Analytic solutions of convolution equations on convex sets with a mixed structure. I

TL;DR

This paper establishes an abstract criterion ensuring the existence of continuous linear right inverses for surjective convolution operators on spaces of analytic functions over convex sets with mixed structures, using subharmonic functions.

Contribution

It introduces a novel criterion based on subharmonic functions for the invertibility of convolution operators on complex convex sets with a mixed structure.

Findings

Proves a criterion linking subharmonic functions to convolution operator invertibility.

Shows that certain convex sets admit a continuous linear right inverse for convolution operators.

Provides conditions under which convolution equations can be solved analytically on complex convex domains.

Abstract

We prove an abstract criterion that a surjective convolution operator in spaces of analytic functions on convex subsets of the complex plane has a continuous linear right inverse. Considered convex sets have a countable neighborhood basis of convex domains. The mentioned criterion is obtained in terms of the existence of a special family of subharmonic functions with global upper bounds and local lower bounds.