On the regularity of p-harmonic functions in the Heisenberg group

Abstract

We describe some recent results obtained in [29], where we prove regularity theorems for sub-elliptic equations in (horizontal) divergence form defined in the Heisenberg group, and exhibiting polynomial growth of order p. The main result tells that when p ε [2,4) solutions to possibly degenerate equations are locally Lipschitz continuous with respect to the intrinsic distance. In particular, such result applies to p-harmonic functions in the Heisenberg group. Explicit estimates are obtained, and eventually applied to obtain the suitable Calderón-Zygmund theory for the associated non-homogeneous problems.