A comparison theorem for subharmonic functions

Abstract

In this article, we prove an extension of the mean value theorem and a comparison theorem for subharmonic functions. These theorems are used to answer the question whether we can conclude that two subharmonic functions which agree almost everywhere on a surface with respect to the surface measure must coincide everywhere on that surface. We prove that this question has a positive answer in the case of hypersurfaces, and we also provide a counterexample in the case of surfaces of higher co-dimension. We also apply these results to Ahlfors-David sets and we prove other versions of the main results in terms of measure densities.