Brezis-Van Schaftingen-Yung Inequalities Beyond the Classical Setting

TL;DR

This paper generalizes Brezis--Van Schaftingen--Yung inequalities to non-doubling measures, variable exponent spaces, and anisotropic settings, with applications to nonlocal operators and PDE regularity.

Contribution

It introduces new fractional and anisotropic inequalities in metric measure spaces with nonstandard growth conditions, extending classical results significantly.

Findings

Established fractional Sobolev inequalities in non-doubling measures

Developed anisotropic and directional inequalities for non-isotropic structures

Analyzed stability and sharpness of inequality constants

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.