Local regularity for nonlocal double phase equations in the Heisenberg group

TL;DR

This paper establishes interior boundedness and Hölder continuity for solutions of nonlocal double phase equations in the Heisenberg group, advancing the understanding of nonlinear integro-differential problems in this setting.

Contribution

It introduces new regularity results for nonlocal double phase equations in the Heisenberg group, including a novel Sobolev-Poincaré inequality and extends De Giorgi-Nash-Moser theory.

Findings

Proved interior boundedness of solutions

Established Hölder continuity of solutions

Developed a new Sobolev-Poincaré inequality

Abstract

We prove interior boundedness and H\"{o}lder continuity for the weak solutions of nonlocal double phase equations in the Heisenberg group $\mathbb{H}^n$. This solves a problem raised by Palatucci and Piccinini et. al. in 2022 and 2023 for nonlinear integro-differential problems in the Heisenberg group $\mathbb{H}^n$. Our proof of the a priori estiamtes bases on the spirit of De Giorgi-Nash-Moser theory, where the important ingredients are Caccioppoli-type inequality and Logarithmic estimate. To achieve this goal, we establish a new and crucial Sobolev-Poincar\'{e} type inequality in local domain, which may be of independent interest and potential applications.