,p_3)^T$, respectively. The matrices $J_i$ also transform like a vector under $O$. The group $O$ is a subgroup of $\\text{SO}(3)$, and so every element $R\\in O$ satisfies $R^TR=\\mathbb{1}_3$ and $\\mbox{det}(R)=1$. Every rotation can be specified by an axis $\\textbf{n}=(n_1,n_2,n_3)^T$ and a rotation angle $\\vphi$ according to\n\\begin{align}\n \\label{cub1} R(\\textbf{n},\\vphi) = \\mathbb{1}_3 + \\sin\\vphi K + (1-\\cos \\vphi) K^2\n\\end{align}\nwith\n\\begin{align}\n\\label{cub2} K = \\begin{pmatrix} 0 & -n_z & n_y \\\\ n_z & 0 & -n_x \\\\ -n_y & n_x & 0 \\end{pmatrix}.\n\\end{align}\nThe inverse of $R(\\textbf{n},\\vphi)$ is $R(-\\textbf{n},\\vphi)$, and the self-inverse elements of $O$ are precisely the ones represented by symmetric matrices, which corresponds to rotations by $0^{o}$ or $180^{o}$. If $R$ is an element of $O$, then $IR$ with inversion $I=\\text{diag}(-1,-1,-1)$ is the corresponding element of $O_h$ that includes an inversion.\n\n\n\n\\begin{widetext}\nWe now summarize the 24 group elements of $O$. The unit element is $R_1=\\mathbb{1}_3$. We further have the following symmetry operations:\\\\\n\\noindent{\\emph{(i) Three rotations by $180^o$ about a 4-fold axis.}} The corresponding rotation axes are the $x,y,z$-axes connecting opposite faces given by $\\textbf{n}=(1,0,0)^T, (0,1,0)^T, (0,0,1)^T$, which leads to the self-inverse group elements\n\\begin{align}\n \\label{cub3} R_2 = \\begin{pmatrix} 1 & 0 & 0 \\\\ 0 & -1 & 0 \\\\ 0 & 0 & -1 \\end{pmatrix},\\ R_3 = \\begin{pmatrix} -1 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & -1 \\end{pmatrix},\\ R_4 = \\begin{pmatrix} -1 & 0 & 0 \\\\ 0 & -1 & 0 \\\\ 0 & 0 & 1 \\end{pmatrix}.\n\\end{align}\n\\emph{(ii) Eight rotations by $120^o$ about a 3-fold axis.} These 3-fold axes are the axes connecting opposite vertices. The sign of $\\textbf{n}$ matters and we have\n\\begin{align} \n \\label{cub4} \\textbf{n}=\\frac{1}{\\sqrt{3}}\\begin{pmatrix} 1 \\\\ 1 \\\\ 1 \\end{pmatrix},\\ \\frac{1}{\\sqrt{3}}\\begin{pmatrix} 1 \\\\ -1 \\\\ 1 \\end{pmatrix},\\ \\frac{1}{\\sqrt{3}}\\begin{pmatrix} -1 \\\\ 1 \\\\ 1 \\end{pmatrix},\\ \\frac{1}{\\sqrt{3}}\\begin{pmatrix} -1 \\\\ -1 \\\\ 1 \\end{pmatrix},\n\\end{align} \nwhich leads to\n\\begin{align}\n \\label{cub5} R_5 = \\begin{pmatrix} 0 & 0 & 1 \\\\ 1 & 0 & 0 \\\\ 0 & 1 & 0 \\end{pmatrix},\\ R_{6} = \\begin{pmatrix} 0 & -1 & 0 \\\\ 0 & 0 & -1 \\\\ 1 & 0 & 0 \\end{pmatrix},\\ R_{7} = \\begin{pmatrix} 0 & -1 & 0 \\\\ 0 & 0 & 1 \\\\ -1 & 0 & 0 \\end{pmatrix},\\ R_{8}=\\begin{pmatrix} 0 & 0 & -1 \\\\ 1 & 0 & 0 \\\\ 0 & -1 & 0 \\end{pmatrix}.\n\\end{align}\nThe inverse elements follow from $\\textbf{n}\\to -\\textbf{n}$ and read\n\\begin{align}\n \\label{cub6} R_{9}=R_{5}^T,\\ R_{10}=R_{6}^T,\\ R_{11}=R_{7}^T,\\ R_{12}=R_{8}^T.\n\\end{align}\n\\emph{(iii) Six rotations by $90^o$ about a 4-fold axis.} These 4-fold axes are again the $x,y,z$-axes connecting opposite faces, but this time the sign of $\\textbf{n}$ matters. We have $\\textbf{n}=(1,0,0)^T, (0,1,0)^T, (0,0,1)^T$, which leads to\n\\begin{align}\n \\label{cub7} R_{13} &= \\begin{pmatrix} 1 & 0 & 0 \\\\ 0 & 0 & -1 \\\\ 0 & 1 & 0 \\end{pmatrix},\\ R_{14} = \\begin{pmatrix} 0 & 0 & 1 \\\\ 0 & 1 & 0 \\\\ -1 & 0 & 0 \\end{pmatrix},\\ R_{15}= \\begin{pmatrix} 0 & -1 & 0 \\\\ 1 & 0 & 0 \\\\ 0 & 0 & 1 \\end{pmatrix},\n\\end{align}\nand their inverses with $\\textbf{n}\\to -\\textbf{n}$ and\n\\begin{align}\n \\label{cub8} R_{16}=R_{13}^T,\\ R_{17}=R_{14}^T,\\ R_{18}=R_{15}^T.\n\\end{align}\n\\emph{(iv) Six rotations by $180^o$ about a 2-fold axis.} The axes are the axes connecting opposite edges. The matrices are their own inverses and the sign of $\\textbf{n}$ does not matter. We have\n\\begin{align}\n \\label{cub9} \\textbf{n} = \\frac{1}{\\sqrt{2}}\\begin{pmatrix} 1 \\\\ 1 \\\\ 0 \\end{pmatrix},\\ \\frac{1}{\\sqrt{2}}\\begin{pmatrix} 1 \\\\ -1 \\\\ 0 \\end{pmatrix},\\ \\frac{1}{\\sqrt{2}}\\begin{pmatrix} 1 \\\\ 0 \\\\ 1 \\end{pmatrix},\\ \\frac{1}{\\sqrt{2}}\\begin{pmatrix} 1 \\\\ 0 \\\\ -1 \\end{pmatrix},\\ \\frac{1}{\\sqrt{2}}\\begin{pmatrix} 0 \\\\ 1 \\\\ 1 \\end{pmatrix},\\ \\frac{1}{\\sqrt{2}}\\begin{pmatrix} 0 \\\\ 1 \\\\ -1 \\end{pmatrix},\n\\end{align}\nwhich leads to\n\\begin{align}\n \\label{cub10} R_{19} &= \\begin{pmatrix} 0 & 1 & 0 \\\\ 1 & 0 & 0 \\\\ 0 & 0 & -1 \\end{pmatrix},\\ R_{20} = \\begin{pmatrix} 0 & -1 & 0 \\\\ -1 & 0 & 0 \\\\ 0 & 0 & -1 \\end{pmatrix},\\ R_{21} = \\begin{pmatrix} 0 & 0 & 1 \\\\ 0 & -1 & 0 \\\\ 1 & 0 & 0 \\end{pmatrix},\\\\\n \\label{cub11} R_{22} &= \\begin{pmatrix} 0 & 0 & -1 \\\\ 0 & -1 & 0 \\\\ -1 & 0 & 0 \\end{pmatrix},\\ R_{23} = \\begin{pmatrix} -1 & 0 & 0 \\\\ 0 & 0 & 1 \\\\ 0 & 1 & 0 \\end{pmatrix},\\ R_{24} = \\begin{pmatrix} -1 & 0 & 0 \\\\ 0 & 0 & -1 \\\\ 0 & -1 & 0 \\end{pmatrix}.\n\\end{align}\n\\end{widetext}\n\nThe largest subgroup of $O$ is the rotational tetrahedral group $T$ with 12 elements. It consists of the rotations that leave a tetrahedron invariant. (Note that the tetrahedral group $T_d$ is commonly defined such that it includes inversion as well, and so it has 24 elements.) The elements of $T$ are precisely the first twelve $R_{1,\\dots,12}$ in our notation. The group $T$ leaves the expressions $J_1J_2J_3+J_3J_2J_1$ and $p_1p_2p_3$ invariant. In contrast, the remaining twelve elements of $O$ change the sign of these expressions.\n\n\n\n\n\n\n\n\n\\end{document}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nThe technique of controlling various operators by so-called sparse operators has proven to be a very useful tool to obtain (sharp) weighted norm inequalities in the past decade.\n The key feature in this approach is that a typically signed and non-local operator is dominated, either in norm, pointwise or in dual form, by a positive and local expression.\n\n The sparse domination technique comes from Lerner's work towards an alternative proof of the $A_2$-theorem, which was first proven by Hyt\\\"onen in \\cite{Hy12}. In \\cite{Le13a} Lerner applied his local mean oscillation decomposition approach to the $A_2$-theorem, estimating the norm of a Calder\\'on-Zygmund operator by the norm of a sparse operator. This was later improved to a pointwise estimate independently by Conde-Alonso and Rey \\cite{CR16} and by Lerner and Nazarov \\cite{LN15}. Afterwards, Lacey \\cite{La17b} obtained the same result for a slightly larger class of Calder\\'on-Zygmund operators by a stopping cube argument instead of the local mean oscillation decomposition approach. This argument was further refined by Hyt\\\"onen, Roncal and Tapiola \\cite{HRT17} and afterwards made strikingly clear by Lerner \\cite{Le16}, where the following abstract sparse domination principle was shown:\n\n If $T$ is a bounded sublinear operator from $L^{p_1}(\\R^n)$ to $L^{p_1,\\infty}(\\R^n)$ and the \\emph{grand maximal truncation operator}\n\\begin{equation*}\n \\mc{M}_T f(s) := \\sup_{Q \\ni s} \\esssup_{s' \\in Q} \\,\\abs{T(f \\ind_{\\R^n\\setminus 3Q})(s')}, \\qquad s \\in \\R^n\n\\end{equation*}\nis bounded from $L^{p_2}(\\R^n)$ to $L^{p_2,\\infty}(\\R^n)$ for some $1 \\leq p_1,p_2 <\\infty$, then there is an $\\eta \\in (0,1)$ such that for every compactly supported $f \\in L^{p}(\\R^n)$ with $p_0 := \\max\\cbrace{p_1, p_2}$ there exists an $\\eta$-sparse family of cubes $\\mc{S}$ such that\n\\begin{equation}\\label{eq:sparsepointwiseintro}\n \\abs{Tf(s)} \\lesssim\n \\sum_{Q \\in \\mc{S}} \\ip{\\abs{f}}_{p_0,Q}\\ind_Q(s),\\qquad s \\in \\R^n.\n\\end{equation}\nHere $\\ip{{f}}_{p,Q}^p:= \\avint_Q {f}^p :=\\frac{1}{\\abs{Q}}\\int_Q{f}^p$ for $p \\in (0,\\infty)$ and positive $f \\in L^p_{\\loc}(\\R^n)$ and we call a family of cubes $\\mc{S}$ $\\eta$-sparse if for every $Q \\in \\mc{S}$ there exists a measurable set $E_Q \\subseteq Q$ such that $\\abs{E_Q} \\geq \\eta\\abs{Q}$ and such that the $E_Q$'s are pairwise disjoint.\n\n This sparse domination principle was further generalized in the recent paper \\cite{LO19} by Lerner and Ombrosi, in which the authors showed that the weak $L^{p_2}$-boundedness of the more flexible operator\n\\begin{equation*}\n \\mc{M}_{T,\\alpha}^{\\#} f(s) := \\sup_{Q \\ni s} \\esssup_{s',s'' \\in Q} \\,\\abs{T(f \\ind_{\\R^n\\setminus \\alpha Q})(s') - T(f \\ind_{\\R^n\\setminus \\alpha Q})(s'')}, \\qquad s \\in \\R^n\n\\end{equation*}\nfor some $\\alpha \\geq 3$ is already enough to deduce the pointwise sparse domination as in \\eqref{eq:sparsepointwiseintro}. Furthermore, they relaxed the weak $L^{p_1}$-boundedness condition on $T$ to a condition in the spirit of the $T(1)$-theorem.\n\n\\subsection{Main result}\nOur main result is a generalization of the main result in \\cite{LO19} in the following four directions:\n\\begin{enumerate}[(i)]\n \\item \\label{it:generalizationhom} We replace $\\R^n$ by a space of homogeneous type $(S,d,\\mu)$.\n \\item \\label{it:generalizationvect} We let $T$ be an operator from $L^{p_1}(S;X)$ to $L^{p_1,\\infty}(S;Y)$, where $X$ and $Y$ are Banach spaces.\n \\item \\label{it:generalizationsub} We use structure of the operator $T$ and geometry of the Banach space $Y$ to replace the $\\ell^1$-sum in the sparse operator by an $\\ell^r$-sum for $r \\geq 1$.\n \\item \\label{it:generalizationlocal} We replace the truncation $T(f \\ind_{\\R^n\\setminus \\alpha Q})$ in the grand maximal truncation operator by an abstract localization principle.\n\\end{enumerate}\nThe extensions \\ref{it:generalizationhom} and \\ref{it:generalizationvect} are relatively straightforward. The main novelty of this paper is \\ref{it:generalizationsub}, which controls the weight characteristic dependence that can be deduced from the sparse domination. Generalization \\ref{it:generalizationlocal} will only make its appearance in Theorem \\ref{theorem:localsparse} and can be used to make the associated grand maximal truncation operator easier to estimate in specific situations.\n\n\\bigskip\n\nLet $(S,d,\\mu)$ be a space of homogeneous type\nand let $X$ and $Y$ be Banach spaces.\nFor a bounded linear operator $T$ from $L^{p_1}(S;X)$ to $L^{p_1,\\infty}(S;Y)$ and $\\alpha\\geq 1$ we define the\nfollowing \\emph{sharp grand maximal truncation operator}\n\\begin{align*}\n \\mc{M}_{T,\\alpha}^{\\#}&f(s) :=\\sup_{B\\ni s} \\esssup_{s',s''\\in B}\\, \\nrmb{T(f\\ind_{S\\setminus {\\alpha B}})(s')-T(f\\ind_{S\\setminus {\\alpha B}})(s'')}_Y,\\qquad s \\in S,\n\\end{align*}\nwhere the supremum is taken over all balls $B\\subseteq S$ containing $s\\in S$. Our main theorem reads as follows.\n\n\\begin{theorem}\\label{theorem:main}\nLet $(S,d,\\mu)$ be a space of homogeneous type and let $X$ and $Y$ be Banach spaces. Take $p_1,p_2,r \\in [1,\\infty)$ and set $p_0:=\\max\\cbrace{p_1, p_2}$. Take $\\alpha \\geq 3c_d^2\/\\delta$, where $c_d$ is the quasi-metric constant and $\\delta$ is as in Proposition \\ref{proposition:dyadicsystem}. Assume the following conditions:\n\\begin{itemize}\n \\item $T$ is a bounded linear operator from $L^{p_1}(S;X)$ to $L^{p_1,\\infty}(S;Y)$.\n \\item $\\mc{M}_{T,\\alpha}^{\\#}$ is a bounded operator from $L^{p_2}(S;X)$ to $L^{p_2,\\infty}(S)$.\n \\item There is a $C_r>0$ such that for disjointly and boundedly supported $f_1,\\ldots,f_n \\in L^{p_0}(S;X)$\n\\begin{equation*}\n \\qquad \\nrms{T\\hab{\\sum_{k=1}^n f_k}(s)}_Y\\leq C_r \\, \\has{\\sum_{k=1}^n \\nrmb{Tf_k(s)}_Y^r}^{1\/r},\n \\qquad s \\in S.\n\\end{equation*}\n\\end{itemize}\n Then there is an $\\eta \\in (0,1)$ such that for any boundedly supported $f \\in L^{p_0}(S;X)$ there is an $\\eta$-sparse collection of cubes $\\mc{S}$ such that\n\\begin{align*}\n \\nrm{ Tf(s)}_Y&\\lesssim_{S,\\alpha} C_T \\,C_r\\, \\has{\\sum_{Q \\in \\mc{S}} \\ipb{\\nrm{f}_X}_{p_0,Q}^r \\ind_{Q}(s)}^{1\/r}, \\qquad s \\in S,\n\\end{align*}\nwhere $C_T={\\nrm{T}_{L^{p_1}\\to L^{p_1,\\infty}} + \\nrm{\\mc{M}_{T,\\alpha}^{\\#}}_{L^{p_2}\\to L^{p_2,\\infty}}}$.\n\\end{theorem}\nAs the assumption in the third bullet of Theorem \\ref{theorem:main} expresses a form of sublinearity of the operator $T$ when $r=1$, we will call this assumption \\emph{$r$-sublinearity}. Note that it is crucial that the constant $C_r$ is independent of $n \\in \\N$. If $C_r = 1$ it suffices to consider $n=2$.\n\n\\subsection{Sharp weighted norm inequalities}\nOne of the main reasons to study sparse domination of an operator is the fact that sparse bounds yield weighted norm inequalities and these weighted norm inequalities are sharp for many operators. Here sharpness is meant in the sense that for $p \\in (p_0,\\infty)$ we have a $\\beta\\geq 0$ such that\n\\begin{equation}\\label{eq:sharp}\n \\nrm{T}_{L^p(S,w;X) \\to L^p(S,w;Y)} \\lesssim [w]_{A_{p\/p_0}}^\\beta, \\qquad w \\in A_{p\/p_0}\n\\end{equation}\nand \\eqref{eq:sharp} is false for any $\\beta' <\\beta$.\n\nThe first result of this type was obtained by Buckley \\cite{Bu93}, who showed that $\\beta = \\tfrac{1}{p-1}$ for the Hardy--Littlewood maximal operator. A decade later, the quest to find sharp weighted bounds attracted renewed attention because of the work of Astala, Iwaniec and Saksman \\cite{AIS01}. They proved sharp regularity results for the solution to the Beltrami equation under the assumption that $\\beta = 1$ for the Beurling--Ahlfors transform for $p\\geq 2$. This linear dependence on the $A_p$ characteristic for the Beurling--Ahlfors transform was shown by Petermichl and Volberg in \\cite{PV02}. Another decade later, after many partial results, sharp weighted norm inequalities were obtained for general Calder\\'on--Zygmund operators by Hyt\\\"onen in \\cite{Hy12} as discussed before.\n\nIn Section \\ref{section:weights} we will prove weighted $L^p$-boundedness for the sparse operators appearing in Theorem \\ref{theorem:main}. As a direct corollary from Theorem \\ref{theorem:main} and Proposition \\ref{proposition:weights} we have:\n\n\\begin{corollary}\\label{corollary:main}\n Under the assumptions of Theorem \\ref{theorem:main} we have for all $p \\in (p_0,\\infty)$ and $w \\in A_{p\/p_0}$\n\\begin{align*}\n \\nrm{T}_{L^p(S,w;X)\\to L^p(S,w;Y)} &\\lesssim C_T\\,C_r\\, [w]_{A_{p\/p_0}}^{\\max\\cbraceb{\\frac{1}{p-p_0},\\frac{1}{r}}},\n\\end{align*}\nwhere the implicit constant depends on $S,p_0,p,r$ and $\\alpha$.\n\\end{corollary}\nAs noted before the main novelty in Theorem \\ref{theorem:main} is the introduction of the parameter $r \\in [1,\\infty)$. The $r$-sublinearity assumption in Theorem \\ref{theorem:main} becomes more restrictive as $r$ increases and the conclusions of Theorem \\ref{theorem:main} and Corollary \\ref{corollary:main} consequently become stronger. In order to check whether the dependence on the weight characteristic is sharp, one can employ e.g. \\cite[Theorem 1.2]{LPR15}, which provides a lower bound for the best possible weight characteristic dependence in terms of the operator norm of $T$ from $L^p(S;X)$ to $L^p(S;Y)$. For some operators, like Littlewood--Paley or maximal operators, sharpness in the estimate in Corollary \\ref{corollary:main} is attained for $r>1$ and thus Theorem \\ref{theorem:main} can be used to show sharp weighted bounds for more operators than precursors like \\cite[Theorem 1.1]{LO19}.\n\n\\subsection{How to apply our main result}\nLet us outline the typical way how one applies Theorem \\ref{theorem:main} (or the local and more general version in Theorem \\ref{theorem:localsparse}) to obtain (sharp) weighted $L^p$-boundedness for an operator $T$:\n\\begin{enumerate}[(i)]\n \\item If $T$ is not linear it is often \\emph{linearizable}, which means that we can linearize it by putting part of the operator in the norm of the Banach space $Y$. For example if $T$ is a Littlewood--Paley square function we take $Y=L^2$ and if $T$ is a maximal operator we take $Y=\\ell^\\infty$. Alternatively one can apply Theorem \\ref{theorem:localsparse}, which is a local and more abstract version of Theorem \\ref{theorem:main} that does not assume $T$ to be linear.\n \\item The weak $L^{p_1}$-boundedness of $T$ needs to be studied separately and is often already available in the literature.\n \\item The operator $\\mc{M}_{T,\\alpha}^{\\#}$ reflects the non-localities of the operator $T$. The weak $L^{p_2}$-boundedness of $\\mc{M}_{T,\\alpha}^{\\#}$ requires an intricate study of the structure of the operator. In many examples $\\mc{M}_{T,\\alpha}^{\\#}$ can be pointwise dominated by the Hardy--Littlewood maximal operator $M_{p_2}$, which is weak $L^{p_2}$-bounded. This is exemplified for Calder\\'on--Zygmund operators in the proof of Theorem \\ref{theorem:A2}.\n Sometimes one can choose a suitable localization in Theorem \\ref{theorem:localsparse} such that the sharp maximal truncation operator is either zero (see Section \\ref{section:maximal} on the Rademacher maximal operator), or pointwise dominated by $T$.\n \\item The $r$-sublinearity assumption on $T$ is trivial for $r=1$, which suffices if one is not interested in quantitative weighted bounds. To check the $r$-sublinearity for some $r>1$ one needs to use the structure of the operator and often also the geometric properties of the Banach space $Y$ like type $r$.\n See, for example, the proofs of Theorems \\ref{theorem:RMF} and \\cite[Theorem 6.4]{LV19} how to check $r$-sublinearity in concrete cases.\n\\end{enumerate}\n\n\\subsection{Applications}\nThe main motivation to generalize the results in \\cite{LO19} comes from the application in the recent work \\cite{LV19} by Veraar and the author, in which Calder\\'on--Zygmund theory is developed for stochastic singular integral operators. In particular, in \\cite[Theorem 6.4]{LV19} Theorem \\ref{theorem:main} is applied with $p_1=p_2=r=2$ to prove a stochastic version of the vector-valued $A_2$-theorem for Calder\\'on--Zygmund operators, which yields new results in the theory of maximal regularity for stochastic partial differential equations. The fact that $r=2$ in \\cite[Theorem 6.4]{LV19} was needed to obtain a sharp result motivated the introduction of the parameter $r$ in this paper.\nIn future work, further applications of Theorem \\ref{theorem:main} to both deterministic and stochastic partial differential equations will be given, for which it is crucial that we allow spaces of homogeneous type instead of just $\\R^n$, as in these applications $S$ is typically $\\R_+\\times \\R^{n}$ with the parabolic metric.\n\nIn this paper we will focus on applications in harmonic analysis. We will provide a few examples that illustrate the sparse domination principle nicely, and comment on further potential applications in Section \\ref{section:further}.\n\\begin{itemize}\n \\item As a first application of Theorem \\ref{theorem:main} we prove an $A_2$-theorem for vector-valued Calder\\'on--Zygmund operators with operator-valued kernel in a space of homogeneous type. The $A_2$-theorem for vector-valued Calder\\'on--Zygmund operators with operator-valued kernel in Euclidean space has previously been proven in \\cite{HH14} and the $A_2$-theorem for scalar-valued Calder\\'on--Zygmund operators in spaces of homogeneous type in \\cite{NRV13,AV14}. Our theorem unifies these two results.\n \\item Using the $A_2$-theorem, we prove a weighted, anisotropic, mixed norm Mihlin multiplier theorem, which is a natural supplement to the recent results in \\cite{FHL18} and is particularly useful in the study of spaces of smooth, vector-valued functions.\n \\item In our second application of Theorem \\ref{theorem:main} we study sparse domination and quantitative weighted norm inequalities for the Rademacher maximal operator, extending the qualitative bounds in Euclidean space in \\cite{Ke13}. The proof demonstrates how one can use the geometry of the Banach space to deduce $r$-sublinearity for an operator. As a corollary, we deduce that the lattice Hardy--Littlewood and the Rademacher maximal operator are not comparable.\n\\end{itemize}\n\n\\subsection{Outline}\nThis paper is organized as follows: After introducing spaces of homogeneous type and dyadic cubes in such spaces in Section \\ref{section:SHT}, we will set up our abstract sparse domination framework and deduce Theorem \\ref{theorem:main} in Section \\ref{section:main}. We also give some further generalizations of our main results. In Section \\ref{section:weights} we introduce weights and state weighted bounds for the sparse operators in the conclusions of Theorem \\ref{theorem:main}, from which Corollary \\ref{corollary:main} follows. To prepare for our application sections, we will discuss some preliminaries on e.g. Banach space geometry in Section \\ref{section:Banachspace}. Afterwards we will\nuse our main result to prove the previously discussed applications in Sections \\ref{section:A2}-\\ref{section:maximal}.\nFinally, in Section \\ref{section:further} we discuss some potential further applications of our main result.\n\n\n\n\\section{Spaces of homogeneous type}\\label{section:SHT}\nA space of homogeneous type $(S,d,\\mu)$, originally introduced by Coifman and Weiss in \\cite{CW71}, is a set $S$ equipped with a quasi-metric $d$ and a doubling Borel measure $\\mu$. That is,\n a metric $d$ which instead of the triangle inequality satisfies\n\\begin{equation*}\n d(s,t) \\leq c_d\\, \\hab{d(s,u)+d(u,t)}, \\qquad s,t,u\\in S\n\\end{equation*}\nfor some $c_d\\geq 1$, and a Borel measure $\\mu$ that satisfies the doubling property\n\\begin{equation*}\n \\mu\\hab{B(s,2\\rho)} \\leq c_\\mu \\,\\mu\\hab{B(s,\\rho)}, \\qquad s \\in S,\\quad \\rho>0\n\\end{equation*}\nfor some $c_\\mu\\geq 1$, where $B(s,\\rho):=\\cbrace{t \\in S:d(s,t)<\\rho}$ is the ball around $s$ with radius $\\rho$. Throughout this paper we will assume additionally that all balls $B\\subseteq S$ are Borel sets and that we have $0 <\\mu(B)<\\infty$.\n\nIt was shown in \\cite[Example 1.1]{St15} that it can indeed happen that balls are not Borel sets in a quasi-metric space. This can be circumvented by taking topological closures and adjusting the constants $c_d$ and $c_\\mu$ accordingly. However, to simplify matters we just assume all balls to be Borel sets and leave the necessary modifications if this is not the case to the reader.\nThe size condition on the measure of a ball ensures that taking the average $\\ip{{f}}_{p,B}$ of a positive function $f \\in L^p_{\\loc}(S)$ over a ball $B\\subseteq S$ is always well-defined.\n\nAs $\\mu$ is a Borel measure, i.e. a measure defined on the Borel $\\sigma$-algebra of the quasi-metric space $(S,d)$, the Lebesgue differentiation theorem holds and as a consequence the continuous functions with bounded support are dense in $L^p(S)$ for all $p \\in [1,\\infty)$. The Lebesgue differentiation theorem and consequently our results remain valid if $\\mu$ is a measure defined on a $\\sigma$-algebra $\\Sigma$ that contains the Borel $\\sigma$-algebra as long as the measure space $(S,\\Sigma,\\mu)$ is Borel semi-regular. See \\cite[Theorem 3.14]{AM15} for the details.\n\n\nThroughout we will write that an estimate depends on $S$ if it depends on $c_d$ and $c_\\mu$. For a thorough introduction to and a list of examples of spaces of homogeneous type we refer to the monographs of Christ \\cite{Ch90} and Alvarado and Mitrea \\cite{AM15}.\n\n\\subsection{Dyadic cubes}\nLet $00$ there is a $j\\in \\cbrace{1,\\ldots,m}$ and a $Q \\in \\ms{D}^j$ such that $$B(s,\\rho) \\subseteq Q, \\qquad\\text{and}\\qquad \\diam(Q) \\leq \\gamma \\rho.$$\n\\end{proposition}\n\nThe following covering lemma will be used in the proof of our main theorem:\n\\begin{lemma}\\label{lemma:covering}\nLet $(S,d,\\mu)$ be a space of homogeneous type and $\\ms{D}$ a dyadic system with parameters $c_0$, $C_0$ and $\\delta$. Suppose that $\\diam(S) = \\infty$, take $\\alpha \\geq {3c_d^2\/\\delta}$ and let $E \\subseteq S$ satisfy $0<\\diam(E)<\\infty$. Then there exists a partition $\\mc{D} \\subseteq \\ms{D}$ of $S$ such that $E \\subseteq \\alpha Q$ for all $Q \\in \\mc{D}$.\n\\end{lemma}\n\n\\begin{proof}\n For $s \\in S$ and $k \\in \\Z$ let $Q_s^k\\in \\ms{D}_k$ be the unique cube such that $s \\in Q_s^k$ and denote its center by $z_{s}^k$. Define\n \\begin{equation*}\n K_s := \\cbraceb{k \\in \\Z: E \\not\\subseteq 2 c_d Q_s^k},\n \\end{equation*}\n where $c_d$ is the quasi-metric constant.\n If $k \\in \\Z$ is such that\n $$\\diam(2c_dQ_s^k) \\leq 4c_d^2C_0\\delta^k < \\diam(E),$$ then $E\\not\\subseteq 2c_dQ_s^k$, i.e. $k \\in K_s$ so is $K_s$ non-empty. On the other hand if $k \\in \\Z$ is such that\n$\n C_0\\delta^k > \\sup_{s' \\in E} d(s,s'),\n$\n then\n \\begin{equation*}\n \\sup_{s'\\in E} d(s',z_{s}^k) \\leq c_d \\,\\hab{\\sup_{s' \\in E} d(s,s') + d(s, z_{s}^k)} \\leq 2c_dC_0\\delta^k\n \\end{equation*}\n so $E \\subseteq 2c_dQ_s^k$ and thus $k \\notin K_s$. Therefore $K_s$ is bounded from below.\n\n Define $k_s := \\min K_s$ and set $\\mc{D}:= \\cbrace{Q_s^{k_s}:s \\in S}$. Then $\\mc{D}$ is a partition of $S$. Indeed, suppose that for $s,s'\\in S$ we have $Q_s^{k_s} \\cap Q_{s'}^{k_{s'}}\\neq \\emptyset$. Then using property \\ref{it:dyadicsystem2} of a dyadic system we may assume without loss of generality that $Q_s^{k_s} \\subseteq Q_{s'}^{k_{s'}}$. Property \\ref{it:dyadicsystem2} of a dyadic system then implies that $k_s \\geq k_{s'}$. In particular $s \\in Q_{s'}^{k_{s'}}$, so by the minimality of $k_s$ we must have $k_s = k_{s'}$. Therefore since the elements of $\\ms{D}_{k_s}$ are pairwise disjoint we can conclude $Q_s^{k_s} = Q_{s'}^{k_{s'}}$.\n\n To conclude note that $z_s^{k_s} \\in Q_s^{k_s} \\subseteq Q_s^{k_s-1}$ by property \\ref{it:dyadicsystem2} of a dyadic system, so $d(z_s^{k_s-1},z_s^{k_s}) \\leq C_0\\delta^{k_s-1}$. Therefore using the minimality of $k_s$ we obtain\n $$E \\subseteq 2c_dQ_s^{k_s-1} = B(z_s^{k_s-1},2c_dC_0\\delta^{k_s-1}) \\subseteq B\\has{z_s^{k_s},\\frac{3c_d^2}{\\delta} \\cdot C_0\\delta^{k_s}} \\subseteq \\alpha Q_s^{k_s},$$\n which finishes the proof.\n\\end{proof}\n\n\\subsection{The Hardy--Littlewood maximal operator}\\label{subsection:HL}\nOn a space of homogeneous type $(S,d,\\mu)$ with a dyadic system $\\ms{D}$ we define the \\emph{dyadic Hardy--Littlewood maximal operator} for $f \\in L^1_{\\loc}(S)$ by\n\\begin{equation*}\n M^{\\ms{D}}f(s):= \\sup_{Q\\in \\ms{D}: s \\in Q} \\ipb{\\abs{f}}_{1,Q}, \\qquad s \\in S.\n\\end{equation*}\nBy Doob's maximal inequality (see e.g. \\cite[Theorem 3.2.2]{HNVW16}) $M^{\\ms{D}}$ is strong $L^p$-bounded for all $p\\in (1,\\infty)$ and weak $L^1$-bounded. We define the (non-dyadic) \\emph{Hardy--Littlewood maximal operator} for $f \\in L^1_{\\loc}(S)$ by\n\\begin{equation*}\n Mf(s):= \\sup_{B \\ni s} \\,\\ipb{\\abs{f}}_{1,Q}, \\qquad s \\in S,\n\\end{equation*}\nwhere the supremum is taken over all balls $B \\subseteq S$ containing $s$. By Proposition \\ref{proposition:dyadicsystem} there are dyadic systems $\\ms{D}^1,\\ldots,\\ms{D}^m$ such that\n\\begin{equation*}\n Mf(s) \\lesssim_S \\sum_{j=1}^m M^{\\ms{D}}f(s), \\qquad s \\in S,\n\\end{equation*}\nso $M$ is also strong $L^p$-bounded for $p\\in(1,\\infty)$ and weak $L^1$-bounded. For $p_0 \\in [1,\\infty)$ and $f \\in L^{p_0}_{\\loc}(S)$ we define\n\\begin{equation*}\n M_{p_0}f(s) := \\sup_{B \\ni s} \\,\\ipb{\\abs{f}}_{p_0,Q} = M\\hab{\\abs{f}^{p_0}}(s)^{1\/p_0}, \\qquad s \\in S,\n\\end{equation*}\nwhich is strong $L^p$-bounded for $p \\in(p_0,\\infty)$ and weak $L^{p_0}$-bounded. This follows from the boundedness of $M$ by rescaling.\n\n\\section{Pointwise \\texorpdfstring{$\\ell^r$}{lr}-sparse domination}\\label{section:main}\nIn this section we will prove a local version of the sparse domination result in Theorem \\ref{theorem:main}, from which we will deduce Theorem \\ref{theorem:main} by a covering argument using Lemma \\ref{lemma:covering}. This local version will use an abstract localization of the operator $T$, since it depends upon the operator at hand as to the most effective localization. For example in the study of a Calder\\'on--Zygmund operator it is convenient to localize the function inserted into $T$, for a maximal operator it is convenient to localize the supremum in the definition of the maximal operator and for a Littlewood--Paley operator it is most suitable to localize the defining integral.\n\n\\begin{definition}\n Let $(S,d,\\mu)$ be a space of homogeneous type with a dyadic system $\\ms{D}$, let $X$ and $Y$ be Banach spaces, $p \\in [1,\\infty)$ and $\\alpha \\geq 1$. For a bounded operator\n$$T\\colon L^{p}(S;X) \\to L^{p,\\infty}(S;Y)$$ we say that a family of operators $\\cbrace{T_Q}_{Q \\in \\ms{D}}$ from $L^p(S;X)$ to $L^{p,\\infty}(Q;Y)$ is an \\emph{$\\alpha$-localization family of $T$} if for all $Q\\in \\ms{D}$ and $f \\in L^{p}(S;X)$ we have\n\\begin{align*}\n T_Q(f \\ind_{\\alpha Q})(s)&=T_Qf(s), & & s \\in Q , & & \\text{(Localization)}\\\\\n \\nrmb{T_Q(f\\ind_{\\alpha Q})(s)}_Y &\\leq \\nrmb{T(f\\ind_{\\alpha Q})(s)}_Y, & &s \\in Q, & & \\text{(Domination)}\n\\end{align*}\nFor $Q,Q'\\in \\ms{D}$ with $Q' \\subseteq Q$ we define the difference operator\n\\begin{align*}\nT_{Q\\setminus Q'}f(s)&:= T_{Q}f(s) - T_{Q'}f(s), \\qquad s \\in Q'.\n\\end{align*}\n and for $Q\\in \\ms{D}$ the \\emph{localized sharp grand maximal truncation operator}\n \\begin{align*}\n \\mc{M}_{T,Q}^{\\#}&f(s) :=\\sup_{\\substack{Q'\\in \\ms{D}(Q):\\\\s \\in Q'}}\\, \\esssup_{s',s'' \\in Q'} \\,\\nrmb{(T_{Q\\setminus Q'}) f(s')-(T_{Q\\setminus Q'}) f(s'')}_Y, \\qquad s \\in S.\n\\end{align*}\n\\end{definition}\n\nIn order to obtain interesting results, one needs to be able to recover the boundedness of $T$ from the boundedness of $T_Q$ uniformly in $Q \\in \\ms{D}$. The canonical example of an $\\alpha$-localization family is\n\\begin{align*}\nT_Qf(s) &:=T(f \\ind_{\\alpha Q})(s), \\qquad s \\in Q.\n\\end{align*}\nfor all $Q \\in \\ms{D}$ and it is exactly this choice that will lead to Theorem \\ref{theorem:main}.\nWe are now ready to prove our main result, which is a local, more general version of Theorem \\ref{theorem:main}.\n\n\\begin{theorem}\\label{theorem:localsparse}\nLet $(S,d,\\mu)$ be a space of homogeneous type with dyadic system $\\ms{D}$ and let $X$ and $Y$ be Banach spaces. Take $p_1,p_2,r \\in [1,\\infty)$, set $p_0:=\\max\\cbrace{p_1, p_2}$ and take $\\alpha \\geq 1$. Suppose that\n\\begin{itemize}\n \\item $T$ is a bounded operator from $L^{p_1}(S;X)$ to $L^{p_1,\\infty}(S;Y)$ with $\\alpha$-localization family $\\cbrace{T_Q}_{Q \\in \\ms{D}}$.\n \\item $\\mc{M}_{T,Q}^{\\#}$ is bounded from $L^{p_2}(S;X)$ to $L^{p_2,\\infty}(S)$ uniformly in $Q \\in \\ms{D}$.\n \\item For all $Q_1,\\ldots,Q_n \\in \\ms{D}$ with $Q_n\\subseteq \\ldots\\subseteq Q_1$ and any $f \\in L^p(S;X)$\n\\begin{equation*}\n \\qquad \\nrmb{T_{Q_1}f(s)}_Y\\leq C_r \\has{\\nrmb{T_{Q_n}{f}(s)}_Y^r+\\sum_{k=1}^{n-1}\\nrmb{T_{Q_{k}\\setminus Q_{k+1}}f(s)}_Y^r}^{1\/r},\\quad s \\in Q_n.\n\\end{equation*}\n\\end{itemize}\n Then for any $f \\in L^{p_0}(S;X)$ and $Q\\in \\ms{D}$ there exists a $\\frac12$-sparse collection of dyadic cubes $\\mc{S}\\subseteq \\ms{D}(Q)$ such that\n\\begin{align*}\n \\nrmb{T_Qf(s)}_Y\\lesssim_{S,\\ms{D},\\alpha} C_T\\,C_r \\,\n \\has{ \\sum_{P \\in \\mc{S}} \\ipb{\\nrm{f}_X}_{p_0,\\alpha P}^r \\ind_P(s)}^{1\/r},\\qquad s \\in Q,\n \\end{align*}\nwith $C_T:={\\nrm{T}_{L^{p_1}\\to L^{p_1,\\infty}} + \\sup_{P \\in \\ms{D} } \\nrm{\\mc{M}_{T,P}^{\\#}}_{L^{p_2}\\to L^{p_2,\\infty}}}.$\n\\end{theorem}\n\nThe assumption in the third bullet in Theorem \\ref{theorem:localsparse} replaces the $r$-sub\\-linearity assumption in Theorem \\ref{theorem:main}. We will call this assumption a \\emph{localized $\\ell^r$-estimate}.\n\n\\begin{proof}\nFix $f \\in L^p(S,X)$ and $Q \\in \\ms{D}$. We will prove the theorem in two steps: we will first construct the $\\frac12$-sparse family of cubes $\\mc{S}$ and then show that the sparse expression associated to $\\mc{S}$ dominates $T_Qf$ pointwise.\n\n\\textbf{Step 1:}\nWe will construct the $\\frac12$-sparse family of cubes $\\mc{S}$ iteratively. Given a collection of pairwise disjoint cubes $\\mc{S}^k$ for some $k \\in \\N$ we will first describe how to construct $\\mc{S}^{k+1}$. Afterwards we can inductively define $\\mc{S}^k$ for all $k \\in \\N$ starting from $\\mc{S}^1 = \\cbrace{Q}$ and set $\\mc{S}:=\\bigcup_{k \\in \\N} \\mc{S}^k$.\n\nFix a $P\\in \\mc{S}^k$ and for $\\lambda\\geq1$ to be chosen later define\n\\begin{align*}\n \\Omega_{P}^1&:= \\cbraces{s \\in P: \\nrm{T_{P} f(s)}_Y> \\lambda\\, C_T \\,\\ipb{\\nrm{f}_X}_{p_0,\\alpha P}}\\\\\n \\Omega_{P}^2&:= \\cbraces{s \\in P: \\mc{M}_{T,P}^{\\#}(f)(s)> \\lambda \\,C_T\\,\\ipb{\\nrm{f}_X}_{p_0,\\alpha P}}\n\\end{align*}\nand $\\Omega_P:= \\Omega_{P}^1 \\cup \\Omega_{P}^2$. Let $c_1\\geq 1$, depending on $S$, $\\ms{D}$ and $\\alpha$, be such that $\\mu(\\alpha P) \\leq c_1\\,\\mu(P)$. By the domination property of the $\\alpha$-localization family we have\n\\begin{align*}\n\\nrm{T_P f(s)}_Y &\\leq \\nrm{T(f\\ind_{\\alpha P})(s)}_Y, & &s \\in P,\n\\intertext{and by the localization property}\n \\mc{M}_{T,P}^{\\#}(f)(s)&=\\mc{M}_{T,P}^{\\#}(f\\ind_{\\alpha P})(s), & &s \\in P.\n\\end{align*}\n Thus by the weak boundedness assumptions on $T$ and $\\mc{M}^{\\#}_{T,P}$ and H\\\"older's inequality we have for $i=1,2$\n\\begin{align}\\label{eq:holdercomp}\n \\mu(\\Omega_{P}^i) &\\leq \\has{\\frac{\\nrm{f \\ind_{\\alpha P}}_{L^{p_i}(S;X)}}{\\lambda \\,\\ipb{\\nrm{f}_X}_{p_0,\\alpha P}}}^{p_i}\n = \\frac{\\ipb{\\nrm{f}_X}_{p_i,\\alpha P}^{p_i}}{\\lambda^{p_i}\\ipb{\\nrm{f}_X}_{p_0,\\alpha P}^{p_i}} \\mu(\\alpha P)\n \\leq \\frac{ c_1}{\\lambda} \\, \\mu(P).\n\\end{align}\nTherefore it follows that\n\\begin{equation}\\label{eq:OmegatoQ}\n \\mu(\\Omega_P) \\leq \\frac{2c_1}{\\lambda} \\mu(P).\n\\end{equation}\nTo construct the cubes in $\\mc{S}^{k+1}$ we will use a local Calder\\'on--Zygmund decomposition (see e.g. \\cite[Lemma 4.5]{FN18}) on\n\\begin{equation*}\n \\Omega_{P,\\rho}:= \\cbrace{s \\in P: M^{\\ms{D}(P)} (\\ind_{\\Omega_P})>\\tfrac1\\rho}, \\qquad \\rho>0\n\\end{equation*}\n which will be a proper subset of $P$ for our choice of $\\lambda$ and $\\rho$. Here $M^{\\ms{D}(P)}$ is the dyadic Hardy--Littlewood maximal operator with respect to the restricted dyadic system $\\ms{D}(P)$.\n The local Calder\\'on--Zygmund decomposition yields a pairwise disjoint collection of cubes $\\mc{S}_P \\subseteq \\ms{D}(P)$ and a constant $c_2 \\geq 2$, depending on $S$ and $\\ms{D}$, such that $\\Omega_{P,c_2}= \\textstyle{\\bigcup_{P'\\in \\mc{S}_P}}P'$ and\n\\begin{equation}\\label{eq:Pjsize}\n \\tfrac{1}{c_2} \\, \\mu(P') \\leq \\mu(P'\\cap \\Omega_{P}) \\leq \\tfrac{1}{2} \\,\\mu(P'),\\qquad P'\\in \\mc{S}_P.\n\\end{equation}\nThen by \\eqref{eq:OmegatoQ}, \\eqref{eq:Pjsize} and the disjointness of the cubes in $\\mc{S}_P$ we have\n\\begin{align*}\n \\sum_{P'\\in \\mc{S}_P} \\mu(P') \\leq c_2 \\, \\sum_{P'\\in \\mc{S}_P} \\mu(P'\\cap \\Omega_P) \\leq c_2 \\,\\mu(\\Omega_P) \\leq \\frac{2c_1c_2}{\\lambda} \\mu(P).\n\\end{align*}\n Therefore, by choosing $\\lambda=4c_1c_2$, we have $\\sum_{P'\\in \\mc{S}_P} \\mu(P') \\leq \\frac12 \\mu(P)$. This choice of $\\lambda$ also ensures that $\\Omega_{P,c_2}$ is a proper subset of $P$ by as claimed before. We define $S^{k+1} := \\bigcup_{P \\in \\mc{S}^k} \\mc{S}_P$.\n\nNow take $\\mc{S}^1 = \\cbrace{Q}$, iteratively define $\\mc{S}^k$ for all $k \\in \\N$ as described above and set $\\mc{S} :=\\bigcup_{k \\in \\N} \\mc{S}^k$. Then $\\mc{S}$ is $\\frac12$-sparse family of cubes, since for any $P\\in \\mc{S}$ we can set $$E_P:= P\\setminus \\bigcup_{P'\\in \\mc{S}_P} P',$$ which are pairwise disjoint by the fact that $\\bigcup_{P'\\in \\mc{S}^{k+1}} P' \\subseteq \\bigcup_{P\\in \\mc{S}^k} P$ for all $k \\in \\N$ and we have\n\\begin{equation*}\n \\mu(E_P) = \\mu(P) - \\sum_{P'\\in \\mc{S}_P} \\mu(P') \\geq \\frac12 \\mu(P).\n\\end{equation*}\n\n\n\n\\textbf{Step 2:}\n We will now check that the sparse expression corresponding to $\\mc{S}$ constructed in Step 1 dominates $T_Qf$ pointwise. Since\n\\begin{equation*}\n\\lim_{k \\to \\infty} \\mu\\hab{\\bigcup_{P\\in \\mc{S}^k}P} \\leq \\lim_{k \\to \\infty} \\frac{1}{2^k}\\, \\mu(Q) =0,\n\\end{equation*}\n we know that there is a set $N_0$ of measure zero such that for all $s \\in Q\\setminus N_0$ there are only finitely many $k \\in \\N$ with $s \\in \\bigcup_{P\\in \\mc{S}^k}P$. Moreover by the Lebesgue differentiation theorem we have for any $P \\in \\mc{S}$ that\n $\\ind_{\\Omega_P} (s) \\leq M^{\\ms{D}(P)}(\\ind_{\\Omega_P})(s)$\nfor a.e. $s \\in P$. Thus\n\\begin{equation}\\label{eq:Npprop}\n \\Omega_P\\setminus N_P \\subseteq \\Omega_{P,1}\\subseteq \\Omega_{P,c_2} = \\bigcup_{P'\\in \\mc{S}_P} P'\n\\end{equation} for some set $N_P$ of measure zero. We define\n $N:= N_0 \\cup\\bigcup_{P \\in \\mc{S}}N_P,$ which is a set of measure zero.\n\n\n\n Fix $s \\in Q\\setminus N$ and take the largest $n \\in \\N$ such that $s \\in \\bigcup_{P\\in \\mc{S}^n}P$, which exists since $s \\notin N_0$. For $k =1,\\ldots,n$ let $P_k \\in \\mc{S}^k$ be the unique cube such that $s \\in P_k$ and note that by construction we have\n $P_{n} \\subseteq \\ldots \\subseteq P_1=Q.$\nUsing the localized $\\ell^r$-estimate of $T$ we split $\\nrm{T_Qf(s)}_Y^r $ into two parts\n\\begin{align*}\n \\nrmb{T_Qf(s)}_Y^r &\\leq C_r^r \\has{\\nrmb{T_{P_{n}}f(s)}_Y^r + \\sum_{k=1}^{n-1}\\nrmb{T_{P_k\\setminus P_{k+1}}f(s)}_Y^r}\\\\\n &=:C_{r}^r \\has{ \\hspace{2pt} \\text{\\framebox[15pt]{A}}+ \\text{\\framebox[15pt]{B}} \\hspace{2pt} }.\n\\end{align*}\n\n\n\nFor \\framebox[15pt]{A} note that $s \\notin N_{P_n}$ and $s \\notin \\bigcup_{P'\\in \\mc{S}^{n+1}}P' $ and therefore by \\eqref{eq:Npprop} we know that $s \\in P_n \\setminus \\Omega_{P_n}$. So by the definition of $\\Omega_{P_n}^1$\n\\begin{equation*}\n \\text{\\framebox[15pt]{A}} \\leq \\lambda^r \\, C_T^r\\, \\ipb{\\nrm{f}_X}_{p_0,\\alpha P_n}^r.\n\\end{equation*}\nFor $1\\leq k \\leq n-1$ we have by \\eqref{eq:OmegatoQ} and \\eqref{eq:Pjsize} that\n\\begin{align}\\label{eq:Pkmorethan14}\n\\begin{aligned}\n \\mu\\hab{P_{k+1} \\setminus (\\Omega_{P_{k+1}} \\cup \\Omega_{P_{k}}) }&\\geq \\mu(P_{k+1}) - \\mu(\\Omega_{P_{k+1}}) - \\mu(P_{k+1} \\cap \\Omega_{P_k})\\\\\n &\\geq \\mu(P_{k+1}) - \\frac{1}{2c_2}\\mu({P_{k+1}}) - \\frac{1}{2}\\mu(P_{k+1})>0,\n \\end{aligned}\n\\end{align}\n so $P_{k+1} \\setminus (\\Omega_{P_{k+1}} \\cup \\Omega_{P_k})$ is non-empty. Take $s' \\in P_{k+1} \\setminus (\\Omega_{P_{k+1}} \\cup \\Omega_{P_k})$, then we have\n\n\\begin{equation*}\n\\begin{aligned}\n \\nrmb{T_{P_{k}\\setminus P_{k+1}}f(s)}_Y &\\leq \\nrmb{T_{P_{k}\\setminus P_{k+1}}f(s)-T_{P_{k}\\setminus P_{k+1}}f(s')}_Y + \\nrmb{T_{P_{k}\\setminus P_{k+1}} f(s')}_Y\\\\\n&\\leq \\mc{M}_{T,P_k}^{\\#}f(s') + \\nrmb{T_{P_k}(s')}_Y+\\nrmb{T_{P_{k+1}}(s')}_Y\\\\\n&\\leq 2\\lambda \\,C_T\\,\\hab{\\ipb{\\nrm{f}_X}_{p_0,\\alpha P_{k}} + \\ipb{\\nrm{f}_X}_{p_0,\\alpha P_{k+1}}},\n\\end{aligned}\n\\end{equation*}\nwhere we used the definition of $\\mc{M}_{T,P_k}^{\\#}$ and $T_{P_{k+1}\\setminus P_k}$ in the second inequality and $s' \\notin \\Omega_{P_{k+1}} \\cup \\Omega_{P_k}$ in the third inequality. Using $(a+b)^r \\leq 2^{r-1}(a^r+b^r)$ for any $a,b>0$ this implies that\n\\begin{align*}\n \\text{\\framebox[15pt]{B}} &\\leq \\sum_{k=1}^{n-1} 2^r 2^{r-1} \\lambda^r \\,C_T^r\\, \\has{ \\ipb{\\nrm{f}_X}^r_{p_0,\\alpha P_k} + \\ipb{\\nrm{f}_X}^r_{p_0,\\alpha P_{k+1}}}\\\\\n &\\leq \\sum_{k=1}^{n} 4^r \\lambda^r \\,C_T^r\\, \\ipb{\\nrm{f}_X}^r_{p_0,\\alpha P_k}.\n\\end{align*}\nCombining the estimates for \\framebox[15pt]{A} and $\\text{\\framebox[15pt]{B}}$ we obtain\n\\begin{align*}\n \\nrmb{T_Qf(s)}_Y\n &\\leq 5\\,\\lambda\\, C_T\\, C_{r}\\, \\has{\\sum_{k=1}^n\\ipb{\\nrm{f}_X}_{p_0,\\alpha P_k}^r }^{1\/r}\\\\\n &= 5\\,\\lambda\\, C_T\\, C_{r}\\, \\has{\\sum_{P \\in \\mc{S}}\\ipb{\\nrm{f}_X}_{p_0,\\alpha P}^r \\ind_{P}(s)}^{1\/r}.\n\\end{align*}\nSince $s \\in Q\\setminus N$ was arbitrary and $N$ has measure zero, this inequality holds for a.e. $s \\in Q$.\nNoting that $\\lambda = 4c_1c_2$ and $c_1$ and $c_2$ only depend on $S$, $\\alpha$ and $\\ms{D}$ finishes the proof of the theorem.\n\\end{proof}\n\nAs announced Theorem \\ref{theorem:main} now follows directly from Theorem \\ref{theorem:localsparse} and a covering argument with Lemma \\ref{lemma:covering}.\n\n\\begin{proof}[Proof of Theorem \\ref{theorem:main}.] We will prove Theorem \\ref{theorem:main} in three steps: we will first show that the assumptions of Theorem \\ref{theorem:main} imply the assumptions of Theorem \\ref{theorem:localsparse}, then we will improve the local conclusion of Theorem \\ref{theorem:localsparse} to a global one and finally we will replace the averages over the dilation $\\alpha P$ in the conclusion of Theorem \\ref{theorem:localsparse} by the average over larger cubes $P'$.\n\nTo start let $\\ms{D}^1,\\ldots,\\ms{D}^m$ be as in Proposition \\ref{proposition:dyadicsystem} with parameters $c_0$, $C_0$, $\\delta$ and $\\gamma$, which only depend on $S$.\n\n\\textbf{Step 1:} For any $Q \\in \\ms{D}^1$ define\n$T_Q$\n by $T_Qf(s) := T(f\\ind_{\\alpha Q})(s)$ for $s \\in Q$. Then:\n \\begin{itemize}\n \\item $\\cbrace{T_Q}_{Q \\in \\ms{D}^1}$ is an $\\alpha$-localization family of $T$.\n \\item For any $Q \\in \\ms{D}^1$ and $f \\in L^{p_1}(S;X)$ we have\n\\begin{align*}\n \\mc{M}^{\\#}_{T,Q}f(s) &\\leq\\mc{M}^{\\#}_{T,\\alpha}(f\\ind_{\\alpha Q})(s), \\qquad s \\in Q.\n\\end{align*}\n So by the weak $L^{p_2}$-boundedness of $\\mc{M}^{\\#}_{T,\\alpha}$ it follows that $\\mc{M}^{\\#}_{T,Q}f$ is weak $L^{p_2}$-bounded uniformly in $Q \\in \\ms{D}^1$.\n \\item For any $f \\in L^p(S;X)$ and $Q_1,\\ldots,Q_n \\in \\ms{D}^1$ with $Q_n \\subseteq \\ldots \\subseteq Q_1$ the functions $f_k:= f\\ind_{\\alpha Q_k \\setminus \\alpha Q_{k+1}}$ for $k=1,\\ldots,n-1$ and $f_n:= f\\ind_{\\alpha Q_n}$ are disjointly supported. Thus by the $r$-sublinearity of $T$\n \\begin{equation*}\n\\nrmb{T_{Q_1}f(s)}_Y\\leq C_r \\has{\\nrmb{T_{Q_n}{f}(s)}_Y^r+\\sum_{k=1}^{n-1}\\nrmb{T_{Q_{k}\\setminus Q_{k+1}}f(s)}_Y^r}^{1\/r}, \\qquad s \\in Q_n.\n\\end{equation*}\n \\end{itemize}\nSo the assumptions of Theorem \\ref{theorem:localsparse} follow from the assumptions of Theorem \\ref{theorem:main}.\n\n\\textbf{Step 2:} Let $f \\in L^p(S;X)$ be boundedly supported. First suppose that $\\diam(S) = \\infty$ and let $E$ be a ball containing the support of $f$. By Lemma \\ref{lemma:covering} there is a partition $\\mc{D} \\subseteq \\ms{D}^1$ such that $E \\subseteq \\alpha Q$ for all $Q \\in \\mc{D}$. Thus by Theorem \\ref{theorem:localsparse} we can\nfind a $\\frac{1}{2}$-sparse collection of cubes $\\mc{S}_Q \\subseteq \\ms{D}^1(Q)$ for every $Q \\in \\mc{D}$ with\n\\begin{align}\n \\notag \\nrmb{Tf(s)}_Y &\\lesssim_{S,\\alpha} C_T\\, C_r\\, \\has{\\sum_{P \\in \\mc{S}_Q} \\ipb{\\nrm{f}_X}_{p_0,\\alpha P}^r\\ind_P(s)}^{1\/r}, \\qquad s \\in Q,\n\\intertext{where we used that $T_Qf = T(f \\ind_{\\alpha Q}) =Tf$ as $\\supp f \\subseteq \\alpha Q$.\nSince $\\mc{D}$ is a partition, $\\mc{S} := \\bigcup_{Q \\in \\mc{D}}S_Q$ is also a $\\frac{1}{2}$-sparse collection of cubes with}\n \\label{eq:sparsedomdilation} \\nrmb{Tf(s)}_Y &\\lesssim_{S,\\alpha} C_T\\, C_r\\, \\has{\\sum_{P \\in \\mc{S}} \\ipb{\\nrm{f}_X}_{p_0,\\alpha P}^r\\ind_P(s)}^{1\/r}, \\qquad s \\in S,\n\\end{align}\nIf $\\diam (S) < \\infty$, then \\eqref{eq:sparsedomdilation} follows directly from Theorem \\ref{theorem:localsparse} since $S \\in \\ms{D}$ in that case.\n\n\\textbf{Step 3:} For any $P \\in \\mc{S}$ with center $z$ and sidelength $\\delta^k$ we can find a $P' \\in \\ms{D}^j$ for some $1\\leq j\\leq m$ such that\n\\begin{equation*}\n\\alpha P = B(z,\\alpha C_0 \\cdot \\delta^k) \\subseteq P', \\qquad \\diam(P') \\leq \\gamma \\alpha C_0 \\cdot\\delta^k.\n\\end{equation*}\nTherefore there is a $c_1>0$ depending on $S$ and $\\alpha$ such that\n$$\\mu(P') \\leq \\mu \\hab{B(z,\\gamma \\alpha C_0 \\cdot \\delta^k)} \\leq c_1\\, \\mu \\hab{ B(z,c_0\\cdot \\delta^k) } \\leq c_1\\, \\mu(P).$$\nSo by defining $E_{P'}:= E_P$ we can conclude that the collection of cubes $\\mc{S}' := \\cbrace{P':P \\in \\mc{S}}$ is $\\frac{1}{2c_1}$-sparse. Moreover since $\\alpha P \\subseteq P'$ and $\\mu(P') \\leq c_1 \\, \\mu(P) \\leq c_1\\,\\mu(\\alpha P)$ for any $P \\in \\mc{S}$, we have\n\\begin{equation*}\n \\ipb{\\nrm{f}_X}_{p_0,\\alpha P} \\leq c_1 \\ipb{\\nrm{f}_X}_{p_0,P'}.\n\\end{equation*}\nCombined with \\eqref{eq:sparsedomdilation} this proves the sparse domination in the conclusion of Theorem \\ref{theorem:main}.\n\\end{proof}\n\n\\begin{remark}~\\label{remark:mainpointwise}\nThe assumption $\\alpha \\geq {3c_d^2\/\\delta}$ in Theorem \\ref{theorem:main} arises from the use of Lemma \\ref{lemma:covering}, which transfers the local sparse domination estimate of Theorem \\ref{theorem:localsparse} to the global statement of Theorem \\ref{theorem:main}. To deduce weighted estimates the local sparse domination estimate of Theorem \\ref{theorem:localsparse} suffices by testing against boundedly supported functions. However the operator norm of $\\mc{M}_{T,\\alpha}^{\\#}$ usually becomes easier to estimate for larger $\\alpha$, so the lower bound on $\\alpha$ is not restrictive.\n\\end{remark}\n\n\n\\subsection*{Further generalizations}\nOur main theorems, Theorem \\ref{theorem:main} and Theorem \\ref{theorem:localsparse}, allow for various further generalizations. One can for instance change the boundedness assumptions on $T$ and $\\mc{M}^{\\#}_{T,\\alpha}$, treat multilinear operators, or deduce domination by sparse forms for operators that do not admit a pointwise sparse estimate. We end this section by sketching some of these possible generalizations.\n\nIn \\cite[Section 3]{LO19} various variations and extensions of the main result in \\cite{LO19} are outlined. In particular they show:\n\\begin{itemize}\n \\item The sparse domination for an individual function follows from assumptions on the same function. This can be exploited to prove a sparse $T(1)$-type theorem, see \\cite[Section 4]{LO19}.\n \\item One can use certain Orlicz estimates to deduce sparse domination with Orlicz averages.\n \\item The method of proof extends to the multilinear setting (see also \\cite{Li18}).\n\\end{itemize}\n Our results can also be extended in these directions, which we leave to the interested reader. In the remainder of this section, we will explore some further directions in which our results can be extended.\n\n \\bigskip\n\nSparse domination techniques have been successfully applied to \\emph{fractional integral operators}, see e.g. \\cite{CB13, CB13b,Cr17,IRV18}. In these works sparse domination and sharp weighted estimates are deduced for e.g.\n the Riesz potentials, which for $0<\\alpha \\lambda\\, C_T \\,\\mu(\\alpha P)^{\\frac1{p_0}-\\frac1{q_0}}\\ipb{\\nrm{f}_X}_{p_0,\\alpha P}}\\\\\n \\Omega_{P}^2&:= \\cbraces{s \\in P: \\mc{M}_{T,P}^{\\#}(f)(s)> \\lambda \\,C_T\\,\\mu(\\alpha P)^{\\frac1{p_0}-\\frac1{q_0}}\\ipb{\\nrm{f}_X}_{p,\\alpha P}}\n\\end{align*}\nand then by the assumptions on $T$ and $\\mc{M}^{\\#}_{T,P}$ we have for $i=1,2$\n\\begin{align*}\n \\mu(\\Omega_{P}^i) &\\leq \\has{\\frac{\\nrm{f \\ind_{\\alpha P}}_{L^{p_0}(S;X)}}{\\lambda \\,\\mu(\\alpha P)^{\\frac1{p_0}-\\frac1{q_0}}\\ipb{\\nrm{f}_X}_{p_0,\\alpha P}}}^{q_0}\n = \\frac{\\ipb{\\nrm{f}_X}_{p_0,\\alpha P}^{q_0}}{\\lambda^{q_0}\\ipb{\\nrm{f}_X}_{p_0,\\alpha P}^{q_0}} \\mu(\\alpha P)\n \\leq \\frac{ c_1}{\\lambda} \\, \\mu(P).\n\\end{align*}\nwhich proves \\eqref{eq:OmegatoQ}. In Step 2 of the proof of Theorem \\ref{theorem:localsparse} one needs to keep track of the factor $\\mu(\\alpha P)^{\\frac1{p_0}-\\frac1{q_0}}$ in the estimates.\n\\end{proof}\n\nIn the celebrated paper \\cite{BFP16} by Bernic\\'ot, Frey and Petermichl, domination by \\emph{sparse forms} was introduced to treat operators falling outside the scope of Calder\\'on--Zygmund theory. This method was later adopted by Lerner in \\cite{Le19} into his framework to prove sparse domination for rough homogeneous singular integral operators. As our methods are based on Lerner's sparse domination framework, our main result can also be generalized to the sparse form domination setting.\n\nLet $(S,d,\\mu)$ be a space of homogeneous type with a dyadic system $\\ms{D}$, let $X$ and $Y$ be Banach spaces, $q \\in (1,\\infty)$, $p \\in [1,q)$ and $\\alpha \\geq 1$. For a bounded operator\n$$T\\colon L^{p}(S;X) \\to L^{p,\\infty}(S;Y)$$ with an $\\alpha$-localization family $\\cbrace{T_Q}_{Q \\in \\ms{D}}$ we define the \\emph{localized sharp grand $q$-maximal truncation operator} for $Q \\in \\ms{D}$ by\n \\begin{align*}\n &\\mc{M}_{T,Q,q}^{\\#}f(s):= \\\\&\\hspace{1cm}\\sup_{\\substack{Q'\\in \\ms{D}(Q):\\\\s \\in Q'}}\\, \\has{\\avint_{Q'}\\avint_{Q'} \\,\\nrmb{(T_{Q\\setminus Q'}) f(s')-(T_{Q\\setminus Q'}) f(s'')}_Y^q\\dd \\mu(s')\\dd \\mu(s'')}^{1\/q}.\n\\end{align*}\nNote that for $q=\\infty$ one formally recovers the operator $\\mc{M}_{T,Q}^{\\#}$.\n\nWe will prove a version of Theorem \\ref{theorem:localsparse} for operators for which the truncation operators $\\mc{M}_{T,Q,q}^{\\#}$ are bounded uniformly in $Q \\in \\ms{D}$ using sparse forms. Of course taking\n \\begin{align*}\nT_Qf(s) &:=T(f \\ind_{\\alpha Q})(s), \\qquad s \\in Q.\n\\end{align*}\nfor $Q \\in \\ms{D}$ as the $\\alpha$-localization family one can easily deduce a statement like Theorem \\ref{theorem:main} in this setting, which we leave to the interested reader.\n\n\\begin{theorem}\\label{theorem:sparseform}\nLet $(S,d,\\mu)$ be a space of homogeneous type with dyadic system $\\ms{D}$ and let $X$ and $Y$ be Banach spaces. Take $q_0 \\in (1,\\infty]$, $r \\in (0,q_0)$, $p_1,p_2 \\in [1,q_0)$, set $p_0:=\\max\\cbrace{p_1, p_2}$ and take $\\alpha \\geq 1$. Suppose that\n\\begin{itemize}\n \\item $T$ is a bounded operator from $L^{p_1}(S;X)$ to $L^{p_1,\\infty}(S;Y)$ with an $\\alpha$-localization family $\\cbrace{T_Q}_{Q \\in \\ms{D}}$.\n \\item $\\mc{M}_{T,Q,q_0}^{\\#}$ is bounded from $L^{p_2}(S;X)$ to $L^{p_2,\\infty}(S)$ uniformly in $Q \\in \\ms{D}$.\n \\item $T$ satisfies a localized $\\ell^r$-estimate.\n\\end{itemize}\n Then for any $f \\in L^{p_0}(S;X)$, $g \\in L^{\\ha{\\frac1r-\\frac1{q_0}}^{-1}}(S)$ and $Q\\in \\ms{D}$ there exists a $\\frac12$-sparse collection of dyadic cubes $\\mc{S}\\subseteq \\ms{D}(Q)$ such that\n \\begin{equation*}\n \\has{\\int_Q \\nrmb{T_Qf}_Y^r\\cdot \\abs{g}^r\\dd\\mu }^{1\/r} \\lesssim_{S,\\ms{D},\\alpha,r} C_T \\, C_r \\has{\\sum_{P \\in \\mc{S}} \\mu(P) \\ipb{\\nrm{f}_X}_{p_0,\\alpha P}^r \\ipb{\\abs{g}}_{\\frac{1}{\\frac1r-\\frac1{q_0}}, P}^r}^{1\/r}\n \\end{equation*}\nwith $C_T:={\\nrm{T}_{L^{p_1}\\to L^{p_1,\\infty}} + \\sup_{P \\in \\ms{D} } \\nrm{\\mc{M}_{T,P,q_0}^{\\#}}_{L^{p_2}\\to L^{p_2,\\infty}}}$ and $C_r$ the constant from the localized $\\ell^r$-estimate.\n\\end{theorem}\n\n\n\n\\begin{proof}\nWe construct the sparse collection of cubes $\\mc{S}$ exactly as in Step 1 of the proof of Theorem \\ref{theorem:localsparse}, using $\\mc{M}_{T,P,q_0}^{\\#}$ instead of $\\mc{M}_{T,P}^{\\#}$ in the definition of $\\Omega_P^2$. We will check that sparse form corresponding to $\\mc{S}$ satisfies the claimed domination property, which will roughly follow the same lines as Step 2 of the proof of Theorem \\ref{theorem:localsparse}.\n\nFix $f \\in L^{p_0}(S;X)$ and $g \\in L^{\\ha{\\frac1r-\\frac1{q_0}}^{-1}}(S)$. Note that for a.e. $s \\in Q$\nthere are only finitely many $k \\in \\N$ with $s \\in \\bigcup_{P\\in \\mc{S}^k}P$. So we can use the localized $\\ell^r$-estimate of $T$ to split\n\\begin{equation}\\label{eq:splitform}\\begin{aligned}\n \\int_Q \\nrmb{T_Qf}_Y^r\\cdot \\abs{g}^r &\\leq C_r^r \\sum_{k\\in \\N} \\sum_{P \\in \\mc{S}^k} \\has{\\int_{P \\setminus \\bigcup_{P' \\in \\mc{S}^{k+1}}P'} \\nrmb{T_Pf}_Y^r\\cdot \\abs{g}^r \\\\&\\hspace{2cm}+\\sum_{P' \\in \\mc{S}^{k+1}:P'\\subseteq P} \\int_{P'}\\nrmb{T_{P\\setminus P'}f}_Y^r\\cdot \\abs{g}^r}\\\\\n &=:C_{r}^r \\sum_{k\\in \\N} \\sum_{P \\in \\mc{S}^k} \\has{ \\hspace{2pt} \\text{\\framebox[20pt]{A$_{P}$}}+ \\text{\\framebox[20pt]{B$_{P}$}} \\hspace{2pt} }.\\end{aligned}\n\\end{equation}\nFix $k \\in \\N$ and $P \\in \\mc{S}^k$. As in the estimate for \\framebox[15pt]{A} in Step 2 of the proof of Theorem \\ref{theorem:localsparse}, we have\n\\begin{align*}\n \\text{\\framebox[20pt]{A$_{P}$}} &\\leq \\lambda^r \\, C_T^r\\, \\ipb{\\nrm{f}_X}_{p_0,\\alpha P}^r \\int_P\\abs{g}^r \\leq \\lambda^r \\, C_T^r\\, \\mu(P) \\ipb{\\nrm{f}_X}_{p_0,\\alpha P}^r \\ip{\\abs{g}}_{\\frac{1}{\\frac1r-\\frac1{q_0}}}^r,\n\\end{align*}\nusing H\\\"older's inequality in the second inequality.\nFor $P' \\in \\mc{S}^{k+1}$ such that $P'\\subseteq P$ we have as in \\eqref{eq:Pkmorethan14} that\n\\begin{align*}\n \\mu\\hab{P' \\setminus (\\Omega_{P'} \\cup \\Omega_{P}) } &\\geq \\frac14 \\mu(P').\n\\end{align*}\nTherefore we can estimate each of the terms in the sum in \\framebox[20pt]{B$_{P}$} as follows\n\\begin{equation*}\n\\begin{aligned}\n \\int_{P'}&\\nrmb{T_{P\\setminus P'}f}_Y^r\\cdot \\abs{g}^r\\\\ &\\leq 2^{r}\\int_{P'} \\avint_{P' \\setminus (\\Omega_{P} \\cup \\Omega_{P'})} \\nrmb{T_{P\\setminus P'}f(s)-T_{P\\setminus P'}f(s')}_Y^r\\cdot \\abs{g(s)}^r \\dd \\mu(s')\\dd \\mu(s)\\\\\n &\\hspace{2.2cm}+ 2^{r}\\int_{P'} \\avint_{P' \\setminus (\\Omega_{P} \\cup \\Omega_{P'})} \\nrmb{T_{P\\setminus P'}f(s')}_Y^r \\cdot \\abs{g(s)}^r \\dd \\mu(s')\\dd \\mu(s)\\\\\n &\\leq 2^{r+2} \\mu(P') \\inf_{s'' \\in P'} \\mc{M}_{T,P,{q_0}}^{\\#}f(s'')^r\\cdot \\ip{\\abs{g}}_{\\frac{1}{\\frac1r-\\frac1{q_0}}, P'}^r\\\\\n &\\hspace{1.5cm}+ 2^{2r} \\mu(P') \\avint_{P' \\setminus (\\Omega_{P} \\cup \\Omega_{P'})} \\nrmb{T_{P}f}_Y^r +\\nrmb{T_{P'}f}_Y^r \\dd \\mu \\cdot \\ip{\\abs{g}}_{r, P'}^r\\\\\n &\\leq 4^{r+2}\\lambda^r C_T^r\\,\\mu(P') \\hab{\\ipb{\\nrm{f}_X}_{p_0,\\alpha P}^r + \\ipb{\\nrm{f}_X}_{p_0,\\alpha P'}^r} \\ip{\\abs{g}}_{\\frac{1}{\\frac1r-\\frac1{q_0}}, P'}^r\n\\end{aligned}\n\\end{equation*}\nwhere we used\nH\\\"older's inequality and the definitions of $\\mc{M}_{T,P,{q_0}}^{\\#}$ and $T_{P\\setminus P'}$ in the second inequality and the definitions of $\\Omega_{P}$ and $\\Omega_{P'}$ in the third inequality. Furthermore we note that by H\\\"olders inequality we have\n\\begin{align*}\n \\sum_{\\substack{P' \\in \\mc{S}^{k+1}:\\\\P'\\subseteq P}} \\mu(P') \\, \\ip{\\abs{g}}_{\\frac{1}{\\frac1r-\\frac1{q_0}}, P'}^r &\\leq \\has{\\sum_{\\substack{P' \\in \\mc{S}^{k+1}:\\\\P'\\subseteq P}} \\int_{P'}\\abs{g}^{\\frac{1}{\\frac1r-\\frac1{q_0}}}\\dd \\mu}^{1-\\frac{r}{{q_0}}} \\cdot \\has{\\sum_{\\substack{P' \\in \\mc{S}^{k+1}:\\\\P'\\subseteq P}} \\mu(P')}^{r\/{q_0}}\\\\\n &\\leq \\has{ \\int_{P}\\abs{g}^{\\frac{1}{\\frac1r-\\frac1{q_0}}}\\dd \\mu}^{1-\\frac{r}{{q_0}}} \\cdot {\\mu(P)}^{r\/{q_0}}= \\mu(P) \\ip{\\abs{g}}_{\\frac{1}{\\frac1r-\\frac1{q_0}}, P}^r\n\\end{align*}\nThus for \\framebox[20pt]{B$_{P}$} we obtain\n\\begin{align*}\n \\text{\\framebox[20pt]{B$_{P}$}}\\leq 4^{r+2}\\lambda^r C_T^r &\\has{ \\mu(P) \\ipb{\\nrm{f}_X}_{p_0,\\alpha P}^r \\ip{\\abs{g}}_{\\frac{1}{\\frac1r-\\frac1{q_0}}, P}^r \\\\&+ \\sum_{P' \\in \\mc{S}^{k+1}:P'\\subseteq P} \\mu(P') \\ipb{\\nrm{f}_X}_{p_0,\\alpha P'}^r \\ip{\\abs{g}}_{\\frac{1}{\\frac1r-\\frac1{q_0}}, P'}^r}\n\\end{align*}\nPlugging this estimate and the estimate for \\framebox[20pt]{A$_{P}$} into \\eqref{eq:splitform} yields\n\\begin{equation*}\n \\int_Q \\nrmb{T_Qf}_Y^r\\cdot \\abs{g}^r \\dd \\mu \\leq 4^{r+3} \\lambda^r \\, C_T^r \\, C_r^r \\sum_{P \\in \\mc{S}} \\mu(P) \\ipb{\\nrm{f}_X}_{p_0,\\alpha P}^r \\ip{\\abs{g}}_{\\frac{1}{\\frac1r-\\frac1{q_0}}, P}^r.\n\\end{equation*}\nSince $\\lambda = 4c_1c_2$ and $c_1$ and $c_2$ only depend on $S$, $\\alpha$ and $\\ms{D}$, this finishes the proof of the theorem.\n\\end{proof}\n\n\n\n\\section{Weighted bounds for sparse operators}\\label{section:weights}\nAs discussed in the introduction, one of the main motivations to study sparse domination for an operator is to obtain (sharp) weighted bounds. In this section we will introduce Muckenhoupt weights and state weighted $L^p$-bounds for the sparse operators in the conclusions of Theorem \\ref{theorem:main} and Theorem \\ref{theorem:localsparse}, which are well-known in the Euclidean setting.\n\nLet $(S,d,\\mu)$ be a space of homogeneous type. A \\emph{weight} is a locally integrable function $w\\colon S \\to (0,\\infty)$. For $p \\in [1,\\infty)$, a Banach space $X$ and a weight $w$ the weighted Bochner space $L^p(S,w;X)$ is the space of all strongly measurable $f:S \\to X$ such that\n\\begin{equation*}\n \\nrm{f}_{L^p(S,w;X)}:= \\has{\\int_S\\nrm{f(s)}_X^pw\\dd \\mu}^{1\/p}<\\infty.\n\\end{equation*}\nFor $p\\in [1,\\infty)$ and a weight $w$ we say that $w$ lies in the \\emph{Muckenhoupt class $A_p$} and write $w\\in A_p$ if its \\emph{$A_p$-characteristic} satisfies\n\\begin{equation*}\n [w]_{A_p}:= \\sup_{B\\subseteq S} \\ip{w}_{1,B}\\ip{w^{-1}}_{\\frac{1}{p-1},B}<\\infty,\n\\end{equation*}\nwhere the supremum is taken over all balls $B \\subseteq S$ and the second factor is replaced by $\\esssup_B w^{-1}$ if $p=1$.\nFor an introduction to Muckenhoupt weights we refer to \\cite[Chapter 7]{Gr14a}.\n\n\\bigskip\n\nLet $p_0, r \\in [1,\\infty)$, $p \\in (p_0,\\infty)$, $w \\in A_{p\/p_0}$. We are interested in the boundedness on $L^p(S,w)$ of sparse operators of the form\n\\begin{equation}\\label{eq:sparseop2}\n f\\mapsto \\has{\\sum_{Q \\in \\mc{S}} \\ipb{\\abs{f}}_{p_0,Q}^r \\ind_{Q}}^{1\/r},\n\\end{equation}\nwhich appear in the conclusions of Theorem \\ref{theorem:main} and Theorem \\ref{theorem:localsparse}.\nIn the Euclidean case such bounds are thoroughly studied and most of the arguments extend directly to spaces of homogeneous type. For the convenience of the reader we will give a self-contained proof of the strong weighted $L^p$-boundedness of these sparse operators in spaces of homogeneous type, following the proof of \\cite[Lemma 4.5]{Le16}. For further results we refer to:\n\\begin{itemize}\n\\item Weak weighted $L^p$-boundedness (including the endpoint $p=p_0$), for the sparse operators in \\eqref{eq:sparseop2} can be found \\cite{HL18, FN18}.\n\\item More precise bounds in terms of two-weight $A_p$-$A_\\infty$-characteristics for various special cases of the sparse operators in \\eqref{eq:sparseop2} can be found in e.g. \\cite{FH18, HL18,HP13,LL16}.\n \\item Weighted bounds for the fractional sparse operators in Theorem \\ref{theorem:fractional} can be found in \\cite{FH18}\n \\item Weighted bounds for the sparse forms in Theorem \\ref{theorem:sparseform} can be found in \\cite{BFP16, FN18}.\n\\end{itemize}\n\n\\begin{proposition}\\label{proposition:weights}\n Let $(S,d,\\mu)$ be a space of homogeneous type, let $\\mc{S}$ be an $\\eta$-sparse collection of cubes and take $p_0, r \\in [1,\\infty)$.\nFor $p \\in (p_0,\\infty)$, $w \\in A_{p\/p_0}$ and $f \\in L^p(S,w)$ we have\n\\begin{align*}\n \\nrms{\\has{\\sum_{Q \\in \\mc{S}} \\ipb{\\abs{f}}_{p_0,Q}^r \\ind_{Q}}^{1\/r}}_{ L^p(S,w)} &\\lesssim [w]_{A_{p\/p_0}}^{\\max\\cbraceb{\\frac{1}{p-p_0},\\frac{1}{r}}} \\nrm{f}_{L^p(S,w)},\n\\end{align*}\nwhere\nthe implicit constant depends on $S,p_0,p,r$ and $\\eta$.\n\\end{proposition}\n\n\\begin{proof}\nWe first note that by Proposition \\ref{proposition:dyadicsystem} we may assume without loss of generality that $\\mc{S} \\subseteq \\ms{D}$, where $\\ms{D}$ is an arbitrary dyadic system in $(S,d,\\mu)$. Furthermore if $p -p_0\\leq r$ we have $\\max\\cbraceb{\\frac{1}{p-p_0},\\frac{1}{r}} = \\frac{1}{p-p_0}$. Since $\\ell^{p-p_0}\\hookrightarrow \\ell^r$, the case $p -p_0\\leq r$ follows from the case $p -p_0=r$, so without loss of generality we may also assume $p\\geq p_0+r$.\n\nFor a weight $u$ and a measurable set $E$ we define $u(E):=\\int_Eu\\dd \\mu$ and we denote the dyadic Hardy--Littlewood maximal operator with respect to the measure $u\\dd \\mu$ by $M^{\\ms{D},u}$, which\nis bounded on $L^p(S,u)$ for all $p \\in (1,\\infty)$ by Doob's maximal inequality (see e.g. \\cite[Theorem 3.2.2]{HNVW16}). Take $f \\in L^p(S,w)$, set $q := (p\/r)'=\\frac{p}{p-r}$ and take\n$$g \\in L^{q}(S,w^{1-q}) = \\hab{L^{p\/r}(S,w)}^*.$$\nThen we have by the disjointness of the $E_Q$'s associated to each $Q \\in \\mc{S}$\n\\begin{equation}\\label{eq:gsparse}\n\\begin{aligned}\n \\sum_{Q \\in \\mc{S}} w(E_Q) \\has{\\frac{\\mu(Q)}{w(Q)}}^{q} \\ipb{\\abs{g}}_{1,Q}^{q}\n &\\leq \\sum_{Q \\in \\mc{S}}\\int_{E_Q} M^{\\ms{D},w}(gw^{-1})^{q} w \\dd\\mu\\\\\n &\\leq \\nrmb{M^{\\ms{D},w}(gw^{-1})}_{L^{q}(S,w)}^{q}\\\\&\\lesssim_{p,r} \\nrm{g}_{L^{q}(S,w^{1-q})}^{q}\n\\end{aligned}\n\\end{equation}\nand similarly, setting $\\sigma:= w^{1-(p\/p_0)'}$, we have\n\\begin{equation}\\label{eq:fsparse}\n\\begin{aligned}\n \\sum_{Q \\in \\mc{S}} \\sigma(E_Q) \\has{\\frac{\\mu(Q)}{\\sigma(Q)}}^{\\frac{p}{p_0}} \\ipb{\\abs{f}^{p_0}}_{1,Q}^{{p}\/{p_0}}\n &\\leq \\nrmb{M^{\\ms{D},\\sigma}(\\abs{f}^{p_0}\\sigma^{-1})}_{L^{p\/p_0}(S,\\sigma)}^{p\/p_0}\\\\\n &\\lesssim_{p,p_0} \\nrm{f}_{L^{p}(S,w)}^{p}\n\\end{aligned}\n\\end{equation}\nusing $\\sigma \\cdot \\sigma^{-p_0\/p}=w$.\nDefine the constant\n\\begin{equation*}\n c_w := \\sup_{Q \\in \\ms{D}} \\frac{w(Q)^{1\/r}}{w(E_Q)^{\\frac1r-\\frac1p}} \\frac{\\sigma(Q)^{1\/p_0}}{\\sigma(E_Q)^{1\/p} } \\frac{1}{\\mu(Q)^{1\/p_0}},\n\\end{equation*}\nThen by H\\\"olders inequality, \\eqref{eq:gsparse} and \\eqref{eq:fsparse} we have\n\\begin{align*}\n \\int_S \\has{\\sum_{Q \\in \\mc{S}} \\ipb{\\abs{f}}_{p_0,Q}^r \\ind_{Q}}\\cdot g \\dd \\mu &= \\sum_{Q \\in \\mc{S}} \\mu(Q) \\ipb{\\abs{f}^{p_0}}_{1,Q}^{r\/p_0} \\ip{\\abs{g}}_{1,Q}\\\\\n &\\leq c_w^r \\sum_{Q \\in \\mc{S}} \\has{ \\sigma(E_Q)^{r\/p} \\has{\\frac{\\mu(Q)}{\\sigma(Q)}}^{r\/p_0} \\ipb{\\abs{f}^{p_0}}_{1,Q}^{r\/p_0}}\\\\&\\hspace{1cm}\\cdot\\has{ w(E_Q)^{{1\/q}}\\frac{\\mu(Q)}{w(Q)} \\ip{\\abs{g}}_{1,Q}}\\\\\n &\\lesssim_{p,p_0,r} c_w^r \\nrmb{f}_{L^{p}(S,w)}^{r} \\nrm{g}_{L^{q}(S,w^{1-q})}.\n\\end{align*}\nSo by duality it remains to show $c_w \\lesssim [w]_{A_{p\/p_0}}^{\\max\\cbraceb{\\frac{1}{p-p_0},\\frac{1}{r}}}$. Fix a $Q \\in \\ms{D}$ and note that by H\\\"olders's inequality we have\n\\begin{equation*}\n \\mu(Q)^{p\/p_0}\\leq \\eta^{p\/p_0} \\has{\\int_{E_Q}w^{p_0\/p} w^{-p_0\/p} \\dd \\mu}^{p\/p_0}\n \\leq \\eta^{p\/p_0} \\,w(E_Q)\\,\\sigma(E_Q)^{p\/p_0-1}.\n\\end{equation*}\nand thus\n\\begin{equation*}\n \\frac{w(Q)}{w(E_Q)}\\has{\\frac{\\sigma(Q)}{\\sigma(E_Q)}}^{{p\/p_0}-1} \\leq \\eta^{p\/p_0} \\frac{w(Q)}{\\mu(Q)}\\has{\\frac{\\sigma(Q)}{\\mu(Q)}}^{{p\/p_0}-1} \\lesssim_S \\eta^{p\/p_0}[w]_{A_{p\/p_0}}.\n\\end{equation*}\nTherefore we can estimate\n\\begin{align*}\n c_w &= \\sup_{Q \\in \\ms{D}} \\bracs{\\frac{w(Q)}{\\mu(Q)}\\has{\\frac{\\sigma(Q)}{\\mu(Q)}}^{\\frac{p}{p_0}-1}}^{\\frac1p} \\cdot \\bracs{\\has{\\frac{w(Q)}{w(E_Q)}}^{\\frac1r-\\frac1p} \\has{\\frac{\\sigma(Q)}{\\sigma(E_Q)}}^{\\frac1p}}\\\\\n &\\lesssim_S [w]_{A_{p\/p_0}}^{\\frac1p} \\, \\sup_{Q \\in \\ms{D}} \\bracs{\\frac{w(Q)}{w(E_Q)}\\has{\\frac{\\sigma(Q)}{\\sigma(E_Q)}}^{\\frac{p}{p_0}-1}}^{\\max \\cbraceb{{\\frac1r-\\frac1p}, \\frac1p\\frac{p_0}{p-p_0}}}\\\\\n &\\lesssim_{S,\\eta} [w]_{A_{p\/p_0}}^{\\frac1p+\\max \\cbraceb{{\\frac1r-\\frac1p}, \\frac1p\\frac{p_0}{p-p_0}} } = [w]_{A_{p\/p_0}}^{\\max\\cbraceb{\\frac{1}{p-p_0},\\frac{1}{r}}},\n\\end{align*}\nwhich finishes the proof.\n\\end{proof}\n\n\n\\section{Banach space geometry and $\\mc{R}$-boundedness}\\label{section:Banachspace}\nBefore turning to applications of Theorem \\ref{theorem:main} and Theorem \\ref{theorem:localsparse} in the subsequent sections, we first need to introduce some geometric properties of a Banach space $X$ and the $\\mc{R}$-boundedness of a family of operators.\n\n\\subsection{Type and cotype}Let $(\\varepsilon_k)_{k=1}^\\infty$ be a sequence of independent \\emph{Rademacher variables} on $\\Omega$, i.e. uniformly distributed random variables taking values in $\\cbrace{z \\in \\K:\\abs{z} = 1}$. We say that a Banach space $X$ has (Rademacher) type $p \\in [1,2]$ if for any $x_1,\\ldots,x_n \\in X$ we have\n\\begin{equation*}\n \\nrms{\\sum_{k=1}^n\\varepsilon_k x_k}_{L^2(\\Omega;X)} \\lesssim_{X,p} \\has{\\sum_{k=1}^n \\nrm{x_k}_X^p}^{1\/p},\n\\end{equation*}\nand say that $X$ has nontrivial type if $X$ has type $p>1$.\nWe say that $X$ has (Rademacher) cotype $q \\in [2,\\infty]$ if for any $x_1,\\ldots,x_n \\in X$ we have\n\\begin{equation*}\n \\has{\\sum_{k=1}^n \\nrm{x_k}_X^q}^{1\/q}\\lesssim_{X,q} \\nrms{\\sum_{k=1}^n\\varepsilon_k x_k}_{L^2(\\Omega;X)},\n\\end{equation*}\n and say that $X$ has finite cotype if $X$ has cotype $q<\\infty$. See \\cite[Chapter 7]{HNVW17} for an introduction to type and cotype.\n\n\\subsection{Banach lattices and $p$-convexity and $q$-concavity.}\nA Banach lattice is a partially ordered Banach space $X$ such that for $x,y \\in X$\n\\begin{equation*}\n \\abs{x} \\leq \\abs{y} \\Rightarrow \\nrm{x}_X \\leq \\nrm{y}_Y.\n\\end{equation*}\nOn a Banach lattice there are two properties that are closely related to type and cotype. We say that a Banach lattice is \\emph{$p$-convex} with $p \\in [1,\\infty]$ if for $x_1,\\ldots,x_n\\in X$\n\\begin{equation*}\n \\nrms{\\has{\\sum_{k=1}^n\\abs{x_k}^p}^{1\/p}}_{X} \\lesssim_{X,p} \\has{\\sum_{k=1}^n \\nrm{x_k}^p}^{1\/p},\n\\end{equation*}\nwhere the sum on the left-hand side is defined through the Krivine calculus. A Banach lattice is called \\emph{$q$-concave} for $q \\in [1,\\infty]$ if for $x_1,\\ldots,x_n\\in X$\n\\begin{equation*}\n \\has{\\sum_{k=1}^n \\nrm{x_k}^q}^{1\/q} \\lesssim_{X,q} \\nrms{\\has{\\sum_{k=1}^n\\abs{x_k}^q}^{1\/q}}_{X}.\n\\end{equation*}\nIf a Banach lattice has finite cotype then $p$-convexity implies type $p$. Conversely type $p$ implies $r$-convexity for all $1\\leq r< p$. Similar relations hold for cotype $q$ and $q$-concavity. We refer to \\cite[Chapter 1]{LT79} for an introduction to Banach lattices, $p$-convexity and $q$-concavity.\n\n\n\\subsection{The \\texorpdfstring{$\\UMD$}{UMD} property} We say that a Banach space $X$ has the {$\\UMD$ property} if the martingale difference sequence of any finite martingale in $L^p(\\Omega;X)$ is unconditional for some (equivalently all) $p \\in (1,\\infty)$. The $\\UMD$ property implies reflexivity, nontrivial type and finite cotype.\nFor an introduction to the theory of $\\UMD$ Banach spaces we refer the reader to \\cite[Chapter 4]{HNVW16} and \\cite{Pi16}.\n\n\\subsection{\\texorpdfstring{$\\mc{R}$}{R}-Boundedness} Let $X$ and $Y$ be Banach spaces and $\\Gamma \\subseteq \\mc{L}(X,Y)$. We say that $\\Gamma$ is $\\mc{R}$-bounded if for any $x_1,\\ldots,x_n$ and $T_1,\\ldots,T_n \\in \\Gamma$ we have\n\\begin{equation*}\n \\has{\\E\\nrmb{\\sum_{k=1}^n \\varepsilon_k T_kx_k}^2}^{1\/2} \\lesssim \\has{\\E\\nrmb{\\sum_{k=1}^n \\varepsilon_k x_k}^2}^{1\/2},\n\\end{equation*}\n where $(\\varepsilon_k)_{k=1}^\\infty$ is a sequence of independent {Rademacher variables}\nThe least admissible implicit constant is denoted by $\\mc{R}(\\Gamma)$. $\\mc{R}$-boundedness is a strengthening of uniform boundedness and is often a key assumption to prove boundedness of operators on Bochner spaces. We refer to \\cite[Chapter 8]{HNVW17} for an introduction to $\\mc{R}$-boundedness.\n\n\\section{The \\texorpdfstring{$A_2$}{A2}-theorem for operator-valued Calder\\'on--Zygmund operators in a space of homogeneous type} \\label{section:A2}\nThe $A_2$-theorem, first proved by Hyt\\\"onen in \\cite{Hy12} as discussed in the introduction, states that a Calder\\'on--Zygmund operator is bounded on $L^2(\\R^d,w)$ with a bound that depends linearly on the $A_2$-characteristic of $w$. From this sharp weighted bounds for all $p\\in (1,\\infty)$ can be obtained by sharp Rubio de Francia extrapolation \\cite{DGPP05}. Since its first proof by Hyt\\\"onen, the $A_2$-theorem has been extended in various directions. We mention two of these extensions relevant for the current discussion:\n\\begin{itemize}\n \\item The $A_2$-theorem for Calder\\'on--Zygmund operators on a geometric doubling metric space\n was first proven by Nazarov, Reznikov and Volberg \\cite{NRV13}, afterwards it was proven on a space of homogeneous type\n by Anderson and Vagharshakyan \\cite{AV14} (see also \\cite{An15}) using Lerner's mean oscillation decomposition method. It was further extended to the setting of ball bases by Karagulyan \\cite{Ka16}.\n \\item The $A_2$-theorem for vector-valued Calder\\'on--Zygmund operators with operator-valued kernel was proven by H\\\"anninen and Hyt\\\"onen \\cite{HH14}, using a suitable adapted version of Lerner's median oscillation decomposition.\n\\end{itemize}\nIn this section we will prove sparse domination for vector-valued Calder\\'on--Zygmund operators with operator-valued kernel on a space of homogeneous type. This yields the $A_2$-theorem for these Calde\\'ron--Zygmund operators, unifying the results from \\cite{AV14} and \\cite{HH14}.\n\nAs an application of this theorem, we will prove a weighted, anisotropic, mixed norm Mihlin multiplier theorem in the next section. We will also use it to study maximal regularity for parabolic partial differential equations in forthcoming work. In these applications $S$ is (a subset of) $\\R^d$ equipped with the anisotropic quasi-norm\n\\begin{equation}\\label{eq:anisotropic}\n \\abs{s}_{\\mbs{a}} := \\has{\\sum_{j=1}^d\\abs{s_j}^{2\/a_j}}^{1\/2}, \\qquad s \\in \\R^d.\n\\end{equation}\nfor some $\\mbs{a} \\in (0,\\infty)^d$ and the Lebesgue measure.\n\nIn a different direction our $A_2$-theorem can be applied in the study of fundamental harmonic analysis operators associated with various discrete and continuous orthogonal expansions, started by Muckenhoupt and Stein \\cite{MS65}. In the past decade there has been a surge of results in which such operators are proven to be vector-valued Calder\\'on--Zygmund operators on concrete spaces of homogeneous type. Weighted bounds are then often concluded using \\cite[Theorem III.1.3]{RRT86} or \\cite{RT88}. With our $A_2$-theorem these results can be made quantitative in terms of the $A_p$-characteristic. We refer to \\cite{BCN12,BMT07,CGRTV17,NS12,NS07} and the references therein for an overview of the recent developments in this field.\n\n\n\n\\bigskip\n\nLet $(S,d,\\mu)$ be a space of homogeneous type, $X$ and $Y$ be Banach spaces and let $$K\\colon(S \\times S)\\setminus\\cbrace{(s,s):s \\in S} \\to \\mc{L}(X,Y)$$\nbe strongly measurable in the strong operator topology.\nWe say that $K$ is a \\emph{Dini kernel} if there is a $c_K \\geq 2$ such that\n \\begin{align*}\n \\nrm{K(s,t)-K(s,t')} &\\leq\n \\omega \\has{\\frac{d(t,t')}{d(s,t)}}\\frac{1}{\\mu\\hab{{B(s,d(s,t))}}}, &&0\\varepsilon}\\omega\\has{\\frac{d(s',s'')}{d(s',t)}}\\frac{1}{\\mu\\hab{{B(s',d(s',t))}}} \\nrm{f(t)}_X\\dd \\mu(t)\\\\\n &\\leq \\sum_{j=0}^\\infty \\omega\\hab{c_K^{-1} 2^{-j}} \\int_{2^j\\varepsilon1$, which is implied by the Dini condition. See \\cite[Section 3]{Li18} for the definition of the $L^r$-H\\\"ormander condition and a comparison between the $L^r$-H\\\"ormander and the Dini condition.\n\\end{remark}\n\nNote that Theorem \\ref{theorem:A2} does not assume anything about the Banach spaces $X$ and $Y$ and is therefore applicable in situations where for example $Y=\\ell^\\infty$. However, in various applications $X$ and $Y$ will need to have the $\\UMD$ property in order to check the assumed\n weak $L^{p_0}$-boundedness of $T$ for some $p_0 \\in [1,\\infty)$. For instance, for a large class of operators the weak $L^{p_0}$-boundedness of $T$ can be checked using theorems like the $T(1)$-theorem or $T(b)$-theorem. See \\cite{Fi90} and \\cite{Hy14} for these theorems in the vector-valued setting, which assume the $\\UMD$ property for the underlying Banach space.\n\n If $S$ is Euclidean space, one can also use an (operator-valued) Fourier multiplier theorem to check the a priori $L^{p_0}$-bound, which we will discuss in the next section.\n\n\\section{The weighted anisotropic mixed-norm Mihlin multiplier theorem}\n Let $X$ and $Y$ be Banach spaces. Denote the space of $X$-valued Schwartz functions by $\\mc{S}(\\R^d;X)$ and the space of $Y$-valued tempered distributions by $\\mc{S}'(\\R^d;Y):= \\mc{L}(\\mc{S}(\\R^d);Y)$. To an $m \\in L^\\infty(\\R^d;\\mc{L}(X,Y))$ we associate the Fourier multiplier operator\n\\begin{equation*}\n T_m\\colon\\mc{S}(\\R^d;X) \\to \\mc{S}'(\\R^d;Y), \\qquad T_m f = (m\\widehat{f}\\vspace{2pt})^\\vee.\n\\end{equation*}\nSince $\\mc{S}(\\R^d;X)$ is dense in $L^p(\\R^d;X)$ and $L^p(\\R^d;Y)$ is continuously embedded into $\\mc{S}'(\\R^d;X)$, one may ask under which conditions on $m$ the operator $T_m$ extends to a bounded operator from $L^p(\\R^d;X)$ to $L^p(\\R^d;Y)$. If this is the case we call $m$ a bounded Fourier multiplier. We refer to \\cite[Chapter 5]{HNVW16} for an introduction to operator-valued Fourier multiplier theory.\n\nOne of the main Fourier multiplier theorems is the Mihlin multiplier theorem, first proven in the operator-valued setting by Weis in \\cite{We01b}.\nThe operator-valued Mihlin multiplier theorem of Weis has since been extended in many directions. Recently Fackler, Hyt\\\"onen and Lindemulder extended the operator-valued Mihlin multiplier theorem to a weighted, anisotropic, mixed norm setting in \\cite{FHL18}. This is for example useful in the study of spaces of smooth, vector-valued functions and has applications to parabolic PDEs with inhomogeneous boundary conditions, see e.g. \\cite{Li17b}.\n In \\cite{FHL18} the Mihlin multiplier theorem is shown using the following two approaches:\n\\begin{itemize}\n \\item Using a weighted Littlewood--Paley decomposition, they show a weighted, anisotropic, mixed-norm\n Mihlin multiplier theorem for rectangular $A_p$-weights, i.e. $A_p$-weights for which the defining supremum is taken over rectangles instead of balls.\n \\item Using Calder\\'on--Zygmund theory, they show a weighted, isotropic, non-mixed-norm\n Mihlin multiplier theorem for cubicular $A_p$-weights, i.e. $A_p$-weights for which the defining supremum is taken over cubes, which is equivalent to the definition using balls we used\nin Section \\ref{section:weights}.\n\\end{itemize}\nBoth approaches have their pros and cons. The result using a Littlewood--Paley decomposition only requires estimates of $\\partial^\\theta m$ for $\\theta \\in \\cbrace{0,1}$, whereas the approach using Calder\\'on--Zygmund theory also requires estimates of higher-order derivatives. On the other hand,\nthe class of rectangular $A_p$-weights is a proper subclass of the class of cubicular $A_p$-weights.\n\nIn applications it is be desirable to have the Mihlin multiplier theorem for cubicular $A_p$-weights in the anisotropic, mixed-norm setting as well. This would remove the need to distinguish between the isotropic and anisotropic setting in e.g. \\cite[(6) on p.64]{Li17b}.\nIn order to obtain the Mihlin multiplier theorem for cubicular $A_p$-weights in the anisotropic, mixed-norm setting one needs Calder\\'on--Zygmund theory in $\\R^d$ equipped with an anisotropic norm. Since this is a special case of a space of homogeneous type, we can use Theorem \\ref{theorem:A2} to supplement the results of \\cite{FHL18}, which will be the main result of this section.\n\n\\bigskip\n\nLet us introduce the anisotropic, mixed-norm setting. For $\\mbs{a} \\in (0,\\infty)^d$ let $\\abs{\\,\\cdot\\,}_{\\mbs{a}}$ be the anisotropic quasi-norm as in \\eqref{eq:anisotropic} and define $$\\R^d_{\\mbs{a}} := (\\R^d, \\abs{\\,\\cdot-\\cdot\\,}_{\\mbs{a}}, \\dd t),$$ where $\\dd t$ denotes Lebesgue measure. Then $\\R^d_{\\mbs{a}}$ is a space of homogeneous type and e.g.\n \\begin{align*}\n \\ms{D}&:= \\cbraces{\\prod_{j=1}^d \\hab{2^{-a_jn}([0,1) +m_j)} :\\mbs{m}\\in \\Z^d, n \\in \\Z}\n\\end{align*}\nis a dyadic system in $\\R^d_{\\mbs{a}}$.\n We write\n$\\abs{\\mbs{a}}_1:=\\sum_{j=1}^d a_j,$ $\\abs{\\mbs{a}}_\\infty := \\max_{j=1,\\ldots,d}a_j,$\nand for $\\theta \\in \\N^d$ we set $\\mbs{a} \\cdot\\theta := \\sum_{j=1}^d a_j \\theta_{j}$.\n\nTake $l \\in \\N$, $\\mathpzc{d} \\in \\N^l$ and consider the $\\mathpzc{d}$-decomposition of $\\R^d$:\n\\begin{equation*}\n\\R^{d}_{\\mz{d}}:= \\R^{\\mathpzc{d}_1} \\times \\ldots \\times \\R^{\\mathpzc{d}_l}.\n\\end{equation*}\nFor a $t \\in \\R^{d}_{\\mz{d}}$ we write $t = (t_1,\\ldots,t_l)$ with $t_j \\in \\R^{\\mz{d}_j}$ for $j=1,\\ldots,l$ and similarly we write $\\mbs{a} = (\\mbs{a}_1,\\ldots,\\mbs{a}_l)$.\nFor $\\mbs{p} \\in [1,\\infty)^l$, a vector of weights $\\mbs{w} \\in \\prod_{j=1}^l A_p(\\R^{\\mathpzc{d}_j}_{\n{\\mbs{a}_j}})$ and a Banach space $X$ we define the weighted mixed-norm Bochner space $L^{\\mbs{p}}(\\R^{d}_{\\mz{d}},\\mbs{w};X)$ as the space of all strongly measurable $f:\\R^{d}_{\\mz{d}} \\to X$ such that\n\\begin{equation*}\n \\nrm{f}_{L^{\\mbs{p}}(\\R^{d}_{\\mz{d}},\\mbs{w};X)}:= \\has{\\int_{\\R^{\\mz{d}_1}}\\ldots\\has{\\int_{\\R^{\\mz{d}_l}}\\nrm{f}_X^{p_l} w_l\\dd t_l}^{\\frac{p_{l-1}}{p_l}}\\ldots w_1 \\dd t_1}^{\\frac{1}{p_1}}\n\\end{equation*}\nis finite.\n\nWe are now ready to state and prove the announced weighted anisotropic, mixed-norm Mihlin multiplier theorem.\n\n\\begin{theorem}\\label{theorem:mihlin}\nLet $X$ and $Y$ be $\\UMD$ Banach spaces, set $N = \\abs{\\mbs{a}}_1+ \\abs{\\mbs{a}}_\\infty+ 1$\n and let $m \\in L^\\infty(\\R^d;\\mc{L}(X,Y))$. Suppose that for all $\\theta \\in \\N^d$ with $\\mbs{a} \\cdot\\theta \\leq N$ the distributional derivative $\\partial^\\theta m$ coincides with a continuous function on $\\R^d \\setminus \\cbrace{0}$ and we have the $\\mc{R}$-bound\n\\begin{equation*}\n \\mc{R}\\hab{\\cbraceb{\\abs{\\xi}_{\\mbs{a}}^{\\mbs{a} \\cdot \\theta}\\cdot \\partial^\\theta m(\\xi): \\xi \\in \\R^d}\\setminus\\cbrace{0}} \\leq C_m.\n\\end{equation*}\nfor some $C_m>0$. Then for every compactly supported $f \\in L^1(\\R^d;X)$ there exists an $\\eta$-sparse collection of anisotropic cubes $\\mc{S}$ such that\n\\begin{equation*}\n \\nrm{T_mf(s)}_Y\\lesssim_{X,Y,\\mbs{a}} \\,C_m\\, \\sum_{Q \\in \\mc{S}} \\ipb{\\nrm{f}_X}_{1,Q} \\ind_{Q}(s), \\qquad s \\in \\R^d.\n\\end{equation*}\nMoreover, for all $\\mbs{p} \\in (1,\\infty)^l$ and $\\mbs{w} \\in \\prod_{j=1}^l A_{p_j}(\\R^{\\mz{d}_j}_{\\mbs{a}_j})$ we have\n\\begin{align*}\n \\nrm{T_m}_{L^{\\mbs{p}}(\\R^d_{\\mz{d}},\\mbs{w};X) \\to L^{\\mbs{p}}(\\R^d_{\\mz{d}},\\mbs{w};Y)} &\\lesssim_{X,Y,\\mz{d},\\mbs{a},\\mbs{p}, \\mbs{w}} C_m.\n\\end{align*}\n\\end{theorem}\n\n\\begin{proof}\n We will check the conditions of Theorem \\ref{theorem:A2}. By \\cite[Theorem 3]{Hy07}, which trivially extends to the case $X \\neq Y$,\n we know that $T_m$ is bounded from $L^2(\\R^d;X)$ to $L^2(\\R^d;Y)$ with\n \\begin{equation*}\n \\nrm{T_m}_{L^2(\\R^d;X) \\to L^2(\\R^d;Y)} \\lesssim_{X,Y,\\mz{d},\\mbs{a}} C_m.\n \\end{equation*}\n By \\cite[Lemma 4.4.6 and 4.4.7]{Li14b} we know that $\\widecheck{m}$ coincides with a continuous function on $\\R^d \\setminus \\cbrace{0}$, which is bounded away from $0$ and $$K(t,s):= \\widecheck{m}(t-s), \\qquad t\\neq s$$ is a Dini kernel on the space of homogeneous type $\\R^d_{\\mbs{a}}$ with\n \\begin{equation*}\n \\omega(r) = C_{\\mbs{a}} \\cdot C_m \\cdot r^{\\min \\mbs{a}}, \\qquad r \\in [0,1].\n \\end{equation*}\n Now let $f \\in L^p(\\R^d;X)$ with compact support. Fix a $c \\in \\R^d \\setminus \\overline{\\supp f}$ and take $r>0$ such that $B(c,2r) \\cap \\overline{\\supp f} = \\varnothing$. Take a sequence $(f_n)_{n=1}^\\infty$ in $\\mc{S}(\\R^d;X)$ such that $\\overline{\\supp f_n} \\cap B(c,r)= \\varnothing$ and $f_n \\to f$ in $L^2(\\R^d;X)$. Then $Tf_n \\to Tf$ in $L^2(\\R^d;X)$ and, by passing to a subsequence if necessary, we have $f_n(t) \\to f(t)$ and $Tf_n(t) \\to Tf(t)$ for a.e. $t \\in \\R^d$. Fix $n \\in \\N$, then we have for all $\\varphi \\in C_c^\\infty(\\R^d \\setminus \\overline{\\supp f_n})$\n \\begin{align*}\n \\ip{T_m f_n, \\varphi} &= \\int_{\\R^d} m(s)\\widehat{f_n}(s) \\widecheck{\\varphi}(s)\\dd s\\\\\n &= \\int_{\\R^d}\\widecheck{m}(s) \\int_{\\R^d}f_n(t-s) \\varphi(t) \\dd t\\dd s \\\\\n &= \\int_{\\R^d} \\int_{\\R^d} K(t,s)f_n(s)\\dd s \\, \\varphi(t) \\dd t\n \\end{align*}\n from which we obtain for a.e. $t \\in B(c,r)$\n \\begin{equation*}\n T_m f(t) = \\lim_{n\\to \\infty} T_m f_n(t) = \\lim_{n \\to \\infty} \\int_{\\R^d}K(t,s)f_n(s)\\dd s = \\int_{\\R^d}K(t,s)f(s)\\dd s\n \\end{equation*}\n Covering $\\R^d \\setminus \\overline{\\supp f}$ by countably many such balls, we conclude that $T_m$ has kernel $K$. Therefore the sparse domination, as well as the weighted estimate in case $l=1$, follows from Theorem \\ref{theorem:A2}.\n\nTo conclude the proof we will show the case $l=2$, the general case follows by iterating the argument. Take $\\mbs{p} \\in (1,\\infty)^2$ and $\\mbs{w} \\in A_{p_1}(\\R^{\\mz{d}_1}_{\\mbs{a}_1}) \\times A_{p_2}(\\R^{\\mz{d}_2}_{\\mbs{a}_2})$. For $v_1 \\in A_{p_2}(\\R^{\\mz{d}_1}_{\\mbs{a}_1})$ note that\n $${v}(t):= v_1(t_1)\\cdot w_2(t_2) , \\qquad t \\in \\R^{\\mz{d}_1} \\times \\R^{\\mz{d}_2}$$ belongs to $A_{p_2}(\\R^d_{\\mbs{a}})$, so by the case $l=1$ we have\n \\begin{equation*}\n \\nrm{T_mf}_{L^{p_2}(\\R^d,{v};Y)} \\lesssim_{X,Y,\\mz{d},\\mbs{a},p_2, {v}} C_m \\cdot \\nrm{f}_{L^{p_2}(\\R^d,{v};X)}\n \\end{equation*}\n for all $f \\in L^{p_2}(\\R^d, {v};X)$.\nSince balls in $\\R^{\\mz{d}_2}$ with respect to the quasi-metric $\\abs{\\,\\cdot-\\cdot\\,}_{\\mbs{a}_2}$ form a Muckenhoupt basis, we can use Rubio de Francia extrapolation as in \\cite[Theorem 3.9]{CMP11} on the extrapolation family\n \\begin{equation*}\n \\cbraces{\\hab{\\nrm{T_mf}_{L^{p_2}(\\R^{\\mz{d}_2},w_2;Y)},\\nrm{f}_{L^{p_2}(\\R^{\\mz{d}_2},w_2;X)}}:f \\colon \\R^d \\to X \\text{ simple}}\n \\end{equation*}\nto deduce\n\\begin{equation*}\n \\nrm{T_mf }_{L^{\\mbs{p}}(\\R^{d}_{\\mz{d}} ,\\mbs{w};Y)} \\lesssim_{X,Y,\\mz{d},\\mbs{a},\\mbs{p},\\mbs{w}} C_m \\nrm{f }_{L^{\\mbs{p}}(\\R^{d}_{\\mz{d}} ,\\mbs{w};X) }\n\\end{equation*}\nfor all simple $f$, which implies the result by density.\n\\end{proof}\n\n\\begin{remark}~\n\\begin{enumerate}[(i)]\n\\item The weight dependence of the implicit constant in Theorem \\ref{theorem:mihlin} in the case $l=1$ is $[w]_{A_{p}(\\R^d_{\\mbs{a}})}^{\\max\\cbrace{\\frac{1}{p-1},1}}$, which is sharp. For $l\\geq 2$ the dependence our proof yields is more complicated and not sharp for all choices of $\\mbs{p} \\in (1,\\infty)^l$.\n \\item In the proof of Theorem \\ref{theorem:mihlin} we only use the $\\mc{R}$-boundedness of the set\n $$\n \\cbraceb{\\abs{\\xi}_{\\mbs{a}}^{\\mbs{a} \\cdot \\theta}\\cdot \\partial^\\theta m(\\xi): \\xi \\in \\R^d\\setminus\\cbrace{0}}\n $$\n for $\\theta \\in \\cbrace{0,1}^d$. For all other $\\theta \\in \\N^d$ with $\\mbs{a} \\cdot \\theta \\leq N$ it suffices to know uniform boundedness of this set.\n\\item One could reduce the number of derivatives necessary in Theorem \\ref{theorem:mihlin}, by arguing as in \\cite{Hy04} instead of using \\cite[Lemma 4.4.6 and 4.4.7]{Li14b}. See also \\cite[Section 6]{FHL18}.\n\\item Using the sparse domination of Theorem \\ref{theorem:mihlin} one can also deduce two-weight estimates for $T_m$ as in \\cite[Section 6]{FHL18}.\n\\end{enumerate}\n\\end{remark}\n\n\\section{The Rademacher maximal function}\\label{section:maximal}\nIn this section we will apply Theorem \\ref{theorem:localsparse} to the Rademacher maximal function. The proofs will illustrate very nicely how the geometry of the Banach space plays a role in deducing the localized $\\ell^r$-estimate for this operator. In particular, we will use the type of a Banach space $X$ to deduce the localized $\\ell^r$-estimate for the Rademacher maximal function.\n\n\n The Rademacher maximal function was introduced by Hyt\\\"onen, McIntosh and Portal in \\cite{HMP08} as a vector-valued generalization of Doob's maximal function that takes into account the different ``directions'' in a Banach space. They used the Rademacher maximal function to prove a Carleson's embedding theorem for vector-valued functions in connection to Kato's square root problem in Banach spaces. The Carleson's embedding theorem for vector-valued functions has since found many other applications, like the local vector-valued $T(b)$ theorem (see \\cite{HV15}).\n\n\nLet $(S,d,\\mu)$ be a space of homogeneous type with a dyadic system $\\ms{D}$ and let $X$ be a Banach space.\nFor $f \\in L^1_{\\loc}(S;X)$ we define the \\emph{Rademacher maximal function} by\n\\begin{align*}\n M_{\\Rad}^{\\ms{D}}f(s)&:= \\sup\\cbraces{\\nrms{\\sum_{Q \\in \\ms{D}:s \\in Q}\\varepsilon_Q\\lambda_Q \\ip{f}_{1,Q}}_{L^2(\\Omega;X)}: \\\\&\\hspace{2cm}(\\lambda_Q)_{Q \\in \\ms{D}} \\text{ finitely non-zero with } \\sum_{Q \\in \\ms{D}}\\abs{\\lambda_Q}^2\\leq 1},\n\\end{align*}\nwhere $(\\varepsilon_Q)_{Q \\in \\ms{D}}$ is a Rademacher sequence on $\\Omega$. One can interpret this maximal function as Doob's maximal function\n \\begin{equation*}\n f^*(s): =\\sup_{Q \\in \\ms{D}:s \\in Q} \\nrmb{\\ip{f}_{1,Q}}_X, \\qquad s\\in S,\n\\end{equation*}\n with the uniform bound over the $\\ip{f}_{1,Q}$'s replaced by the $\\mc{R}$-bound. Here the $\\mc{R}$-bound of a set $U\\subseteq X$ is the $\\mc{R}$-bound of the family of operators $T_x:\\C \\to X$ given by $\\lambda \\mapsto \\lambda x$ for $x \\in U$.\n\nWe say that the Banach space $X$ has the $\\RMF$ property if $M_{\\Rad}^{\\ms{D}[0,1)}$ is a bounded operator on $L^p([0,1);X)$ for some $p \\in (1,\\infty)$, where\n$$\\ms{D}[0,1) := \\cbraceb{2^{-k} [j-1,j): k\\in \\N\\cup\\cbrace{0},\\, j=1,\\ldots,2^k}$$\nis the standard dyadic system in $[0,1)$.\nIt was shown by Hyt\\\"onen, McIntosh and Portal \\cite[Proposition 7.1]{HMP08} that this implies boundedness for all $p \\in (1,\\infty)$ and by Kemppainen \\cite[Theorem 5.1]{Ke11} that this implies boundedness of $M_{\\Rad}^{\\ms{D}}$ on $L^p(S;X)$ for any space of homogeneous type $(S,d,\\mu)$ with a dyadic system $\\ms{D}$.\n\nThe relation of $\\RMF$ property to other Banach space properties is not yet fully understood. However, we do have some necessary and sufficient conditions:\n\\begin{itemize}\n \\item The $\\mc{R}$-bound of a set $U\\subseteq X$ is equivalent to the uniform bound of that set if and only if $X$ has type $2$ (see \\cite[Proposition 8.6.1]{HNVW17}). Therefore if $X$ has type $2$ we have for any $f \\in L^1_{\\loc}([0,1);X)$ that $M_{\\Rad}^{\\ms{D}[0,1)}f \\lesssim M^{\\ms{D}[0,1)}(\\nrm{f}_X)$, so $X$ has the $\\RMF$ property.\n \\item Any $\\UMD$ Banach lattice has the $\\RMF$ property, see also the discussion related to the Hardy--Littlewood maximal operator at the end of this section.\n \\item Non-commutative $L^p$-spaces for $p \\in (1,\\infty)$ have the $\\RMF$ property, see \\cite[Corollary 7.6]{HMP08}.\n \\item The $\\RMF$ property implies nontrivial type, see \\cite[Proposition 4.2]{Ke11}.\n\\end{itemize}\nIt is an open problem whether nontrivial type or even the $\\UMD$ property implies the $\\RMF$ property.\n\n\\bigskip\n\nWeighted bounds for the Rademacher maximal function in the Euclidean setting were studied by Kemppainen \\cite[Theorem 1]{Ke13}. The proof was based on a good-$\\lambda$ inequality, which does not give sharp quantitative estimates in terms of the weight characteristic. Using Theorem \\ref{theorem:localsparse} we can prove sharp quantitative weighted estimates for the Rademacher maximal function through sparse domination. We will not consider the situation in which $X$ has type $2$, as this case follows directly from\n$M_{\\Rad}^{\\ms{D}[0,1)}f \\lesssim M^{\\ms{D}[0,1)}(\\nrm{f}_X)$ and the well-known sparse domination for the Hardy--Littlewood maximal operator.\n\nWe will need a version of the Rademacher maximal function for finite collections of cubes. For a subcollection of cubes $\\mc{D} \\subseteq \\ms{D}$ we define $M_{\\Rad}^{\\mc{D}}$ analogous to $M_{\\Rad}^{\\ms{D}}$.\n\n\\begin{theorem}\\label{theorem:RMF}\n Let $(S,d,\\mu)$ be a space of homogeneous type with a dyadic system $\\ms{D}$ and let $X$ be a Banach space with the $\\RMF$ property. Assume that $X$ has type $r$ for $r \\in [1,2)$. For any finite collection of cubes $\\mc{D} \\subseteq \\ms{D}$ and $f \\in L^1(S;X)$ there exists an $\\frac12$-sparse collection of cubes $\\mc{S}\\subseteq \\ms{D}$ such that\n \\begin{equation*}\n M_{\\Rad}^{\\mc{D}}f(s) \\lesssim_{X,S,\\ms{D},r} \\has{\\sum_{Q \\in \\mc{S}}\\ipb{\\nrm{f}_X}_{1,Q}^{(\\frac1r-\\frac12)^{-1}}\\ind_Q(s)}^{\\frac1r-\\frac12}, \\qquad s\\in S\n \\end{equation*}\nMoreover, for all $p \\in (1,\\infty)$ and $w\\in A_p$ we have\n \\begin{align*}\n \\nrmb{M_{\\Rad}^{\\ms{D}}}_{L^p(S,w;X) \\to L^p(S,w;X)} &\\lesssim_{X,S,\\ms{D},p,r} [w]_{A_p}^{\\max\\cbraceb{\\frac{1}{p-1},\\frac{1}{r}-\\frac{1}{2}}}.\n \\end{align*}\n\\end{theorem}\n\n\\begin{proof}\nFix a finite collection of cubes $\\mc{D} \\subseteq \\ms{D}$. By \\cite[Proposition 6.1]{Ke11} $M_{\\Rad}^{\\mc{D}}$ is weak $L^1$-bounded.\nWe will view $M_{\\Rad}^{\\mc{D}}$ as a bounded operator\n \\begin{equation*}\n M_{\\Rad}^{\\mc{D}}: L^1(S;X) \\to L^{1,\\infty}(S; \\mc{L}(\\ell^2(\\mc{D}),L^2(\\Omega;X)))\n \\end{equation*}\n given by\n \\begin{equation*}\n M_{\\Rad}^{\\mc{D}}f(s) = \\has{(\\lambda_Q)_{Q \\in \\mc{D}} \\mapsto \\sum_{Q \\in \\ms{D}:s \\in Q} \\varepsilon_Q \\lambda_Q \\,\\ip{f}_{1,Q}}, \\qquad s \\in S,\n \\end{equation*}\n where $(\\varepsilon_Q)_{Q \\in \\mc{D}}$ is a Rademacher sequence on $\\Omega$.\n\n\n For $Q \\in \\ms{D}$ set\n $$\\mc{D}(Q):=\\cbrace{P \\in\\mc{D}:P \\subseteq Q}$$\n and define $T_Q:= M_{\\Rad}^{\\mc{D}(Q)}$. Then $\\cbrace{T_Q}_{Q \\in \\ms{D}}$ is a $1$-localization family for $M_{\\Rad}^{\\mc{D}}$.\n Furthermore we have for $f \\in L^1(S;X)$ and $s \\in Q \\in \\ms{D}$ that\n \\begin{align*}\n \\mc{M}^{\\#}_{M_{\\Rad}^{\\mc{D}},Q}f(s) &= \\sup_{\\substack{Q' \\in \\ms{D}(Q):\\\\s \\in Q'}} \\esssup_{s' ,s'' \\in Q'} \\nrmb{T_{Q\\setminus Q'}f(s')-T_{Q\\setminus Q'}f(s'')}_{\\mc{L}(\\ell^2(\\ms{D}),L^2(\\Omega;X))}\\\\&=0\n \\end{align*}\n where the second step follows from the fact that $T_{Q\\setminus Q'}f = M_{\\Rad}^{\\mc{D}(Q)\\setminus \\mc{D}(Q')}f$ is constant on $Q'$. So $\\mc{M}^{\\#}_{M_{\\Rad}^{\\mc{D}},Q}$ is trivially bounded from $L^1(S;X)$ to $L^{1,\\infty}(S)$.\n\nSet $q:=(\\frac1r-\\frac12)^{-1}$. To check the localized $\\ell^q$-estimate for $M_{\\Rad}^{\\mc{D}}$ take $Q_1,\\ldots,Q_n \\in \\ms{D}$ with $Q_n\\subseteq \\ldots\\subseteq Q_1$. Let $(\\lambda_Q)_{Q \\in \\mc{D}} \\in \\ell^2(\\mc{D})$ be of norm one and let $(\\varepsilon_Q)_{Q \\in \\mc{D}}$ and $(\\varepsilon'_k)_{k=1}^n$ be Rademacher sequences on $\\Omega$ and $\\Omega'$ respectively. Define for $k=1,\\ldots,n-1$\n\\begin{equation*}\n \\lambda_k := \\has{\\sum_{Q \\in \\mc{D}\\ha{Q_{k+1}}\\setminus \\mc{D}\\ha{Q_k}}\n\\abs{\\lambda_Q}^2}^{1\/2}, \\qquad \\lambda_{n} := \\has{\\sum_{Q \\in \\mc{D}\\ha{Q_n}} \\abs{\\lambda_Q}^2}^{1\/2}\n\\end{equation*}\n Then for $f \\in L^1(S;X)$, setting $f_Q := \\varepsilon_Q \\lambda_Q\\ip{f}_{1,Q}$, we have\n \\begin{align*}\n \\nrms{\\sum_{Q \\in \\mc{D}\\ha{Q_1}} &\\varepsilon_Q \\lambda_Q\\ip{f}_{1,Q}}_{L^2(\\Omega;X)}\\\\\n &= \\nrms{\\varepsilon_n'\\sum_{Q \\in \\mc{D}\\ha{Q_n}} f_Q+ \\sum_{k=1}^{n-1}\\varepsilon_{k}'\\sum_{Q\\in \\mc{D}\\ha{Q_{k+1}}\\setminus \\mc{D}\\ha{Q_k}} f_Q}_{L^2(\\Omega\\times \\Omega';X)}\\\\\n &\\lesssim_{X,r} \\has{\\lambda_n^r \\nrms{\\sum_{Q \\in \\mc{D}\\ha{Q_n}} \\lambda_n^{-1}f_Q}^r_{L^2(\\Omega;X)} \\\\&\\hspace{1cm}+ \\sum_{k=1}^{n-1}\\lambda_{k}^r \\nrms{\\sum_{Q\\in \\mc{D}\\ha{Q_{k+1}}\\setminus \\mc{D}\\ha{Q_k}} \\lambda_{k}^{-1}f_Q}_{L^2(\\Omega;X)}^r}^{1\/r}\\\\\n &\\leq\\has{ \\nrms{\\sum_{Q \\in \\mc{D}\\ha{Q_n}} \\varepsilon_Q \\lambda_n^{-1} \\lambda_Q\\ip{f}_{1,Q}}^{q}_{L^2(\\Omega;X)} \\\\&\\hspace{1cm}+ \\sum_{k=1}^{n-1} \\nrms{\\sum_{Q\\in \\mc{D}\\ha{Q_{k+1}}\\setminus \\mc{D}\\ha{Q_k}} \\varepsilon_Q \\lambda_{k}^{-1}\\lambda_Q\\ip{f}_{1,Q}}_{L^2(\\Omega;X)}^{q}}^{1\/q},\n \\end{align*}\nusing randomization (see \\cite[Proposition 6.1.11]{HNVW17}) in the first step, type $r$ of $X$ in the second step, and H\\\"older's inequality and $\\sum_{k=1}^n \\lambda_k^2=1$ in the last step. Noting that for $k=1,\\ldots,n-1$\n\\[\\sum_{Q\\in \\mc{D}(Q_{k+1})\\setminus \\mc{D}(Q_k)} \\abs{\\lambda_{k}^{-1}{\\lambda_Q}}^2=1,\\qquad \\sum_{Q\\in {\\mc{D}(Q_n)} } \\abs{\\lambda_{n}^{-1}{\\lambda_Q}}^2=1,\\] this implies the localized $\\ell^{q}$-estimate for $M_{\\Rad}^{\\mc{D}}$.\n\nHaving checked all assumptions of Theorem \\ref{theorem:localsparse} for $M_{\\Lat}^{\\mc{D}}$ it follows that for any $Q \\in \\mc{D}$ there is a $\\frac{1}{2}$-sparse collection of cubes $\\mc{S}_Q\\subseteq \\ms{D}(Q)$ such that\n\\begin{align*}\n \\nrmb{T_Q(s)}_Y\\lesssim_{X,S,\\ms{D},r} ,\n \\has{ \\sum_{P \\in \\mc{S}} \\ipb{\\nrm{f}_X}_{p,\\alpha P}^r \\ind_P(s)}^{1\/r},\\qquad s \\in Q.\n \\end{align*}\nLet $\\mc{D}'$ be the maximal cubes (with respect to set inclusion) of $\\mc{D}$, which are pairwise disjoint. Then $\\mc{S}:=\\bigcup_{Q \\in \\mc{D}'} \\mc{S}_Q$ is a $\\frac{1}{2}$-sparse collection of cubes that satisfies the claimed sparse domination\nas $T_Q(s) = M_{\\Rad}^{\\mc{D}}f(s)$ for any $s\\in Q \\in \\mc{D}'$ and $M_{\\Rad}^{\\mc{D}}f$ is zero outside $\\bigcup_{Q \\in \\mc{D}'}Q$. The weighted bounds follow from Proposition \\ref{proposition:weights} and the monotone convergence theorem.\n\\end{proof}\n\nLet us check that the weighted estimate in Theorem \\ref{theorem:RMF}, and consequently also the sparse domination in Theorem \\ref{theorem:RMF}, is sharp. We take $X=\\ell^r$ for $r \\in (1,2)$, a prototypical Banach space with type $r$. Since $\\mc{R}$-bounds are stronger than uniform bounds, we note that for any strongly measurable $f \\colon [0,1) \\to \\ell^q$ we have\n\\begin{equation*}\n f^*(s) \\leq M_{\\Rad}^{\\ms{D}[0,1)}f(s),\\qquad s \\in [0,1).\n\\end{equation*}\nThus by the corresponding result for Doob's maximal operator (see \\cite[Proposition 3.2.4]{HNVW16}), we have for $p \\in (1,\\infty)$\n\\begin{equation}\\label{eq:radlowerbound}\n \\nrmb{M_{\\Rad}^{\\ms{D}[0,1)}}_{L^p([0,1);\\ell^r) \\to L^p([0,1);\\ell^r) } \\geq \\frac{p}{p-1}\n\\end{equation}\nNow let $(e_n)_{n=1}^\\infty$ be the canonical basis of $\\ell^r$ and define\n\\begin{equation*}\n f(s):= \\sum_{n=1}^\\infty \\ind_{[2^{-n},2^{-n+1})}(s) e_n, \\qquad s \\in [0,1).\n\\end{equation*}\nFor $p \\in (1,\\infty)$ we have\n\\begin{equation*}\n \\nrm{f}_{L^p([0,1);\\ell^r)} = 1.\n\\end{equation*}\nTo compute $\\nrm{M_{\\Rad}^{\\ms{D}[0,1)}f}_{L^p([0,1);\\ell^r)}$ set $I_j:= [0,2^{-j+1}]$, take $s \\in $ and let $m \\in \\N$ be such that $2^{-m}\\leq s \\leq 2^{-m+1}$. Then we have, using $\\lambda_{I_j} = m^{-1\/2}$ for $j=1,\\ldots,m$ and the Khintchine--Maurey inequalities (see \\cite[Theorem 7.2.13]{HNVW17}), that\n\\begin{align*}\n M_{\\Rad}^{\\ms{D}[0,1)}f(s) &\\geq \\frac{1}{m^{1\/2}} \\nrms{\\sum_{j=1}^m \\varepsilon_j\\ip{f}_{1,I_j}}_{L^2(\\Omega;\\ell^r)}\n \\gtrsim \\frac{1}{m^{1\/2}}\\nrms{\\has{\\sum_{j=1}^m \\ip{f}_{1,I_j}^2}^{1\/2}}_{\\ell^r}\\\\\n &\\gtrsim \\frac{1}{m^{1\/2}}\\nrms{\\sum_{j=1}^m e_j}_{\\ell^r}\n \\gtrsim m^{1\/r-1\/2} \\gtrsim {\\log(1\/s)}^{1\/r-1\/2}.\n\\end{align*}\nTherefore we obtain\n\\begin{align*}\n \\nrmb{M_{\\Rad}^{\\ms{D}[0,1)}f}_{L^p([0,1);\\ell^r)}&\\gtrsim \\has{\\int_0^1 {\\log(1\/s)}^{p\/r-p\/2} \\dd s}^{1\/p}\n \\\\&= \\has{\\int_1^\\infty x^{p\/r-p\/2} \\ee^{-x}\\dd x}^{1\/p} \\\\&\\geq \\has{\\sum_{n=2}^\\infty n^{p\/r-p\/2}\\ee^{-n}}^{1\/p} \\\\&\\gtrsim p^{1\/r-1\/2},\n\\end{align*}\nwhere we drop all terms except $n = \\ceil{p}$ in the last step. Thus combined with \\eqref{eq:radlowerbound} we find\n\\begin{equation*}\n \\nrmb{M_{\\Rad}^{\\ms{D}[0,1)}}_{L^p([0,1);\\ell^r) \\to L^p([0,1);\\ell^r) } \\gtrsim \\max\\cbraces{\\frac{1}{p-1}, p^{1\/r-1\/2}},\n\\end{equation*}\nwhich implies that the weighted estimate in Theorem \\ref{theorem:RMF} is sharp by \\cite[Theorem 1.2]{LPR15}.\n\n\\bigskip\n\nTo finish this section we will compare the sparse domination for the Rademacher maximal operator in Theorem \\ref{theorem:RMF} with the sparse domination for the lattice Hardy--Littlewood maximal operator obtained by H\\\"anninnen and the author in \\cite[Theorem 1.3]{HL17}. Let $X$ be a Banach lattice with finite cotype and $\\ms{D}$ the standard dyadic system in $\\R^d$. For a simple function $f:\\R^d \\to X$ define \\emph{dyadic lattice Hardy--Littlewood maximal operator} (see e.g. \\cite{GMT93}) by\n\\begin{equation}\\label{eq:maxop}\n M_{\\Lat}^{\\ms{D}}f(s) : = \\sup_{Q \\in \\ms{D}: s \\in Q} \\ipb{\\abs{f}}_{1,Q}, \\qquad s\\in \\R^d,\n\\end{equation}\nwhere the absolute value and the supremum are taken in the lattice sense. By the Khintchine--Maurey inequalities (see e.g. \\cite[Theorem 7.2.13]{HNVW17}) we have\n $$M_{\\Rad}^{\\ms{D}}f \\lesssim M_{\\Lat}^{\\ms{D}}f$$\n for any simple $f\\colon \\R^d \\to X$.\nBy \\cite{Bo84,Ru86} we know that $X$ has the $\\UMD$ property if and only if\n $M_{\\Lat}^{\\ms{D}}$ is bounded on $L^p(\\R^d;X)$ and $L^p(\\R^d;X^*)$ for some (all) $p \\in (1,\\infty)$, which implies that any $\\UMD$ Banach lattice has the $\\RMF$ property.\n\n Comparing the sparse domination result in Theorem \\ref{theorem:RMF} with the corresponding sparse domination result for the dyadic lattice Hardy--Littlewood maximal operator, we see that the sparse operator in Theorem \\ref{theorem:RMF} is smaller than the sparse operator in \\cite[Theorem 1.3]{HL17}. Moreover, the sparse domination for the lattice Hardy--Littlewood maximal operator is sharp, as shown in \\cite[Theorem 1.2]{HL17}. Therefore on any $\\RMF$ Banach lattice that is not $\\infty$-convex, the operators $M_{\\Rad}^{\\ms{D}}$ and $M_{\\Lat}^{\\ms{D}}$ are incomparable, i.e. the (dyadic) lattice Hardy--Littlewood maximal operator is strictly larger than the Rademacher maximal operator. As the only $\\infty$-convex $\\RMF$ Banach lattices are the finite dimensional ones, we have the following corollary.\n\n\\begin{corollary}\n Let $X$ be an infinite dimensional $\\RMF$ Banach lattice. Then there is no $C>0$ such that for all simple $f\\colon \\R^d \\to X$\n $$\n M_{\\Lat}^{\\ms{D}}f\\leq C \\, M_{\\Rad}^{\\ms{D}}f.$$\n\\end{corollary}\n\n\n\\section{Further Applications}\\label{section:further}\n In this final section we comment on some further applications of our main theorems, for which we leave the details to the interested reader.\n\\begin{itemize}\n\\item Sparse domination and weighted bounds for variational truncations of Calder\\'on--Zygmund operators were studied in \\cite{FZ16, HLP13b, MTX15, MTX17}. The arguments presented in these references also imply the boundedness of our sharp grand maximal truncation operator and thus by Theorem \\ref{theorem:main} yield sparse domination of the variational truncations of Cal\\-der\\'on--Zygmund operators.\n\\item In \\cite{LOR17} Lerner, Ombrosi and Rivera-R\\'ios show sparse domination for commutators of a $\\BMO$ function $b$ with a Calder\\'on--Zygmund operator using sparse operators adapted to the function $b$. By a slight adaptation of the arguments presented in the proof of Theorem \\ref{theorem:localsparse}, one can prove the main result of \\cite{LOR17} in our framework and extend it to the vector-valued setting and to spaces of homogeneous type.\n\\item H\\\"ormander--Mihlin type conditions as in \\cite[Theorem IV.3.9]{GR85} imply the weak $L^{p_1}$-boundedness of our maximal truncation operator for $p_1>n\/a$ and thus sparse domination for the associated Fourier multiplier operator by Theorem \\ref{theorem:main}. Vector-valued extensions under Fourier type assumptions can be found in \\cite{GW03,Hy04} and Theorem \\ref{theorem:main} may therefore also be used to prove weighted results in that setting.\n\\item In \\cite{Le11} Lerner used his local mean oscillation decomposition to deduce sparse domination and sharp weighted norm inequalities for various Littlewood--Paley operators. These results are also an almost immediate consequence of Theorem \\ref{theorem:localsparse} with $r=2$, using a truncation of the cone of aperture in the definition of a Littlewood--Paley operator in order to make the localized $\\ell^2$-estimate checkable. Using similar arguments one can also treat the dyadic square function with Theorem \\ref{theorem:localsparse}, which yields the sharp weighted norm inequalities as obtained by Cruz-Uribe, Martell and Perez \\cite{CMP12}.\n\n Very recently Bui and Duong \\cite{BD19} extended the results in \\cite{Le11} to square functions of a general operator $L$ which has a Gaussian heat kernel bound and a bounded holomorphic functional calculus on $L^2(S)$, where $(S,d,\\mu)$ is a space of homogeneous type. The arguments they present can also be used to estimate our sharp grand maximal truncation operator, so their result is also be treated by Theorem \\ref{theorem:localsparse}.\n\\item Fackler, Hyt\\\"onen and Lindemulder \\cite{FHL18} proved weighted vector-valued Littlewood-Paley theory on a $\\UMD$ Banach space in order to prove their weighted, anisotropic, mixed-norm Mihlin multiplier theorems. Using Theorem \\ref{theorem:main} and Proposition \\ref{proposition:weights} on the Littlewood--Paley square function with smooth cut-offs one can prove sparse domination and weighted estimates in the smooth cut-off case. This can then be transferred to sharp cut-offs by standard arguments, recovering \\cite[Theorem 3.4]{FHL18}.\n\\item In \\cite{PSX12} Potapov, Sukochev and Xu proved extrapolation upwards of unweighted vector-valued Littlewood--Paley--Rubio de Francia inequalities. Using \\cite[Lemma 4.5]{PSX12} one can check the weak $L^2$-boundedness of our sharp grand maximal truncation operator, which by Theorem \\ref{theorem:main} and Proposition \\ref{proposition:weights} yields sparse domination and weighted estimates for vector-valued Littlewood--Paley--Rubio de Francia estimates. In the scalar case sparse domination was shown by Garg, Roncal and Shrivastava \\cite{GRS18} using time-frequency analysis.\n \\item Theorem \\ref{theorem:fractional} can be used to show sparse domination and sharp weighted estimates for fractional integral operators as in \\cite{CB13, CB13b,Cr17,IRV18}. The boundedness of the sharp grand maximal truncation operator associated to these operators can be shown using a similar argument as we used in the proof of Theorem \\ref{theorem:A2}.\n\\item In \\cite{BFP16} Bernicot, Frey and Petermichl show that the sparse domination\nprinciple is also applicable to non-integral singular operators falling\noutside the scope of Calder\\'on--Zygmund operators. Sparse domination for square functions related to these operators was studied in \\cite{BBR20}. The methods developed in these papers actually show the boundedness of the localized sharp grand $q$-maximal truncation operator used in Theorem \\ref{theorem:sparseform}, so these results also fit in our framework.\n\\end{itemize}\n\n\\subsection*{Acknowledgement} The author would like to thank Dorothee Frey, Bas Nieraeth and Mark Veraar for their helpful comments on the draft version of this paper. Moreover the author would like to thank Luz Roncal for bringing one of the applications in Section \\ref{section:A2} under the author's attention and Olli Tapiola for his remarks on the Lebesgue differentiation theorem in spaces of homogeneous type.\n\n\\newcommand{\\etalchar}[1]{$^{#1}$}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\n\n\n\n\nHigh energy elastic proton-proton and proton-antiproton cross\nsections reveal very complicated dynamics which is rather\ndifficult to explain within the framework of Quantum\nChromodynamics (QCD) (see the discussion in\n\\cite{Krisch:2010hr,Fiore:2008tp,Dremin:2013xza,\nUzhinsky:2011qu,Bourrely:2012hp,Selyugin:2013cia,Martynov:2013ana,Donnachie:2013xia,Khoze:2013jsa}).\nIn a conventional approach at small transfer momentum experimental\ndata can be described quite well by the diffractive scattering\ninduced by Pomeron exchange between hadrons. At large $-t \\gg 1$\nGeV$^2$ in the popular Donnachie-Landshoff (DL) model the dominant\ncontribution comes from the exchange by Odderon which is the\n$P=C=-1$ partner of Pomeron. It was suggested that this effective\nexchange originated from the perturbative three gluon exchange in\nthe proton-proton and proton-antiproton scattering\n\\cite{Donnachie:1979yu}.\n The experimental support for the existence of such exchange comes\nfrom high energy ISR data on the difference in the dip structure\naround $\\mid t\\mid \\approx 1.4 $ GeV$^2$ in the proton-proton and\nproton-antiproton differential cross sections at $\\sqrt{s}=53 $\nGeV \\cite{landshoff}.\n However, there is no any signal for Odderon at very small transfer\nmomentum. We would like to emphasize that one cannot expect the\nperturbative QCD DL approach to be valid even at the largest\ntransfer momentum $-t\\sim 14$ GeV$^2$ accessible at ISR\nenergies. This is related to the fact that in the three-gluon\nexchange model, which is applied to describe elastic cross\nsections in the interval $-t= 3-14 $ GeV$^2$, the average\nvirtuality of exchanged gluons $\\hat{t}\\approx t\/9 $ is quite\nsmall -$\\hat{t}=0.3-1.6 $ GeV$^2$. Therefore, in this kinematic\nregion nonperturbative QCD effects should be taken into account.\n\nThe attempt to include some of the nonperturbative effects into\nthe DL model was made in \\cite{landshoff2}. In that paper the\nstrength of three-gluon exchange with perturbative quark-gluon\nvertices was considered as a free parameter and its value was\nfound from the fit of the data. Therefore, a good description of\nthe large $-t$ cross sections in the paper is not the result of\ncalculation but rather of the fine tuning to experimental data.\n\n\nOne of the successful models of nonperturbative effects is the\ninstanton liquid model for QCD vacuum \\cite{shuryak,diakonov}.\nInstantons describe nontrivial topological gluon field excitations\nin vacuum and their existence leads to the spontaneous chiral\nsymmetry breaking in QCD. One of the manifestations of this\nphenomenon is the appearance of dynamical quark mass and\nnonperturbative helicity-flip quark-gluon interaction\n\\cite{Kochelev:1996pv,diakonov}. Such new interaction can be\ntreated as a nonperturbative anomalous quark chromomagnetic moment\n(AQCM). It was shown that AQCM gives a very important contribution\nthe to quark-quark scattering at large energies for both polarized\nand nonpolarized cases\n\\cite{Kochelev:1996pv,diakonov,Kochelev:2013zoa,Kochelev:2009rf,Kochelev:2006ny}.\nOne of the applications of these results is a new model for\nPomeron based on AQCM and nonperturbative two gluon exchange\nbetween hadrons suggested in \\cite{diakonov,Kochelev:2009rf}.\n\n\nIn this paper, we extend this model to the case of the three\ngluon colorless exchange between nucleons. It will be shown that\na nonperturbative version of the Donnachie-Landshoff Odderon model\nbased on AQCM describes well high energy data for the elastic\nproton-proton, proton-antiproton cross sections at large transfer\nmomentum. The spin effects in elastic scattering are also under\ndiscussion.\n\n\n\\section{Anomalous quark chromomagnetic moment and Odderon exchange}\n\n\nThe interaction vertex of a massive quark with a gluon can be\nwritten in the following form:\n\\begin{equation}\nV_\\mu(k_1^2,k_2^2,q^2)t^a = -g_st^a[ \\gamma_\\mu\nF_1(k_1^2,k_2^2,q^2) -\n\\frac{\\sigma_{\\mu\\nu}q_\\nu}{2M_q}F_2(k_1^2,k_2^2,q^2)],\n \\label{vertex}\n \\end{equation}\nwhere the form factors $F_{1,2}$ describe nonlocality of the\ninteraction, $k_{1,2}$ is the momentum of incoming and outgoing\nquarks, respectively, $ q=k_1-k_2$, $M_q$ is the quark mass, and\n$\\sigma_{\\mu\\nu}=(\\gamma_\\mu \\gamma_\\nu-\\gamma_\\nu \\gamma_\\mu)\/2$.\n Within the instanton\nmodel the shape of the form factor $F_2(k_1^2,k_2^2,q^2)$ is\n \\be\n F_2(k_1^2,k_2^2,q^2) =\\mu_a\n\\Phi_q(\\mid k_1\\mid\\rho\/2)\\Phi_q(\\mid k_2\\mid\\rho\/2)F_g(\\mid\nq\\mid\\rho) \\ , \\label{form1} \\ee where \\ba\n\\Phi_q(z)&=&-z\\frac{d}{dz}(I_0(z)K_0(z)-I_1(z)K_1(z)), \\nonumber\\\\\nF_g(z)&=&\\frac{4}{z^2}-2K_2(z) \\label{form3} \\ea are the\nFourier-transformed quark zero-mode and instanton fields,\nrespectively, $I_{\\nu}(z)$ and $K_{\\nu}(z)$ are the modified\nBessel functions and $\\rho$ is the instanton size.\n\nAQCM is defined by formula\n\\begin{equation}\n\\mu_a=F_2(0,0,0). \\nonumber\n\\end{equation}\n\nFor our estimation below we will use the value of AQCM $\\mu_a=-1$\nwhich is within the interval $-\\mu_a\\sim 0.4 -1.6$ given by the\ninstanton model \\cite{Kochelev:2009rf}. This value is also\nsupported by hadron spectroscopy (see \\cite{Ebert:2009ub} and\nreferences therein). Recently, a similar value of AQCM was also\nobtained within the Dyson-Schwinger equation approach with\nnonperturbative quark and gluon propagators \\cite{Cloet:2013jya}.\n In Fig.1, the Donnachie-Landshoff perturbative QCD (pQCD)\nand nonperturbative AQCM-induced three gluon exchange between two\nnucleons are presented.\n\n\n\\begin{figure}[htb]\n\\includegraphics[scale=1.0]{fig1.eps}\n\\caption{ The left panel is the Donnachie-Landshoff mechanism for\nthe large $-t$ proton-proton scattering. The right panels are the\nexample of the AQCM contribution induced by the second term in\nEq.\\ref{vertex}.}\n\\end{figure}\nWithin the DL model the differential cross-section of the\nproton-proton and proton-antiproton scattering is given by the\nformula\n\\begin{equation}\n\\frac{d\\sigma}{dt}\\approx \\frac{244P^4}{s^6t^2R^{12}}\\mid\nM_{qq}(\\theta)\\mid^6\n \\label{sigma}\n\\end{equation}\nwhere $M_{qq}$ is the matrix element at the quark level, $\\theta$\nis the scattering angle in the center of mass, $P$ is the\nprobability of the three quark configuration in a proton, and $R$\nis the proton radius. In the pQCD DL approach at the quark level\n\\begin{equation}\n\\mid\nM^{pQCD}_{qq}(\\theta)\\mid^2=\\frac{128\\pi^2\\alpha_s^2}{9}\\frac{\\hat{s}^2}{\\hat{t}^2},\n\\label{pQCD}\n\\end{equation}\n where $\\hat{s}\\approx s\/9$, at $\\hat{s}\\gg-\\hat{t}$\\ \\ $\\hat{t}\/\\hat{s} \\sim -\\sin^2\\theta\/4$,\nand the following values of the parameters were taken {\\it ad hoc} :\\\\\n\\begin{equation}\nP=1\/10,\\ \\ \\alpha_s=0.3,\\ \\ R=0.3 fm.\n\\end{equation}\nWe should emphasize that DL assumed a very small proton radius\nwhich is far away from the real proton size $R\\approx 1$ fm.\n For more suitable values $P=1$ and $R=1$ fm, we got\n$d\\sigma\/dt\\sim 8\\cdot 10^{-4}\/t^8$ mb\/GeV$^2$. It is about two\norders of magnitude less than high energy data $d\\sigma\/dt\\approx\n9\\cdot 10^{-2}\/t^8$ mb\/GeV$^2$ at large $-t$, Fig.2 .\n\\begin{figure}[h] \\vspace*{-0.0cm}\n\\centerline{\\epsfig{file=elasticpp.eps,width=10cm,height=10cm,angle=0}}\\\n\\caption{ \\small The contribution of pQCD exchange (dashed line)\nand AQCM contribution (solid line) to the elastic proton-proton\nscattering at large energy and large momentum transfer in\ncomparison with data \\cite{Nagy:1978iw}.}\n\\end{figure}\nFor the AQCM contribution at the quark level we have\n\\begin{eqnarray}\n\\mid M^{AQCM}_{qq}(\\hat{s},\\hat{t})\\mid^2&=&\\frac{16\\pi^3}{3}\n\\alpha_s (\\mid \\hat t\\mid) \\mid\\mu_a\\mid\n \\rho_c^2 F_g^2(\\sqrt{\\mid \\hat{t}\\mid}\\rho_c)\\frac{\\hat{s}^2}{\\mid\\hat{t}\\mid}\n+\\frac{\\pi^4}{2}\\mu_a^2\\rho_c^4 F_g^4(\\sqrt{\\mid\n\\hat{t}\\mid}\\rho_c)\\hat{s}^2. \\label{AQCM2}\n\\end{eqnarray}\nFor estimation, we use $R=1$ fm, $P=1$\\footnote{The value of\nthe strong proton radius $R\\approx 1$ fm is related to the\nconfinement scale. The probability of the three quark\nconfiguration in the proton $P=1$ is a natural assumption in our\nthree quarks on three quarks scattering model for large $-t$.},\n dynamical quark mass\n$M_q=280$ MeV, average instanton size $\\rho_c =1\/3$ fm and the\nstrong coupling constant\n\\begin{equation}\n\\alpha_s(q^2)=\\frac{4\\pi}{9\\ln((q^2+m_g^2)\/\\Lambda_{QCD}^2)},\n\\end{equation}\nwith $\\Lambda_{QCD}=0.280$ GeV and $m_g=0.88$ GeV\n\\cite{Kochelev:2009rf}. To get Eq.\\ref{AQCM2} the approximation\n$F_1(k_1^2,k_2^2,q^2)\\approx 1$ was used and we neglected nonzero\nvirtuality of quarks in a proton.\n The final result for the AQCM\ncontribution to the proton-proton and proton-antiproton cross\nsection is presented by the solid line in Fig.2. We should mention\n that the AQCM contribution asymptotically decays as $1\/t^{11}$ due\nto the form factor Eq.\\ref{form3}. Therefore, asymptotically at\nvery large transfer momentum perturbative $1\/t^8$ should give the\ndominating contribution. However, in the kinematic region\naccessible at the present time in experiments $-t \\leq 14$\nGeV$^2$, the nonperturbative AQCM contribution\n describes the available large\n$-t$ data very well, Fig.2.\n\\begin{figure}[htb]\n\\centerline{\\epsfig{file=fig3.eps,width=14cm,height=4cm,angle=0}}\\\n\\caption{ The interference between a) DL-type AQCM diagram and b)\nPomeron spin-flip induced by AQCM.}\n\\end{figure}\nFinally, some part of the difference between the structure of the\ndip around $-t\\approx 1-2 $ GeV$^2$ in the proton-proton and\nproton-antiproton elastic scatterating at ISR energies might be\nrelated to the difference in the sign of the interference between\nthe AQCM Odderon and Pomeron spin-flip amplitudes, Fig.3.\n\n\nIn our approach the spin-flip component, which is proportional to\n$t$, gives the dominating contribution to the negative charge\nparity Odderon amplitude. In the region of small transfer momentum\nthis contribution to the amplitude of the $PP$ and $P\\bar P$\nscattering has the dependence\n\\begin{equation}\n M\\sim \\frac{\\sqrt{-t}}{(m_g^2-t)^3}, \\label{small}\n\\end{equation}\ndue to quark spin-flip induced by AQCM. In Eq.\\ref{small},\n$m_g\\approx 0.4$ GeV is the dynamical gluon mass\n\\cite{Aguilar:2013hoa}.\n Therefore, the difference in the $PP$\nand $P\\bar P$ differential cross sections at small $-t$ and the\ndifference in the total $PP$ and $P\\bar P$ cross sections should\nbe very small at high energies. This is in agreement with\nexperimental data.\n\nOf course, one can describe $PP$ and $\\bar PP$ large $-t> 3.5 $\nGeV$^2$ data by using the assumption about a specific $t$\ndependence of the Pomeron trajectory (see, for example\n\\cite{Bugrii:1979zh}). However, in anyway, it is necessary to\nintroduce the additional $C=-1$ exchange with high intercept to\ndescribe the difference in the $PP$ and $\\bar PP$ elastic cross\nsections at $\\sqrt{s}=53$ GeV. A natural candidate for such\nexchange is the nonperturbative three gluon DL-type exchange. We\nwould like to mention that the sizable contribution from the\nconventional Pomeron exchange at large $-t>3.5$ GeV$^2$ is not\nexpected due to the huge suppression factor at large energies,\n$(s\/s_0)^{2\\alpha^\\prime_Pt}$, which comes from the nonzero slope\nof the Pomeron trajectory $\\alpha^\\prime_P\\approx 0.25$\nGeV$^{-2}$.\n\nIn the estimation above we assume, as in the DL model, that\nmomenta of exchanged gluons are approximately equal. The\njustification of this assumption is quite clear. To keep a\nproton as a bound state of three quarks at large transfer\nmomentum, all quarks in the proton should scatter approximately to\nthe same angle. In fact, one can also consider more complicated\nmultigluon contributions to elastic scattering, but we believe\nthat such contribution will be suppressed by either additional\nfactors $\\alpha_s$ or by extra factors $1\/t^n$ coming from gluon\npropagators and\/or from form factors in the quark-gluon vertices.\n\n\n\n\\section{\n Single-spin asymmetry $A_N$ in $PP$ and $P\\bar P$ elastic\n scattering}\n\nOne of the long-standing problems of QCD is the understanding of\nthe large spin effects observed in the different high energy\nreactions \\cite{Krisch:2010hr}, \\cite{Leader:2001gr}. Recently, we\nhave shown that the AQCM contribution leads to very large\nsingle-spin asymmetry (SSA) in the quark-quark scattering\n\\cite{Kochelev:2013zoa} and, therefore, it can be considered as a\nfundamental mechanism for explanation of anomalously large SSA\nobserved in different inclusive and exclusive reactions at high\nenergy. In elastic scattering, large SSA was found in the\nproton-proton scattering at AGS energies at large transfer\nmomentum, Fig.4.\n\\begin{figure}[htb] \\vspace*{-0.0cm}\n\\centerline{\\epsfig{file=ANpp.eps,width=8cm,height=8cm,angle=0}}\\\n\\caption{ \\small Single-spin asymmetry in the elastic\n$PP\\rightarrow PP$ scattering at large momentum transfer at AGS\n\\cite{Crabb:1990as}.}\n\\end{figure}\nIn the bases of the c.m. helicity amplitudes SSA is given by the\nformula\n\\begin{equation}\nA_N=-\\frac{2Im[\\Phi_5^*(\\Phi_1+\\Phi_2+\\Phi_3-\\Phi_4)]}{\\mid\\Phi_1\\mid^2+\\mid\\Phi_2\\mid^2+\\mid\\Phi_3\\mid^2+\n\\mid\\Phi_4\\mid^2+4\\mid\\Phi_5\\mid^2},\\label{SSA}\n\\end{equation}\n where the helicity amplitudes $\\Phi_1=<++\\mid++>$,\n$\\Phi_2=<++\\mid-->$, $\\Phi_3=<+-\\mid+->$, $\\Phi_4=<++\\mid-->$ and\n$\\Phi_5=<++\\mid-+>$. It is evident that due to the negative charge\nparity Odderon contribution, the helicity-flip amplitude $\\Phi_5$\nshould have a different sign for the proton-proton and\nproton-antiproton scattering. Therefore, SSA in the case of the\nelastic proton-antiproton scattering flips the sign in comparison\nwith the proton-proton scattering. This prediction can be tested\nby the PAX Collaboration at HESR \\cite{Barone:2005pu}. Due to the\ndominance of spin one $t$-channel gluon exchanges in the structure\nof Pomeron and Odderon, we can also expect that single-spin\nasymmetry at large $-t$ should have a weak energy dependence. This\nprediction can be checked in the polarized proton-proton elastic\nscattering in the pp2pp experiment at RHIC in case of extending\ntheir kinematics to the large transfer momentum region\n\\cite{pp2pp} \\footnote{We would like to thank Jacek Turnau for the\ndiscussion of this problem.}. However, the calculation of\nabsolute value of SSA in the elastic $PP$\n and $P\\bar P$ scattering at large transfer momenta is a very difficult task, because one\n needs\n to know spin-flip and non-spin flip components of both Odderon\n and Pomeron exchanges. Furthermore, in the region of small\n transfer momenta and low energies it is needed to include the effects\n of secondary Reggion exchanges as well.\n\n\n\n\n\n\n\n\n\\section{Conclusion}\n\nIn summary, it is shown that the anomalous quark-gluon\nnonperturbative vertex gives a large contribution to the elastic\nproton-proton and proton-antiproton scattering at large momentum\ntransfer. One can treat three-gluon exchange induced by this\nvertex as effective Odderon exchange with the spin-flip dominance\nin its amplitude.\n We should mention that the anomalous quark\nchromomagnetic moment is proportional to $1\/\\alpha_s$\n\\cite{Kochelev:1996pv}. Therefore, non-spin flip component in\nOdderon due to perturbative vertex should be suppressed by\n$\\alpha_s$ factor.\n We argue that a strong spin\ndependence of the Odderon amplitude might lead to the large spin\neffects in the proton-proton and proton-antiproton scattering at\nlarge momentum transfer.\n\n Our approach is based on the existence of two\nquite different scales in hadron physics. One of them is related\nto the confinement radius $ R\\approx 1$ fm and it is consistent,\nas well, with an average distance between instanton and\nantiinstanton within the instanton liquid model, $R_{I\\bar\nI}\\approx 1$ fm \\cite{shuryak,diakonov} . This scale is\nresponsible for the diffractive type scattering at small momentum\ntransfer. Another one is fixed by the scale of spontaneous\nsymmetry breaking. Within the instanton model it is given by an\naverage instanton size in QCD vacuum $\\rho_c\\approx 1\/3$ fm. This\nscale leads to the appearance of a large dynamical quark mass and\nlarge anomalous quark chromomagnetic moment and is responsible for\nthe dynamics of the hadron-hadron elastic scattering at large\nmomentum transfer. We would like to mention that the two scale\nmodel for the hadron structure was discussed in different aspects\nin the papers \\cite{Dorokhov:1993fc,Schweitzer:2012hh}.\n\n\\section{Acknowledgments}\nThe authors are very grateful to A.E. Dorokhov, A.V. Efremov,\nS.~V.~Goloskokov, E.A.~Kuraev, N.N.~Nikolaev, L.N. Lipatov, O.V.\nSelyugin and V.V. Uzhinsky for useful discussions.\n The work of N.K. was supported in part by a visiting\nscientist grant from the University of Valencia and by the MEST\nof the Korean Government (Brain Pool Program No. 121S-1-3-0318).\nWe also acknowledge that this work was initiated through the\nseries of APCTP-BLTP JINR Joint Workshops.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{ Abelian Lie algebras}\nIn this section, we show by an elementary argument that any abelian Lie algebra ( i.e. any vector space over $\\mathbb{Q}$), is potentially one dimensional, i.e. for any vector space $L$ over $\\mathbb{Q}$, there exists a field $E$ of characteristic zero, such that $\\dim_E L=1$. Then we show that this can not be generalized for nilpotent Lie algebras of class $2$.\n\n\\begin{proposition}\nLet $L$ be a vector space over $\\mathbb{Q}$. Then there exists a field $E$ of characteristic zero, such that $L$ is a vector space over $E$ and $\\dim_E L=1$.\n\\end{proposition}\n\n\\begin{proof}\nFirst, let $\\dim_{\\mathbb{Q}}L=n$ and $x_1, \\ldots, x_n$ be a basis of $L$. Suppose\n$$\nq(t)=a_0+a_1t+\\cdots+a_{n-1}t^{n-1}+t^n\\in \\mathbb{Q}[t]\n$$\nbe an irreducible polynomial. Let $\\lambda\\in \\mathbb{C}$ be any root of $q(t)$ and $E=\\mathbb{Q}(\\lambda)$. Define an action of $E$ on $L$ by the $\\mathbb{Q}$-bilinear extension of following rules\n\\begin{eqnarray*}\n\\lambda.x_1&=&x_2\\\\\n\\lambda.x_2&=&x_3\\\\\n &\\vdots&\\\\\n\\lambda.x_n&=&-a_0x_1-a_1x_2-\\cdots-a_{n-1}x_n.\n\\end{eqnarray*}\nIt is easy to check that $L$ is a vector space over $E$ of dimension $1$. Now, suppose $\\dim_\\mathbb{Q}L=\\alpha$ is infinite. Let $X$ be any set with the cardinality $\\alpha$ and suppose $\\mathbb{Q}(X)$ is the field rational functions with variables in the set $X$. Note that\n$$\n\\mathbb{Q}(X)=\\bigcup_{Y\\subset X}\\mathbb{Q}(Y),\n$$\nwhere $Y$ varies in the set of all finite subsets of $X$. Since every $\\mathbb{Q}(Y)$ is countable so we have $|\\mathbb{Q}(X)|=|X|$. Hence as $\\mathbb{Q}$-spaces, we have $L\\cong\\mathbb{Q}(X)$. Let $f:L\\to \\mathbb{Q}(X)$ be a $\\mathbb{Q}$-isomorphism. Now for $q\\in \\mathbb{Q}(X)$ and $x\\in L$, define\n$$\nq.x=f^{-1}(qf(x)).\n$$\nSo, by assuming $E=\\mathbb{Q}(X)$, $L$ is an $E$-space with $\\dim_E L=1$.\n\\end{proof}\n\nOne may wants to generalize this proposition for the case of nilpotent Lie algebras of higher nilpotency classes. But this is not true in general, since the three dimensional Heisenberg algebra $H=\\langle x, y, z\\rangle$ with $[x,y]=z$ and $z\\in Z(H)$ is a nilpotent rational Lie algebra of class two but there is no field $E$ with $\\dim_E H=2$. In general, the extension degree $[E:\\mathbb{Q}]$ is restricted by the numbers $\\dim_{\\mathbb{Q}}C_L(x)$, $x\\in L$. To see this, let $L$ be any Lie algebra over $E$. Then clearly for any $x\\in L$, we have $Span_E(x)\\subseteq C_L(x)$, so\n$$\n[E:\\mathbb{Q}]=\\dim_{\\mathbb{Q}}Span_E(x)\\leq \\dim_{\\mathbb{Q}}C_L(x).\n$$\nTherefore, we have\n\n\\begin{lemma}\nLet $L$ be a rational Lie algebra and $E$ be any field such that $(L, E)$ is a completion for $L$. Then\n$$\n[E:\\mathbb{Q}]\\leq \\min_{x\\in L}\\ \\dim_{\\mathbb{Q}}C_L(x).\n$$\n\\end{lemma}\n\nBy a similar argument, we can prove the following proposition.\n\n\\begin{proposition}\nLet $L$ be an infinite dimensional rational Lie algebra which has a finite dimensional completion of the form $(L, E)$. Then\n\n\\ i- either $L$ is solvable or for any ordinal $\\alpha$ we have $\\dim_{\\mathbb{Q}}L^{(\\alpha)}=\\infty$.\n\nii- either $L$ is nilpotent or for any ordinal $\\alpha$ we have $\\dim_{\\mathbb{Q}}\\gamma_{\\alpha}(L)=\\infty$.\n\\end{proposition}\n\n\\section{The class of completions}\nIn this section, we show that the class of completions of any finite dimensional rational Lie algebra is elementary ( axiomatizable in some first order language). We also show that this is not true for the class of all fields $E$ in which $L$ is a Lie algebra over $E$. Remember that a pair $(K, E)$ is a completion of $L$, if\n\n\\ \\ i- $E$ is a field of characteristic zero,\n\n\\ ii- $K$ is a Lie algebra over $E$,\n\niii- $L\\leq_{\\mathbb{Q}}K$ ( a $\\mathbb{Q}$-Lie subalgebra),\n\niv- $K=Span_E(L)$.\n\nLet $\\mathfrak{X}_L$ be the class of all completions of $L$.\n\n\\begin{theorem}\nIf $\\dim_{\\mathbb{Q}}L$ is finite, then the class $\\mathfrak{X}_L$ is elementary.\n\\end{theorem}\n\n\\begin{proof}\nLet $v_1, \\ldots, v_n$ be a $\\mathbb{Q}$-basis of $L$. We introduce a first order language\n$$\n\\mathcal{L}=(0, 1, +, \\times, q, (a^{\\ast}, a\\in L))\n$$\nwhich contains constant symbols $0$, $1$ and $a^{\\ast}$ for all $a\\in L$. It also has two binary functional symbols $+$ and $\\times$, and $q$ is a unary predicate symbol which will be interpreted as \"{\\em to be scalar}\". Let $\\Sigma$ be the following set of axioms:\\\\\n\n\\ 1- $\\forall x\\forall y\\ (q(x)\\wedge q(y))\\Rightarrow (q(x+y)\\wedge q(xy))$.\n\n\\ 2- $\\forall x\\forall y\\ (q(x)\\wedge q(y))\\Rightarrow (x+y=y+x \\wedge x\\times y=y\\times x)$.\n\n\\ 3- $\\forall x\\forall y\\forall z\\ (q(x)\\wedge q(y)\\wedge q(z))\\Rightarrow ((x+y)+z=x+(y+z)\\wedge (x\\times y)\\times z=x\\times (y\\times z))$.\n\n\\ 4- $\\forall x\\ q(x)\\Rightarrow x+0=x$.\n\n\\ 5- $\\forall x\\ 1\\times x=x$.\n\n\\ 6- $\\forall x\\ q(x)\\Rightarrow (\\exists y\\ q(y)\\wedge x+y=0)$.\n\n\\ 7- $\\forall x\\forall y\\forall z \\ (q(x)\\wedge q(y)\\wedge q(z))\\Rightarrow (x\\times (y+z)=x\\times y+x\\times z)$.\n\n\\ 8- $\\forall x\\ (q(x)\\wedge x\\neq 0)\\Rightarrow (\\exists y\\ q(y)\\wedge x\\times y=1)$.\n\n\\ 9- $\\forall x\\forall y\\ (\\neg q(x)\\wedge \\neg q(y))\\Rightarrow(\\neg q(x+y)\\wedge q(x\\times y))$.\n\n10- $\\forall x\\forall y\\ (\\neg q(x)\\wedge \\neg q(y))\\Rightarrow (x+y)=y+x$.\n\n11- $\\forall x\\ \\neg q(x)\\Rightarrow x\\times x=0^{\\ast}$.\n\n12- $\\forall x\\forall y\\forall z\\ (\\neg q(x)\\wedge \\neg q(y)\\wedge \\neg q(z))\\Rightarrow (x+y)+z=x+(y+z)$.\n\n13- $\\forall x\\ \\neg q(x)\\Rightarrow x+0^{\\ast}=x$.\n\n14- $\\forall x\\ \\neg q(x)\\Rightarrow(\\exists y\\ \\neg q(y)\\wedge x+y=0^{\\ast})$.\n\n15- $\\forall x\\forall y\\forall z( \\neg q(x)\\wedge \\neg q(y)\\wedge \\neg q(z))\\Rightarrow (x\\times y)\\times z+(y\\times z)\\times x+(z\\times x)\\times y=0^{\\ast}$.\n\n16- $\\forall x\\forall y\\forall z(\\neg q(y)\\wedge \\neg q(z))\\Rightarrow x\\times(y+z)=x\\times y+x\\times z$.\n\n\n17- $\\forall x\\forall y(q(x)\\wedge \\neg q(y))\\Rightarrow \\neg q(x\\times y)$.\n\n18- $\\forall x\\forall y\\forall z( q(x)\\wedge \\neg q(y)\\wedge \\neg q(z))\\Rightarrow x\\times (y\\times z)=(x\\times y)\\times z= y\\times (x\\times z)$.\n\n19- $\\forall x\\forall y\\forall z( q(x)\\wedge q(y)\\wedge \\neg q(z))\\Rightarrow (x\\times y)\\times z= x\\times (y\\times z)$.\n\n20- $\\forall x\\forall y\\forall z( q(x)\\wedge q(y)\\wedge \\neg q(z))\\Rightarrow (x+ y)\\times z= x\\times z+ y\\times z$.\n\n21- for all $a\\in L$ the axiom: $\\neg q(a^{\\ast})$.\n\n22- for all $a, b\\in L$ the axiom: $a^{\\ast}\\neq b^{\\ast}$.\n\n23- for all $a, b\\in L$ the axioms: $(a+b)^{\\ast}=a^{\\ast}+b^{\\ast}$ and $[a^{\\ast}, b^{\\ast}]=a^{\\ast}\\times b^{\\ast}$.\n\n24- for all natural number $m$ the axiom: $\\forall x\\ (q(x)\\wedge mx=0)\\Rightarrow x=0 $.\n\n25- $\\forall x\\ \\neg q(x)\\Rightarrow (\\exists x_1\\exists x_2 \\ldots \\exists x_n\\ (\\bigwedge_{i=1}^n q(x_i))\\wedge (x=\\sum_{i=1}^nx_i\\times v_i))$.\\\\\n\nNow, let $M\\in Mod(\\Sigma)$ be a model of $\\Sigma$. Define\n$$\nK=\\{ x\\in M: \\neg q(x)\\}\n$$\nand\n$$\nE=\\{x\\in M: q(x)\\}.\n$$\nThen it is easy to verify that $(K, E)$ is a completion of $L$. Conversely if $(K, E)$ is a completion of $L$ then we let $M=K\\cup E$. For all $x, y\\in M$, define $x+y$ to be their sum in $E$, if $x, y\\in E$, their sum in $K$ if, $x, y\\in K$, and $0^{\\ast}$ otherwise. Similarly, define $x\\times y$ to be $xy$, if $x, y\\in E$, $[x,y]$, if $x, y\\in K$, and the scalar product otherwise. It can be verified that $M$ is a model for $\\Sigma$ and so $\\Sigma$ axiomatizes the class $\\mathfrak{X}_L$.\n\\end{proof}\n\nIt is clear that the class of all Lie algebras over a fixed field $E$ is elementary. We show that the class of all fileds for a fixed rational Lie algebra is not elementary. Let $L$ be such a Lie algebra. By $\\mathfrak{F}_L$ we denote the class of all fields $E$ such that $(L, E)$ is a completion of $L$. Note that every element of $\\mathfrak{F}_L$ is a sub-ring of the centroid $\\Gamma(L)$.\n\n\\begin{proposition}\nLet $L$ be a finite dimensional Lie algebra over $\\mathbb{Q}$. Then $\\mathfrak{F}_L$ is not elementary.\n\\end{proposition}\n\n\\begin{proof}\nLet $\\mathcal{L}$ be a first order language and $\\Sigma$ be a set of axioms of $\\mathfrak{F}_L$. Since $L$ is countable, we may assume that $\\mathcal{L}$ is also countable. Now $\\mathbb{Q}$ is a model for $\\Sigma$ and $|\\mathbb{Q}|\\geq |\\mathcal{L}|$. So by the Lowenheim-Skolm theorem, for any infinite cardinal $\\alpha$, the set $\\Sigma$ has a model $E$ of size $\\alpha$. Now\n$$\n\\dim_{\\mathbb{Q}}L=[E:\\mathbb{Q}]\\dim_E L,\n$$\nand the left hand side is finite. So $[E:\\mathbb{Q}]$ is finite, contradicting $|E|=\\alpha$.\n\\end{proof}\n\nSimilarly, one can prove that the class $\\mathfrak{F}_L$ is not elementary for any abelian Lie algebra $L$.\n\n\\section{Entangled ideals}\nIn this section, we introduce the concept of {\\em entangled} ideals in a tensor product $E\\otimes_{\\mathbb{Q}}L$ and we show that they have close connections with the completions of $L$. For a rational Lie algebra $L$ and any field $E$ of characteristic zero, we say that a Lie algebra $K$ is an $E$-completion of $L$, if $(K,E)$ is a completion for $L$.\n\n\\begin{definition}\nAn ideal $N$ of the $E$-algebra $E\\otimes_{\\mathbb{Q}}L$ is called entangled, if $N\\cap (1\\otimes L)=0$.\n\\end{definition}\n\n\\begin{proposition}\nLet $K$ be an $E$-completion of $L$. Then there exists an entangled ideal $N\\unlhd E\\otimes_{\\mathbb{Q}}L$ such that\n$$\nK=\\frac{E\\otimes_{\\mathbb{Q}}L}{N}.\n$$\nConversely, for any entangled ideal $N$, the Lie algebra $K$ defined by the above equality is an $E$-completion.\n\\end{proposition}\n\n\\begin{proof}\nDefine a map $\\varphi:E\\times L\\to K$ by $\\varphi(x, a)=xa$. This map is $\\mathbb{Q}$-bilinear and so there is a $\\mathbb{Q}$-linear map $\\varphi^{\\ast}:E\\otimes_{\\mathbb{Q}}L\\to K$ with $\\varphi^{\\ast}(x\\otimes a)=xa$. Since $K=Span_E(L)$, so $\\varphi^{\\ast}$ is surjective. Also for any $\\lambda\\in E$, we have\n$$\n\\varphi^{\\ast}(\\lambda(x\\otimes a))=\\varphi^{\\ast}((\\lambda x)\\otimes a)= \\lambda \\varphi^{\\ast}(x\\otimes a),\n$$\nhence $\\varphi^{\\ast}$ is $E$-linear. Further\n\\begin{eqnarray*}\n\\varphi^{\\ast}([x\\otimes a, y\\otimes b])&=& \\varphi^{\\ast}((xy)\\otimes [a,b])\\\\\n &=&xy[a,b]\\\\\n &=&[\\varphi^{\\ast}(x\\otimes a), \\varphi^{\\ast}(y\\otimes b)].\n\\end{eqnarray*}\nSo, $\\varphi^{\\ast}$ is an epimorphism of $E$-Lie algebras. Let $N=\\ker \\varphi^{\\ast}$. Then\n$$\nK\\cong \\frac{E\\otimes_{\\mathbb{Q}}L}{N}.\n$$\nNote that\n$$\nN=\\{ \\sum_i x_i\\otimes a_i\\in E\\otimes_{\\mathbb{Q}}L:\\ \\sum_ix_ia_i=0\\},\n$$\nso if $1\\otimes a\\in N$, then $a=0$ and therefore $N$ is an entangled ideal.\n\nConversely, let $N\\unlhd E\\otimes_{\\mathbb{Q}}L$ be an entangled ideal. Let $K$ be defined as above. Then the map $a\\mapsto 1\\otimes a+N$ is a $\\mathbb{Q}$-embedding of $L$ in $K$. So, we may assume that $L\\leq_\\mathbb{Q}K$. Now clearly $K=Span_E(L)$ and so $K$ is an $E$-completion of $L$.\n\\end{proof}\n\n\\begin{example}\nLet $L$ be a Lie algebra over $E$. Then in the same time $L$ is a Lie algebra over $\\mathbb{Q}$. Let $I\\unlhd_{\\mathbb{Q}}L$, $f\\in Aut_{\\mathbb{Q}}(L)$, and $\\sigma\\in Aut(E)$. Define\n$$\nN(I,f,\\sigma)=\\{ \\sum_i x_i\\otimes a_i\\in E\\otimes_{\\mathbb{Q}}I:\\ \\sum_i\\sigma(x_i)f(a_i)=0\\}.\n$$\nNote that the map $\\varphi:E\\times I\\to L$ defined by $\\varphi(x,a)=\\sigma(x)f(a)$ is $\\mathbb{Q}$-bilinear and so $N(I,f,\\sigma)$ is well-defined. It can be easily seen that $N(I,f,\\sigma)$ is an entangled ideal. So we have a completion\n$$\nK(I,f,\\sigma)=\\frac{E\\otimes_{\\mathbb{Q}}L}{N(I,f,\\sigma)}.\n$$\n\\end{example}\n\n\n\\begin{proposition}\nLet $E$ be a finite extension of $\\mathbb{Q}$ and $L$ be a finite dimensional Lie algebra over $E$. With the above notations, we have\n$$\n\\dim_E K(I,f,\\sigma)=\\dim_E K(f(I), 1,1),\n$$\nand in the special case,\n$$\ndim_E K(L, f, \\sigma)=\\dim_E L.\n$$\n\\end{proposition}\n\n\\begin{proof}\nDefine a $\\mathbb{Q}$-linear map $\\sigma\\otimes f:E\\otimes_{\\mathbb{Q}}L\\to E\\otimes_{\\mathbb{Q}}L$ by $(\\sigma\\otimes f)(x\\otimes a)=\\sigma(x)f(a)$. This is an invertible map and we have\n$$\n(\\sigma\\otimes f)(N(I,f,\\sigma))=N(f(I),1,1).\n$$\nTherefore $\\dim_{\\mathbb{Q}}N(I,f,\\sigma)=\\dim_{\\mathbb{Q}}N(f(I), 1,1)$. Now, since $[E:\\mathbb{Q}]$ is finite, so\n$$\n\\dim_EK(I,f,\\sigma)=\\dim_EK(f(I),1,1).\n$$\nAs an special case, when $I=L$, we have $\\dim_E K(L,f,\\sigma)=\\dim_E K(L,1,1)$. But by an easy argument, one can see that\n$$\n\\dim_E N(L, 1,1)=([E:\\mathbb{Q}]-1)\\dim_E L,\n$$\nso we have\n$$\n\\dim_E K(L,f,\\sigma)=\\dim_E L.\n$$\n\\end{proof}\n\nWe can say more about the structure of $K(L,f,\\sigma)$. Let $a\\to \\bar{a}$ be the embedding of $L$ in $K(L,f,\\sigma)$. We have,\n\n\\begin{proposition}\nBy the above notations,\n$$\nK(L,f,\\sigma)=\\{ \\bar{a}:\\ a\\in L\\}\n$$\nand for all $x\\in E$, $x.\\bar{a}=\\overline{f^{-1}(\\sigma(x)f(a))}$.\n\\end{proposition}\n\n\\begin{proof}\nLet $N=N(L,f,\\sigma)$ and $X=\\sum x_i\\otimes a_i+N\\in K(L,f,\\sigma)$. We know that $\\sum\\sigma(x_i)f(a_i)\\in L$, so for some $a\\in L$, we have\n$f(a)=\\sum \\sigma(x_i)f(a_i)$. Therefore $1\\otimes a-\\sum x_i\\otimes a_i\\in N$ and hence $X=\\bar{a}$. Now, if $x\\in E$, then\n$$\nx.\\bar{a}=x\\otimes a+N=1\\otimes b+N,\n$$\nfor some $b$. Clearly, $f(b)=\\sigma(x)f(a)$, so $b=f^{-1}(\\sigma(x)f(a))$. Hence\n$$\nx.\\bar{a}=\\overline{f^{-1}(\\sigma(x)f(a))}.\n$$\n\\end{proof}\n\nNow, we show that nilpotency (solvablity) of $L$ implies nilpotency (solvablity) of its completions.\n\n\\begin{proposition}\nLet $L$ be a rational Lie algebra and $K$ be a completion for $L$ over some field $E$. If $L$ is nilpotent, then $K$ is nilpotent of the same class. If $L$ is solvable, then $K$ is also solvable of the same derived length.\n\\end{proposition}\n\n\\begin{proof}\nWe know that there is an entangled ideal $N$ such that\n$$\nK=\\frac{E\\otimes_{\\mathbb{Q}}L}{N}.\n$$\nSo, for any $n$ we have\n$$\n\\gamma_n(K)=\\frac{E\\otimes_{\\mathbb{Q}}\\gamma_n(L)+N}{N}, \\ \\ K^{(n)}=\\frac{E\\otimes_{\\mathbb{Q}}L^{(n)}+N}{N}.\n$$\nThis shows that, if $\\gamma_n(L)=0$ ($L^{(n)}=0$), then $\\gamma_n(K)=0$ ($K^{(n)}=0$). On the other hand, if $\\gamma_d(K)=0$ ($K^{(d)}=0$), then since $L\\subseteq K$, we must have $\\gamma_d(L)=0$ ($L^{(d)}=0$).\n\\end{proof}\n\nA Lie algebra $L$ over $\\mathbb{Q}$ is potentially finite dimensional, if it has a finite dimensional completion. Using a theorem of Ado (see \\cite{Jac}), it is easy to prove that $L$ is potentially finite dimensional, if and only if $L\\leq_{\\mathbb{Q}}\\mathfrak{g}\\mathfrak{l}_n(E)$ for some field $E$ of characteristic zero and a natural number $n$. By the above proposition, we have\n\n\\begin{corollary}\nLet $L$ be a potentially finite dimensional nilpotent rational Lie algebra. Then $L\\leq_{\\mathbb{Q}}\\mathfrak{n}^+_n(E)$, for some field $E$ and $n\\geq 1$, where $\\mathfrak{n}^+_n(E)$ is the Lie algebra of strictly upper triangular $n\\times n$ matrices over $E$.\n\\end{corollary}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}