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We establish anisotropic uncertainty principles (UPs) for general metaplectic operators acting on $L^2(\mathbb{R}^d)$, including degenerate cases associated with symplectic matrices whose $B$-block has nontrivial kernel. In this setting,…

Analysis of PDEs · Mathematics 2026-01-26 Elena Cordero , Gianluca Giacchi , Edoardo Pucci

Multipartite nonlocality has been extensively investigated within one-dimensional quantum lattices. Previous research has primarily focused on the nonlocality measure $S$, which quantifies the violation of Bell-type inequalities. However,…

Quantum Physics · Physics 2026-04-27 Jia Bao , Bin Guo , Shu Qu , Fanqin Xu , Xueyi Lei , Zhaoyu Sun

We address an open problem posed by H. Brezis, M. Marcus and A.C. Ponce in: Nonlinear elliptic equations with measures revisited. In: Mathematical Aspects of Nonlinear Dispersive Equations (J. Bourgain, C. Kenig, S. Klainerman, eds.),…

Analysis of PDEs · Mathematics 2023-11-14 Tomasz Klimsiak

We prove necessary density conditions for sampling in spectral subspaces of a second order uniformly elliptic differential operator on $R^d$ with slowly oscillating symbol. For constant coefficient operators, these are precisely Landaus…

Functional Analysis · Mathematics 2024-03-13 Karlheinz Gröchenig , Andreas Klotz

In this paper, we characterize Carleson measure and vanishing Carleson measure on Bergman spaces with admissible weights in terms of {\it t-Berezin transform} and {\it averaging function} as key tools. Moreover, power bounded and power…

Complex Variables · Mathematics 2022-12-12 Aakriti Sharma , Ajay K. Sharma , M. Mursaleen

Let $(X,d,\mu)$ be a metric measure space satisfying a $Q$-doubling condition, $Q>1$, and an $L^2$-Poincar\'{e} inequality. Let $\mathscr{L}=\mathcal{L}+V$ be a Schr\"odinger operator on $X$, where $\mathcal{L}$ is a non-negative operator…

Classical Analysis and ODEs · Mathematics 2020-06-30 Renjin Jiang , Bo Li

This paper establishes convergence rates for learning elliptic pseudo-differential operators, a fundamental operator class in partial differential equations and mathematical physics. In a wavelet-Galerkin framework, we formulate learning…

Statistics Theory · Mathematics 2026-01-09 Jiaheng Chen , Daniel Sanz-Alonso

In relatively nice geometric settings, in particular, on Lipschitz domains, absolute continuity of elliptic measure with respect to the surface measure is equivalent to Carleson measure estimates, to square function estimates, and to…

Analysis of PDEs · Mathematics 2024-11-06 Steve Hofmann , José María Martell , Svitlana Mayboroda

We establish $L^p$ solvability of the Dirichlet problem, for some finite $p$, in a 1-sided chord-arc domain $\Omega$ (i.e., a uniform domain with Ahlfors-David regular boundary), for elliptic equations of the form \[ Lu=-\text{div}(A\nabla…

Analysis of PDEs · Mathematics 2026-01-05 Steve Hofmann

We provide resolvent asymptotics as well as various operator-norm estimates for the system of linear partial differential equations describing the thin infinite elastic rod with material coefficients which periodically highly oscillate…

Analysis of PDEs · Mathematics 2023-04-12 Kirill Cherednichenko , Igor Velčić , Josip Žubrinić

In this paper, we derive a local Carleman estimate for the complex second order elliptic operator with Lipschitz coefficients having jump discontinuities. Combing the result in [BL] and the arguments in [DcFLVW], we present an elementary…

Analysis of PDEs · Mathematics 2020-01-14 E. Francini , S. Vessella , J. -N. Wang

In this article we perform an asymptotic analysis of Bayesian parallel kernel density estimators introduced by Neiswanger, Wang and Xing (2014). We derive the asymptotic expansion of the mean integrated squared error for the full data…

Statistics Theory · Mathematics 2020-11-09 Alexey Miroshnikov , Evgeny Savelev

We establish Dahlberg's perturbation theorem for non-divergence form operators L = A\nabla^2. If L_0 and L_1 are two operators on a Lipschitz domain such that the L^p Dirichlet problem for the operator L_0 is solvable for some p in…

Analysis of PDEs · Mathematics 2011-01-28 Martin Dindos , Treven Wall

In $L_2({\mathbb R}^d;{\mathbb C}^n)$, we study a selfadjoint strongly elliptic operator $A_\varepsilon$ of order $2p$ given by the expression $b({\mathbf D})^* g({\mathbf x}/\varepsilon) b({\mathbf D})$, $\varepsilon >0$. Here $g({\mathbf…

Analysis of PDEs · Mathematics 2015-11-16 Andrey Kukushkin , Tatiana Suslina

Given a general symmetric elliptic operator $$ L\_{a} := \sum\_{k,,j=1}^d \p\_k (a\_{kj} \p\_j) + \sum\_{k=1}^d a\_k \p\_k - \p\_k(\overline{a\_k} .) + a\_0$$we define the associated Dirichlet-to-Neumann (D-t-N) operator with partial data,…

Analysis of PDEs · Mathematics 2016-04-14 El Maati Ouhabaz

The main purpose of this paper is to capture the asymptotic behavior for solutions to a class of nonlinear elliptic and parabolic equations with the anisotropic weights consisting of two power-type weights of different dimensions near the…

Analysis of PDEs · Mathematics 2024-04-23 Changxing Miao , Zhiwen Zhao

This work is concerned with an optimal control problem governed by a non-smooth quasilinear elliptic equation with a nonlinear coefficient in the principal part that is locally Lipschitz continuous and directionally but not G\^ateaux…

Optimization and Control · Mathematics 2021-09-28 Christian Clason , Vu Huu Nhu , Arnd Rösch

This thesis is devoted to asymptotic norm estimates for oscillatory integral operators acting on the L^2 space of functions of one real variable. The operators in question have compact support and an oscillatory kernel of the form exp(i…

Classical Analysis and ODEs · Mathematics 2007-05-23 Vyacheslav S. Rychkov

We consider divergence form elliptic operators $L=-\dv A(x)\nabla$, defined in $\mathbb{R}^{n+1}=\{(x,t)\in\mathbb{R}^{n}\times\mathbb{R}\}, n \geq 2$, where the $L^{\infty}$ coefficient matrix $A$ is $(n+1)\times(n+1)$, uniformly elliptic,…

Classical Analysis and ODEs · Mathematics 2007-05-23 S. Hofmann

Under sharp conditions, we prove the existence and refined asymptotic behaviour near zero (resp., at infinity) for all positive radial solutions to elliptic equations such as \begin{equation}\label{eq11} \tag{*} \mathbb…

Analysis of PDEs · Mathematics 2026-03-26 Florica C. Cîrstea , Maria Fărcăşeanu