$H{W}_{\mathrm{l}\mathrm{o}\mathrm{c}}^{2,2}$-regularity for $p$-harmonic functions in Heisenberg groups

Let $1<p\le 4$ when $n=1$ and $1<p<3+\frac{1}{n-1}$ when $n\ge 2$. We obtain the second-order horizontal Sobolev $H{W}_{\mathrm{l}\mathrm{o}\mathrm{c}}^{2,2}$-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.