Recovering functions via doubly homogeneous nonlocal gradients

TL;DR

This paper studies a special class of nonlocal gradient operators with different behaviors at small and large scales, providing a representation formula, Sobolev inequalities, and proposing open questions for future research.

Contribution

It introduces a new class of doubly homogeneous nonlocal gradients, establishes their representation formulas, and derives related Sobolev inequalities, advancing understanding of nonlocal operators.

Findings

Representation formula for doubly homogeneous nonlocal gradients

Sobolev-type inequalities for these operators

Open questions for future research in nonlocal analysis

Abstract

We investigate a class of nonlocal gradients featuring distinct homogeneities at zero and infinity. We establish a representation formula for such doubly homogeneous operators and derive associated Sobolev-type inequalities. We also propose open questions linked to our results, suggesting directions for future research inspired by the work of Haim Brezis.