H\"older continuity and boundedness estimates for nonlinear fractional equations in the Heisenberg group

TL;DR

This paper extends regularity theory to nonlinear fractional equations on the Heisenberg group, proving boundedness and H"older continuity of solutions using nonlocal integro-differential operators.

Contribution

It generalizes De Giorgi-Nash-Moser theory to a broad class of nonlocal, possibly degenerate, equations on the Heisenberg group, including fractional p-Laplacian models.

Findings

Solutions are bounded and H"older continuous.

Established fractional Caccioppoli-type estimates with tail.

Derived logarithmic-type estimates for solutions.

Abstract

We extend the celebrate De Giorgi-Nash-Moser theory to a wide class of nonlinear equations driven by nonlocal, possibly degenerate, integro-differential operators, whose model is the fractional $p$-Laplacian operator on the Heisenberg-Weyl group $\mathbb{H}^n$. Amongst other results, we prove that the weak solutions to such a class of problems are bounded and H\"older continuous, by also establishing general estimates as fractional Caccioppoli-type estimates with tail and logarithmic-type estimates.