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

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.