Regularity theory for nonlocal equations with general growth in the Heisenberg group

TL;DR

This paper develops a regularity theory for nonlocal G-Laplace equations within the Heisenberg group, establishing boundedness, Hölder continuity, and Harnack inequalities for weak solutions using a unified approach.

Contribution

It introduces a unified method to analyze local properties of solutions to nonlocal G-Laplace equations in the Heisenberg group, extending classical regularity results.

Findings

Proved boundedness and Hölder continuity of solutions.

Established Harnack inequality for weak solutions.

Developed an improved nonlocal Caccioppoli estimate.

Abstract

We deal with a wide class of generalized nonlocal $p$-Laplace equations, so-called nonlocal $G$-Laplace equations, in the Heisenberg framework. Under natural hypotheses on the $N$-function $G$, we provide a unified approach to investigate in the spirit of De Giorgi-Nash-Moser theory, some local properties of weak solutions to such kind of problems, involving boundedness, H\"{o}lder continuity and Harnack inequality. To this end, an improved nonlocal Caccioppoli-type estimate as the main auxiliary ingredient is exploited several times.