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

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 combined with the 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\cite{MZ21} to prove the $C^{1,α}$-regularity of weak solutions.

💡 Research Summary

The paper addresses the regularity problem for a class of degenerate elliptic equations that combine a local p‑sub‑Laplacian and a non‑local fractional p‑sub‑Laplacian on the Heisenberg group (\mathbb H^{n}). The model equation is
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