$C^{1,α}$-regularity for Mixed Local and Nonlocal Degenerate Elliptic Equations in the Heisenberg Group

TL;DR

This work studies $C^{1,α}$ regularity for weak solutions to mixed local and nonlocal degenerate elliptic equations on the Heisenberg group $\mathbb{H}^n$. It builds a Morrey-type iterative framework that couples $X$-direction difference quotients with Nirenberg differences and a fractional Sobolev inequality on $\mathbb{H}^n$, establishing Hölder continuity of solutions and subsequently Hölder regularity of the horizontal gradient. A key novelty is obtaining a parameter-free Hölder exponent for the solution and deriving gradient estimates that feed into a known regularity result to yield $C^{1,α}$ regularity (via Theorem $1.2$ in Mukherjee and Zhong). The results extend the sub-Riemannian regularity theory to mixed local-nonlocal diffusion, providing a unified approach for degenerate equations in the Heisenberg setting.

Abstract

The regularity theory for equations combining both local and nonlocal operators in sub-Riemannian geometries is a huge challenge. In this paper, we investigate the $C^{1,α}$-regularity of weak solutions to mixed local and nonlocal degenerate elliptic equations on the Heisenberg group. We first derive a sophisticated iteration scheme of Morrey-type by leveraging horizontal difference quotient combined with the Nirenberg difference quotient and fractional Sobolev-type inequality on the Heisenberg group. Then, the Hölder continuity of the weak solutions is established by applying the local boundedness, the iteration scheme of Morrey-type, an iterative method and the Morrey inequality. Finally, we use the Hölder continuity in conjunction with Theorem 1.2 from Mukherjee and Zhong[18] to prove the $C^{1,α}$-regularity of weak solutions.

Theorems & Definitions (25)

• Theorem 1.1
• Remark 1.2
• Theorem 1.3
• Lemma 2.1: CL97
• Lemma 2.2: Morrey inequality (F75, Theorem 5.15)
• Definition 2.3
• Lemma 2.4
• proof
• Lemma 2.5
• proof