A second order differentiability technique of Bojarski-Iwaniec in the Heisenberg group

Abstract

We adapt a technique developed by Bojarski and Iwaniec in their celebrated 1983 paper [2] to prove second order differentiability results for $p$-harmonic functions to the case of the Heisenberg group. We prove that for $2\le p<4$ we have $X_i ( |Xu|^{(p-2)/p}\, X_j u) \in L^2_{\rm loc} (\Omega )$ and $X_i (|Xu|^p ) \in L^2_{\rm loc} (\Omega)$, where $u$ is a $p$-harmonic function in the Heisenberg group $\mathbb{H}^n$.