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Brezis-Van Schaftingen-Yung Inequalities Beyond the Classical Setting

Saeed Hashemi Sababe

TL;DR

The paper broadens the BVY finite difference framework beyond the classical Euclidean setting by establishing BVY-type inequalities under non-doubling (polynomial-growth) measures, and by incorporating variable exponent and Orlicz growth. It develops nonlocal and anisotropic extensions, including a nonlocal $p$-Laplacian and an anisotropic metric $\rho_A$, along with corresponding Poincaré-type and Sobolev-type results and regularity theory. A rigorous analysis of stability, sharpness of constants, and interpolation between fractional and classical Sobolev spaces is provided, culminating in a Bourgain–Brezis–Mironescu-type limit as the fractional parameter approaches 1. Collectively, these results generalize BVY-type characterizations to a wide range of geometric and functional-analytic settings, with implications for PDEs, harmonic analysis, and numerical analysis on irregular spaces.

Abstract

In this paper, we extend the framework of Brezis--Van Schaftingen--Yung type inequalities in metric measure spaces by exploring several novel directions. First, we establish finite difference characterizations and fractional Sobolev-type inequalities in settings where the underlying measure is non-doubling or only satisfies a weak doubling condition. Second, we incorporate variable exponent and Orlicz space frameworks to capture nonstandard growth phenomena. Third, we derive anisotropic and directional versions of these inequalities to better address non-isotropic structures, and we apply our results to study regularity properties of nonlocal operators. Finally, we investigate the stability and sharpness of the associated constants as well as interpolation and limiting behaviors that bridge classical and fractional settings. These developments not only generalize existing results but also open new avenues for applications in partial differential equations and numerical analysis.

Brezis-Van Schaftingen-Yung Inequalities Beyond the Classical Setting

TL;DR

The paper broadens the BVY finite difference framework beyond the classical Euclidean setting by establishing BVY-type inequalities under non-doubling (polynomial-growth) measures, and by incorporating variable exponent and Orlicz growth. It develops nonlocal and anisotropic extensions, including a nonlocal -Laplacian and an anisotropic metric , along with corresponding Poincaré-type and Sobolev-type results and regularity theory. A rigorous analysis of stability, sharpness of constants, and interpolation between fractional and classical Sobolev spaces is provided, culminating in a Bourgain–Brezis–Mironescu-type limit as the fractional parameter approaches 1. Collectively, these results generalize BVY-type characterizations to a wide range of geometric and functional-analytic settings, with implications for PDEs, harmonic analysis, and numerical analysis on irregular spaces.

Abstract

In this paper, we extend the framework of Brezis--Van Schaftingen--Yung type inequalities in metric measure spaces by exploring several novel directions. First, we establish finite difference characterizations and fractional Sobolev-type inequalities in settings where the underlying measure is non-doubling or only satisfies a weak doubling condition. Second, we incorporate variable exponent and Orlicz space frameworks to capture nonstandard growth phenomena. Third, we derive anisotropic and directional versions of these inequalities to better address non-isotropic structures, and we apply our results to study regularity properties of nonlocal operators. Finally, we investigate the stability and sharpness of the associated constants as well as interpolation and limiting behaviors that bridge classical and fractional settings. These developments not only generalize existing results but also open new avenues for applications in partial differential equations and numerical analysis.
Paper Structure (6 sections, 17 theorems, 285 equations)

This paper contains 6 sections, 17 theorems, 285 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Extensions of Finite Difference Characterizations
  4. Applications to Nonlocal Operators and Anisotropic Settings
  5. Stability, Sharpness, and Interpolation
  6. Concluding Remarks and Open Problems

Key Result

Lemma 2.2

Let $(X,\rho)$ be a metric space and let $\mathcal{B}$ be a collection of balls in $X$ with uniformly bounded diameters. Then there exists a disjoint subcollection $\{B_i\}\subset \mathcal{B}$ such that This version of the Vitali covering lemma is standard in analysis on metric spaces; see, for example, Heinonen and Dai-Lin-Yang-Yuan-Zhang.

Theorems & Definitions (44)

  • Definition 2.1: Metric Measure Space and Doubling Measure
  • Lemma 2.2: Vitali Covering Lemma in Metric Spaces
  • Definition 2.3: $(q,p)$-Poincaré Inequality
  • Definition 2.4: Variable Exponent Lebesgue Spaces
  • Definition 2.5: Orlicz Spaces
  • Theorem 2.6: Finite Difference Characterization on Metric Measure Spaces
  • Definition 3.1: Polynomial Growth Measure
  • Lemma 3.2: Modified Vitali Covering Lemma
  • proof
  • Theorem 3.3: Finite Difference Characterization under Polynomial Growth
  • ...and 34 more