A Kuran type regularity criterion for sub-Laplacians. A necessary and sufficient condition
Let \(G\) be a stratified Lie group, \(\mathcal L\) its sub-Laplacian and \(\Gamma_x\) the fundamental solution of \(\mathcal L\) with pole at \(x.\) If \(\Omega\subset G\) is an open set, let \(\bar{H}_\varphi^\Omega\) and \(\underline{H}_\varphi^\Omega\) be the upper and the lower solution to the problem \(\mathcal Lu=0\) in \(\Omega\) and \(u|_{\partial\Omega}=\varphi\). If \(\overline r{H}_\varphi^\Omega=\underline{H}_\varphi^\Omega\) and they are \(\mathcal L\)-harmonic, then one writes \(H_\varphi^\Omega:=\overline{H}_\varphi^\Omega=\underline{H}_\varphi^\Omega.\) A point \(x\in\partial\Omega\) is called \(\mathcal L\)-regular if \(\lim_{\Omega\ni z\to x}H_\psi^\Omega=\psi(x)\) for every \(\psi\in C(\partial\Omega).\) A non-negative \(\mathcal L\)-harmonic function \(h\) in \(\Omega\) is said to be quasi-bounded in \(\Omega\) if there is an increasing sequence \(h_n\geq 0\) of bounded \(\mathcal L\)-harmonic functions in \(\Omega\) such that \(h=\lim_{n\to\infty}h_n\) in \(\Omega.\) The main theorem of the paper says that if \(\Omega\subset G\) is a bounded open set and \(x\in\partial\Omega\), then \(x\) is \(\mathcal L\)-regular if and only if \(\Gamma_x\) is quasi-bounded in \(\Omega.\)
scientific article; zbMATH DE number 5664593
scientific article