Asymptotic mean value formula for sub-\(p\)-harmonic functions on the Heisenberg group
The authors investigate an asymptotic mean value formula of sub-\(p\)-harmonic functions in the viscosity sense on the Heisenberg group. It is known that \(p\)-harmonic functions have an asymptotic mean value property in the Euclidean case. More precisely, \(u\) is a \(p\)-harmonic function in a domain \(\Omega \subset \mathbb{R}^n\) ( that is, \(-\triangle_{p}u= -\operatorname{div}(|\nabla u|^{p-2}\nabla u)=0\) in \(\Omega\) ) in the viscosity sense if and only if it satisfies the asymptotic mean value formula \[ u(x)=\frac{\alpha}{2}\{ \max_{\overline{B_{\epsilon}(x)}}u+ \min_{\overline{B_{\varepsilon}(x)}}u \} +\beta \frac{1}{|B_{\varepsilon}(x)|} \int_{B_{\varepsilon}(x)}u(y)\;dy+o(\varepsilon^2) \] as \(\varepsilon \rightarrow 0\) in the viscosity sense for \(x \in \Omega\) where the constants \(\alpha\) and \(\beta\) are given by \(\alpha =\displaystyle\frac{p-2}{p+n}\) and \(\beta = \displaystyle\frac{2+n}{p+n}\). This result follows from the Taylor expansion and the theory of viscosity solutions. The goal of the paper is to extend this analysis to sub-\(p\)-harmonic functions on the Heisenberg group. As an application, they give a new proof of the Harnack inequality for sub-\(p\)-harmonic functions on the Heisenberg group.
scientific article; zbMATH DE number 6161962