Functional and variational aspects of nonlocal operators associated with linear PDEs

TL;DR

This paper introduces a general framework for non-local operators linked to linear PDEs, establishing new estimates, localization principles, and invariance properties, with applications to measure theory.

Contribution

It presents a novel difference quotient representation for non-local operators and generalizes known gradient estimates to broader function spaces.

Findings

Established new local to non-local estimates

Proved invariance of quasiconvexity in the non-local setting

Discussed applications to $\\mathcal{A}$-gradient measures

Abstract

We introduce a general difference quotient representation for non-local operators associated with a first-order linear operator. We establish new local to non-local estimates and strong localization principles in various spaces of functions, measures and distributions, which fully generalize those known for gradients. Under suitable assumptions, we also establish the invariance of quasiconvexity within the proposed local-nonlocal setting. Applications to the fine properties of $\mathcal A$-gradient measures are further discussed.