A Kuran Type Regularity Criterion for Sub-Laplacians

The aim of this paper is to show the strong connection between regularity of bounded open set boundary points and quasi-boundedness, on the same set, of the fundamental solution of stratified Lie group sub-Laplacians. In the euclidean case the theorem was proved by Kuran (J Lond Math Soc 2(19):301–311, 1979). We later give two examples using some direct consequences of main theorem.