$$p$$ -Harmonic functions in the Heisenberg group: boundary behaviour in domains well-approximated by non-characteristic hyperplanes

Abstract

In this paper we study, for given \(p,~1<p<\infty \), the boundary behaviour of non-negative \(p\)-harmonic functions in the Heisenberg group \(\mathbb{H }^n\), i.e., we consider weak solutions to the non-linear and potentially degenerate partial differential equation $$\begin{aligned} \sum _{i=1}^{2n}X_i(|Xu|^{p-2}\,X_i u)=0 \end{aligned}$$ where the vector fields \(X_1,\ldots ,X_{2n}\) form a basis for the space of left-invariant vector fields on \(\mathbb{H }^n\). In particular, we introduce a set of domains \(\Omega \subset \mathbb{H }^n\) which we refer to as domains well-approximated by non-characteristic hyperplanes and in \(\Omega \) we prove, for \(2\le p<\infty \), the boundary Harnack inequality as well as the Holder continuity for ratios of positive \(p\)-harmonic functions vanishing on a portion of \(\partial \Omega \).