Rad\'{o}- type theorem for subharmonic and plurisubharmonic functions

TL;DR

This paper generalizes a recent Radó-type theorem for subharmonic functions to plurisubharmonic functions using viscosity techniques and studies the complexity of critical and non-differentiability sets.

Contribution

It extends Radó-type theorems to a broader class of functions and improves viscosity methods, also analyzing the Borel complexity of critical sets.

Findings

Generalized Radó-type theorems for plurisubharmonic functions

Enhanced viscosity techniques for function analysis

Determined Borel complexity of critical and non-differentiability sets

Abstract

We observe that a recent result by Gardiner and Sj\"odin, solving a problem of Kr\'{a}l on subharmonic functions, can be easily generalized to yield a somewhat stronger result. This can be combined with a viscosity technique of ours, which we slightly improve, to obtain Rad\'{o}- type theorems for plurisubharmonic functions. Finally, we study the Borel complexity of the critical set and the set where the gradient does not exist finitely of subharmonic functions and general real valued functions.