Regularity properties of distributions through sequences of functions

TL;DR

This paper establishes criteria to determine when a distribution is smooth or uniformly Hölder continuous based on approximation sequences by smooth functions, especially those from regularizations.

Contribution

It provides necessary and sufficient conditions for distribution regularity using approximation sequences, advancing understanding of distribution smoothness criteria.

Findings

Criteria for distribution smoothness via approximation sequences

Characterization of Hölder continuity in distributions

Conditions involving regularizations $(T*\phi_{n})$

Abstract

We give necessary and sufficient criteria for a distribution to be smooth or uniformly H\"{o}lder continuous in terms of approximation sequences by smooth functions; in particular, in terms of those arising as regularizations $(T\ast\phi_{n})$.