On the exactness of the conditions of embedding theorems for spaces of functions with mixed logarithmic smoothness

TL;DR

This paper investigates the precise conditions under which certain function spaces with mixed logarithmic smoothness, defined on multi-variable periodic functions, can be embedded into each other, clarifying the exactness of these embedding theorems.

Contribution

It establishes necessary and sufficient conditions for the embeddings between spaces of functions with mixed logarithmic smoothness, providing exact criteria.

Findings

Derived precise embedding conditions for the function spaces.

Clarified the exactness of the embedding theorems.

Extended understanding of spaces with mixed logarithmic smoothness.

Abstract

The article considers the Lorentz space $L_{p,\tau}(\mathbb{T}^{m})$, $2\pi$ of periodic functions of many variables and $S_{p,\tau,\theta}^{0, \overline{b}}\mathbf{B}$, $S_{p, \tau, \theta}^{0, \overline{b}}B$ -- spaces of functions with mixed logarithmic smoothness. The article establishes necessary and sufficient conditions for embedding the spaces $S_{p, \tau, \theta}^{0, \overline{b}}\mathbf{B}$ and $S_{p, \tau, \theta}^{ 0, \overline{b}}B$ into each other.