Affine Balayage of Measures and Distribution of Riesz Measures of Subharmonic Functions

TL;DR

This paper advances potential theory by exploring balayage, duality, and distribution of Riesz measures of subharmonic functions, with applications to zero sets of holomorphic functions and new theoretical results.

Contribution

It introduces new theorems on gluing subharmonic functions, describes duality between measures and potentials, and generalizes the Poisson-Jensen formula.

Findings

Theorems on gluing subharmonic functions and Green's functions.

Internal description of measure-potential duality.

Generalized Poisson-Jensen formula for subharmonic functions.

Abstract

We develop and use some key concepts of potential theory, such as balayage and duality between measures and their potentials, to study the distribution of masses of subharmonic functions while restrictions to their growth near the boundary of their domain of definition. They are formulated and proved in terms of the Riesz measure for a subharmonic function. In this article, applications to holomorphic functions concern the distribution of their zero sets under restrictions to growth of this functions. Auxiliary results of our article may also be of independent interest. This is, first of all, 1) theorems on gluing of subharmonic functions and on gluing a subharmonic function with Green's function; 2) an internal description of the duality between balayage of measures and their potentials; 3) the duality between the Jensen and Arens-Singer measures on the one hand and the Jensen and Arens-Singer potentials of on the other; 4) a generalization of classical Poisson-Jensen formula connecting subharmonic functions, their Riesz measures, harmonic measures, and Green's functions.