On embedding theorems of spaces of functions with mixed logarithmic smoothness

TL;DR

This paper investigates the properties of Lorentz spaces with mixed logarithmic smoothness, establishing equivalent norms and proving embedding theorems for these function spaces.

Contribution

It introduces new equivalent norms for spaces with mixed logarithmic smoothness and proves embedding theorems, advancing the understanding of these function spaces.

Findings

Equivalent norms for spaces with mixed logarithmic smoothness established

Embedding theorems for these spaces proved

Enhanced understanding of Lorentz spaces with mixed smoothness

Abstract

The article considers the Lorentz space $L_{p,\tau}(\mathbb{T}^{m})$, $2\pi$ of periodic functions of many variables and spaces with mixed logarithmic smoothness. Equivalent norms of a space with mixed logarithmic smoothness are found and embedding theorems are proved.