We prove a Fefferman--Stein good-$\lambda$ inequality for the Dunkl Poisson semigroup associated with a finite reflection group and a non-negative multiplicity function. For arbitrary complex-valued $f\in C_c^\infty(\mathbb R^N)$, with no $G$-invariance assumption, it compares the orbit-conical non-tangential maximal function $\mathcal N_P^\beta f$ with the area function $\mathcal S_Pf$ formed from the full space-time Dunkl carr\'e du champ, including its reflection-difference energy. The main obstruction is that a general cut-off creates wall differences not controlled by the local Euclidean-gradient product identity. The good set $E=\{x:\mathcal N_P^\beta f(x)\le\lambda\}$ is $G$-invariant; by the equivariance of the Poisson semigroup, so is $a=\varphi(P_t \mathbf 1_E)$, and hence all reflection differences of the cut-off vanish. Poisson maximal and tail estimates, together with the $L^2$ Littlewood--Paley estimate for $P_t \mathbf 1_{E^c}$, then yield the desired distribution inequality. Its integrated form gives maximal-to-area estimates for every $0<p<2$ and endpoint $H^1$-to-$L^1$ bounds for the orbit-conical and Euclidean-conical intrinsic area functions. For chamber lifts of globally smooth data, the inequality has an equivalent formulation on a fundamental chamber, where orbit cones become Euclidean cones and the reflection energy becomes a finite wall coupling. Combined with the known semigroup square-function characterization, these bounds characterize the Dunkl Poisson maximal Hardy space among $L^1(d\omega)$ data.
We prove a fixed-outer-domain content--capacity estimate for every positive codimensional gain on an unbounded complete $2$-PI space. As its principal measure-theoretic consequence, a one-sided ball-growth condition for an arbitrary positive Radon measure $\pi$ yields \[ \pi(K)\lesssim\operatorname{cap}_{2,\omega}(K;\Lambda_0B) \] uniformly for compact $K\subset B$. No doubling or lower-dimensional regularity is imposed on $\pi$. This capacitary domination gives a representative-independent mean-zero trace inequality. On normalized balls, it is equivalent, modulo the ambient Poincar\'e energy, to the mixed $d\omega\,d\pi$ oscillation required in Fefferman--Phong reductions. A standard bounded-overlap localization then yields two-sided global Fefferman--Phong inequalities, cover-independent energy norms, and a concrete realization of the associated homogeneous energy completion. The abstract results are verified for Euclidean $A_2$ weights with generalized Schr\"odinger measure potentials, reverse-H\"older function potentials, Carnot groups, and lower-dimensional singular measures. We compare the local trace conclusion with existing two-weighted Poincar\'e and Sobolev embedding theorems: those routes apply under additional doubling or dimension assumptions on the target measure, whereas our capacity conclusion also supplies absolute continuity with respect to variational capacity and a fixed outer domain. The Euclidean application yields form-domain equivalence, smooth form cores, self-adjoint realization, resolvent energy estimates, and local critical-multiplier bounds. Finally, the method supplies the mixed-measure step missing from a previously published $A_2$ generalized Schr\"odinger argument and gives a fixed-dilate finite-scale $A_2$ extension, for the naturally augmented measure, of a later $A_1$ theory.
We develop a pluripotential approach to complex Hessian equations on compact Hermitian manifolds. In this setting, the lack of closedness of the background metric introduces torsion terms that prevent a direct extension of the K\"ahler theory. Our main result is a uniform $L^\infty$ estimate for bounded $\omega$-$m$-subharmonic solutions of the equation \[ (\omega + dd^c u)^m \wedge \omega^{n-m} = cf\,\omega^n, \] under the assumption that $f \in L^p$, $f \ge 0$ for some $p>1$. The proof combines a weak comparison principle with torsion error, a capacity theory adapted to the Hermitian setting, and a nonlinear iteration scheme controlling the decay of sublevel sets. As applications, we obtain existence, stability and compactness results for weak solutions with $L^p$ densities. These results extend several aspects of the pluripotential theory. of complex Hessian equations beyond the K\"ahler framework.
For every $N\ge 2$, we prove that the Bergman metric on the regular locus of a finite ball quotient $\mathbb{B}^N/\Gamma$, where $\Gamma\subset \mathrm{U}(N)$ is finite and fixed-point-free, is K\"ahler-Einstein if and only if $\Gamma$ is trivial. Consequently, if $\Omega$ is an $N$-dimensional normal Stein space with isolated singularities and compact, smooth, strongly pseudoconvex boundary admitting a real-algebraic CR realization, then the Bergman metric on $\Omega_{\mathrm{reg}}$ is K\"ahler-Einstein if and only if $\Omega$ is biholomorphic to $\mathbb{B}^N$. This proves an algebraic version of the Cheng-Huang-Xiao conjecture in every complex dimension $N\ge 2$.
We study the Calder\'on problem for the equation $-\Delta_\infty u+q(x)u=0$ in a bounded convex domain with $C^2$ boundary. We prove that a positive potential $q$ is uniquely determined and explicitly reconstructible from the measurements $\Lambda_q\bigl(t(e\cdot x+b)|_{\partial\Omega}\bigr)$, where $e\in\mathbb S^{n-1}$, the offset $b>\sup_{\overline\Omega}|x|$ is fixed, and $t\to\infty$. The first potential-dependent term in the large amplitude asymptotics determines weighted integrals of $q$ over the chords parallel to $e$. Combining the measurements in the directions $e$ and $-e$ gives the X-ray transform of the zero extension of $q$, and the Fourier slice identity yields reconstruction and uniqueness.
Let $H_0$ be the standard discrete Laplacian on $\mathbb Z^d$, $d\geq4$, let $R_0(z)=(H_0-z)^{-1}$, and let $p'$ denote the H\"older conjugate of $p$, with $p'=\infty$ when $p=1$. We establish uniform diagonal resolvent estimates from $\ell^p(\mathbb Z^d)$ to $\ell^{p'}(\mathbb Z^d)$ by proving local Fourier-decay bounds for surface measure on the Fermi surfaces of the lattice dispersion relation. At a regular point, the decay is governed by the number of coordinate directions in which the quadratic term vanishes: after a local change of variables, these directions produce cubic one-dimensional phases, while the remaining directions are quadratic. For $d\geq5$, we prove the sharp global range \[ 1\leq p\leq\frac{2(d+2)}{d+5}. \] Away from the threshold energies, this range remains sharp in odd dimensions, whereas for even $d\geq6$ the sharp range improves to \[ 1\leq p\leq\frac{2(2d+5)}{2d+11}. \] In dimension four, we obtain uniform estimates for $1\leq p<4/3$ and an $\ell^{4/3}$-to-$\ell^4$ endpoint estimate with a square-root logarithmic loss. Matching anisotropic Knapp-type constructions give the corresponding necessary conditions. We also derive Birman--Schwinger and Kato smoothing bounds, complex-eigenvalue estimates, and improved thin spectral projection estimates.
The well-known characterization of Hardy spaces $\mathrm{H}_{p}(\mathbb D)$, $0<p<\infty$, in terms of the Littlewood-Paley g-function $$ S f (\zeta) = \left(\int_0^1 |f'(r \zeta)|^2 (1 - r) \mathrm dr\right)^{1/2} \in \mathrm{L}_{p} $$ is generalized to Hardy-type spaces $X_A$ corresponding to quasi-Banach lattices $X$ on the unit circle $\mathbb T$ under the assumption that the Hardy-Littlewood maximal operator $M$ is bounded in $(X^\delta)'$ with some $\delta>0$. As an application to composition operators $C_\varphi$, we derive an exact criterion for the boundedness and compactness of $C_\varphi : \mathcal{B}^\omega \to X_A$, where $\mathcal{B}^\omega = \{f \mid \sup |f'|/\omega<\infty\}$ is the weighted Bloch space with a log-convex radial weight $\omega$, generalizing recent results in the one-dimensional setting.
E. Abakumov, Evgueni Doubtsov, D. Rutsky· 0 citations
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