Ultradistributional boundary values of harmonic functions on the sphere

TL;DR

This paper develops a comprehensive theory of ultradistributional boundary values for harmonic functions on the unit ball, linking ultradifferentiable functions and ultradistributions on the sphere through spherical harmonic analysis.

Contribution

It introduces explicit derivative estimates for spherical harmonics and characterizes ultradistributions on the sphere using Abel summability, advancing boundary value theory.

Findings

Derived explicit derivative bounds for spherical harmonics

Characterized ultradifferentiable functions via spherical harmonic expansions

Connected support of ultradistributions with Abel summability

Abstract

We present a theory of ultradistributional boundary values for harmonic functions defined on the Euclidean unit ball. We also give a characterization of ultradifferentiable functions and ultradistributions on the sphere in terms of their spherical harmonic expansions. To this end, we obtain explicit estimates for partial derivatives of spherical harmonics, which are of independent interest and refine earlier estimates by Calderon and Zygmund. We apply our results to characterize the support of ultradistributions on the sphere via Abel summability of their spherical harmonic expansions.