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
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 Hölder continuity for ratios of positive \(p\)-harmonic functions vanishing on a portion of \(\partial \Omega \).
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Nyström, K. \(p\)-Harmonic functions in the Heisenberg group: boundary behaviour in domains well-approximated by non-characteristic hyperplanes. Math. Ann. 357, 307–353 (2013). https://doi.org/10.1007/s00208-013-0896-3
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DOI: https://doi.org/10.1007/s00208-013-0896-3