On an Application of Hypoellipticity to Solutions of Functional Equations

TL;DR

This paper explores the regularity of solutions to generalized mean value type functional equations using hypoellipticity, providing sufficient conditions and confirming a related conjecture, advancing the understanding of solution smoothness.

Contribution

It introduces a novel approach applying hypoellipticity to establish regularity conditions for solutions of functional equations, and confirms a conjecture by H. Swiatak.

Findings

Established sufficient conditions for solution regularity

Confirmed H. Swiatak's conjecture on functional equations

Extended the application of hypoellipticity in functional analysis

Abstract

We study the regularity of solutions of functional equations of a generalized mean value type. In this paper we give sufficient conditions for the regularity by using hypoellipticity which is a concept of the theory of partial differential equations. Further we also give an affirmative answer to a conjecture of H.\'Swiatak. A part of the results was announced in the comprehensive paper [8] on series of our joint works. To prove the regularity of solutions of functional equation is in general one of central problem in the theory of functional equations.