Quasianalytic functionals and ultradistributions as boundary values of harmonic functions

TL;DR

This paper investigates boundary values of harmonic functions within quasianalytic and ultradistribution spaces, introducing a new approach to Hörmander's support theorem and describing ultradifferentiable functions through almost harmonic functions.

Contribution

It presents a novel method linking ultradifferentiable functions to harmonic functions and extends Hörmander's support theorem to quasianalytic functionals.

Findings

New description of ultradifferentiable functions via almost harmonic functions

Extension of Hörmander's support theorem to quasianalytic functionals

Unified results for classes defined by weight sequences and functions

Abstract

We study boundary values of harmonic functions in spaces of quasianalytic functionals and spaces of ultradistributions of non-quasianalytic type. As an application, we provide a new approach to H\"ormander's support theorem for quasianalytic functionals. Our main technical tool is a description of ultradifferentiable functions by almost harmonic functions, a concept that we introduce in this article. We work in the setting of ultradifferentiable classes defined via weight matrices. In particular, our results simultaneously apply to the two standard classes defined via weight sequences and via weight functions.