A note on estimates of level sets and their role in demonstrating regularity of solutions to nonlocal double phase equations

TL;DR

This paper establishes an estimate on the level sets of solutions to nonlocal double phase equations with $(p, q)$ growth, highlighting a self-improving property that contributes to understanding their regularity.

Contribution

It introduces a novel estimate on level sets for solutions to nonlocal double phase equations, linking it to their regularity and self-improving properties.

Findings

Level set estimate depends on difference quotient of solutions

Self-improving property demonstrated for solutions

Enhances understanding of regularity in nonlocal double phase equations

Abstract

In this note we prove an estimate on the level sets of a function with $(p, q)$ growth that depends on the difference quotient of a bounded weak solution to a nonlocal double phase equation. This estimate is related to a self improving property of these solutions.