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 $\mathbb{H}^n$.
@article{1238418798,
author = {Domokos, Andr\`as and Manfredi, Juan J.},
title = {A second order differentiability technique of Bojarski-Iwaniec in the Heisenberg group},
journal = {Funct. Approx. Comment. Math.},
volume = {40},
number = {1},
year = {2009},
pages = { 69-74},
language = {en},
url = {http://dml.mathdoc.fr/item/1238418798}
}
Domokos, Andràs; Manfredi, Juan J. A second order differentiability technique of Bojarski-Iwaniec in the Heisenberg group. Funct. Approx. Comment. Math., Tome 40 (2009) no. 1, pp. 69-74. http://gdmltest.u-ga.fr/item/1238418798/