Optimal Calder\'on Spaces for generalized Bessel potentials

TL;DR

This paper studies the properties of Calderón spaces with generalized smoothness and characterizes the optimal spaces for embedding generalized Bessel potentials, including those with non-power singularities.

Contribution

It provides new criteria for embeddings of generalized Bessel potentials into Calderón spaces and identifies the optimal spaces for these embeddings.

Findings

Established order-sharp estimates for moduli of continuity of potentials.

Derived embedding criteria for potentials into Calderón spaces.

Identified optimal spaces for embedding generalized Bessel potentials.

Abstract

In the paper we investigate the properties of spaces with generalized smoothness, such as Calder\'on spaces that include the classical Nikolskii-Besov spaces and many of their generalizations, and describe differential properties of generalized Bessel potentials that include classical Bessel potentials and Sobolev spaces. Kernels of potentials may have non-power singularity at the origin. With the help of order-sharp estimates for moduli of continuity of potentials, we establish the criteria of embeddings of potentials into Calder\'on spaces, and describe the optimal spaces for such embeddings.