A comparison theorem for subharmonic functions

TL;DR

This paper extends classical theorems for subharmonic functions, establishing conditions under which functions that agree almost everywhere on a surface must coincide, with applications to geometric measure theory.

Contribution

It introduces new comparison and mean value theorems for subharmonic functions and explores their implications for functions agreeing on surfaces of various codimensions.

Findings

Positive answer for hypersurfaces

Counterexample for higher co-dimension surfaces

Applications to Ahlfors-David sets

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.