$HW^{2,2}_{\rm loc}$-regularity for $p$-harmonic functions in Heisenberg groups

TL;DR

This paper proves second-order horizontal Sobolev regularity for p-harmonic functions in Heisenberg groups, extending the known range of p-values and improving previous results by Domokos and Manfredi.

Contribution

It establishes new second-order regularity results for p-harmonic functions in Heisenberg groups, broadening the p-range compared to prior work.

Findings

Second-order horizontal Sobolev regularity for p-harmonic functions in Heisenberg groups.

Extended the p-range for regularity results beyond previous bounds.

Improved understanding of regularity properties in sub-Riemannian geometry.

Abstract

Let $1<p\le4$ when $n=1$ and $1<p< 3+\frac{1}{n-1} $ when $n\ge2$. We obtain the second-order horizontal Sobolev $HW^{2,2}_{\rm loc} $-regularity of $p$-harmonic functions in the Heisenberg group $\mathbb H^n$. This improves the known range of $p$ obtained by Domokos and Manfredi in 2005.