Local second order horizontal Sobolev regularity for p-harmonic functions with Hörmander vector fields of step two

Abstract

In this paper, we establish a trace inequality for any real symmetric square matrix and apply it to Hörmander vector fields of step two, which are denoted by $X_1,\dots ,X_m$. Let $1<p\le 4$ when $m=2,3$ and $1<p<3+\frac{2}{m-2}$ when $m\ge 4$. Then we utilize the trace inequality to prove the horizontal Sobolev $W_{X,\mathrm{loc}}^{2,2}$-regularity of the weak solution u to the degenerate subelliptic p-harmonic equation ${△}_{X,p}u\left(x\right)=\sum _{i=1}^m{X}_i^{⁎}\left(\right|Xu{|}^{p-2}{X}_{i}u)=0$, namely, $XXu\in L_{\mathrm{loc}}^2$. Compared to the case of Euclidean spaces $R^m$ ($m\ge 4$), the range of this determined p is already optimal.