On generalized Besov and Triebel-Lizorkin spaces of regular distributions

TL;DR

This paper characterizes when generalized Besov and Triebel-Lizorkin spaces contain only regular distributions and explores their embeddings into Lebesgue spaces, providing necessary and sufficient conditions.

Contribution

It establishes necessary and sufficient conditions for these function spaces to contain only regular distributions and links this to their embedding properties.

Findings

Conditions for regular distributions in generalized spaces

Embedding criteria into Lebesgue spaces

Characterization of space properties based on parameters

Abstract

We establish conditions on the parameters which are both necessary and sufficient in order that Besov and Triebel-Lizorkin spaces of generalized smoothness contain only regular distributions. We also connect this with the possibility of embedding such spaces in some particular Lebesgue spaces.