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 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.
Yuying Chen, Yanchang Han, Yong-sheng Han et al.· 0 citations
Let $\omega$ be a radial $\widehat{\mathcal D}$-weight and $u$ be a bounded function on the unit disk $\mathbb D$. We prove that the Toeplitz operator \(T_{\omega,u}\) is compact on \(A_\omega^2\) if and only if its Berezin transform vanishes at the boundary. Our approach is based on a polynomial frame for $A_\omega^2$ and a detailed localization analysis of the resulting infinite matrix representation of $T_{\omega,u}$. Even in the unweighted Bergman space \(A^2\), our argument is new and does not rely on the classical translation operators. We further show that this Axler--Zheng compactness characterization does not extend, in general, to products of Toeplitz operators on \(\widehat{\mathcal D}\)-weighted Bergman spaces, and hence to the corresponding Toeplitz algebra generated by bounded symbols. More precisely, we construct a radial log-subharmonic \(\widehat{\mathcal D}\)-weight \(\omega\) and bounded symbols \(u,v\) such that the product \(T_{\omega,v}T_{\omega,u}\) is noncompact, whereas its Berezin transform vanishes at the boundary.
We study bounded finite-interaction-range operators $H$ on $\ell^2(\Gamma)$, where $\Gamma$ is a uniformly discrete subset of a left-invariant doubling metric measure space $(X,d,\mu)$. Our goal is to approximate spectral information of $H$ from finite sections $H_{L,x}$ supported on balls $B_L(x)\cap\Gamma$. The main technical input is a commutator estimate for Lipschitz''tent''localisations $W_{L,x}$, which depends only on geometric properties (doubling) and a uniform interaction degree. As a consequence, we obtain explicit two-sided pseudospectral inclusion bounds of the form \[ \gamma_{L,\varepsilon}(H)\subset \sigma_\varepsilon(H)\subset \gamma_{L,\varepsilon+C_0/L}(H), \] and Hausdorff convergence of window pseudospectra to the (global) pseudospectrum as $L\to\infty$. In the self-adjoint/ normal case this yields computable gap tests and spectral sampling schemes with rigorous $O(1/L)$ error control, while in the non-normal case it leads to corresponding approximation results for pseudospectra. The framework isolates the geometric core behind earlier approaches on $\mathbb{R}^n$ and on countable Abelian groups, and covers irregular geometries arising in quasicrystal models, as well as new cases such as discrete (non-Abelian) nilpotent groups (including for example the discrete Heisenberg group).
Let $(X,\omega_0)$ be a compact K\"ahler manifold, and let $U\Subset X$ have $C^{3,1}$ uniformly strongly pseudoconvex boundary. For the equilibrium envelope $u_0$ associated with $X\setminus U$, the normalized Monge--Amp\`ere measure vanishes on $U$ and equals the background measure $V^{-1}\omega_0^n$ on $X\setminus\overline U$; its remaining component is a singular measure supported on $\partial U$, where $V=\int_X\omega_0^n$. We identify this boundary component as the negative outward Anzellotti trace of a divergence-measure flux current. We then prove a one-sided Hadamard formula for the normalized Monge--Amp\`ere energy along the outward parallel family $U_\varepsilon=\{\rho<\varepsilon\}$. The nonlinear telescoping identity gives the mixed Bedford--Taylor boundary traces that sum up to the trace of the current \[ \mathcal J_{\mathrm{tot}} =\frac1V\mathrm{d}^c u_0\wedge\sum_{p=0}^{n-1}(p+1)\omega_{u_0}^p\wedge\omega_0^{n-1-p}, \qquad \omega_{u_0}=\omega_0+\mathrm{d}\mathrm{d}^c u_0. \] These results provide a local weak formulation of boundary flux and normal variation for regular interface problems related to Darcy/Hele--Shaw type problems and Monge--Amp\`ere growth.
We prove that for every fixed inverse temperature $\beta<1 / 2$, with high probability over the disorder, the single-site Glauber dynamics for the $n$-spin Sherrington-Kirkpatrick model mixes from every initial configuration to within total variation distance $\varepsilon$ in $O_{\beta}\left(n \log\left(n / \varepsilon\right)\right)$ steps. The bound holds uniformly over all external fields and is optimal up to constants depending only on $\beta$. The main ingredient is a deterministic criterion for optimal-order Poincar\'e inequalities in general Ising models, established via the integrated Bakry-\'Emery criterion together with a new two-spin estimate. A standard application of the localization-scheme framework of Chen and Eldan then upgrades the Poincar\'e inequality to a modified log-Sobolev inequality, yielding the optimal mixing-time bound. The main ideas underlying the proof of the Poincar\'e inequality were generated by GPT-5.6 Sol Ultra.
It is known that the energy measures of any two nonconstant harmonic functions on the standard Sierpi\'nski gasket are mutually absolutely continuous. Strichartz and Tse reported numerical evidence for $L^p$-integrability of the corresponding Radon--Nikodym densities in the range \[ 1<p<\frac{\log 15}{\log 9}. \] For arbitrary ordered pairs of nonconstant harmonic functions, we prove uniform boundedness of the associated density-ratio power sums, and hence $L^p$-integrability, in the subinterval \[ 1<p<\frac{\log(35/3)}{\log 9}. \] When the denominator harmonic direction is represented by the boundary values $(0,-1,1)$, we prove boundedness throughout the full conjectured interval.
Konstantinos Tsougkas· 0 citations
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