Nonlocal Meyers' Example

TL;DR

This paper introduces nonlocal variants of Meyers' example, exploring fractional Laplacians and their limits, to study higher integrability and differentiability properties in nonlocal contexts.

Contribution

It develops nonlocal versions of Meyers' example using fractional Laplacians, bridging local and nonlocal analysis and examining their limiting behavior as s approaches 1.

Findings

Nonlocal Meyers' examples exhibit limited higher integrability.

In the limit s→1, nonlocal examples recover classical Meyers' behavior.

Different fractional Laplacian definitions are analyzed for their properties.

Abstract

We present nonlocal variants of the famous Meyers' example of limited higher integrability and differentiability. In the limit $s \nearrow 1$ we recover the standard Meyers' example. We consider the fractional Laplacian based on differences as well as the one based on fractional derivatives defined by Riesz potentials.