Mixed-norm Herz-type Besov-Triebel-Lizorkin spaces

TL;DR

This paper introduces a new class of function spaces called mixed-norm Herz-type Besov-Triebel-Lizorkin spaces, providing their characterization and embedding properties, which extend existing results in the field.

Contribution

It defines and characterizes mixed-norm Herz-type Besov-Triebel-Lizorkin spaces and establishes improved embedding theorems compared to prior mixed-norm spaces.

Findings

Established $ ext{φ}$-transform characterization of the new spaces.

Proved Sobolev, Franke, and Jawerth embeddings for these spaces.

Extended and improved classical embeddings for mixed-norm Besov and Triebel-Lizorkin spaces.

Abstract

Based on mixed-norm Herz spaces, Besov and Triebel-Lizorkin spaces, we introduce the so called mixed-norm Herz-type Besov-Triebel-Lizorkin spaces. We present the $\varphi $-transform characterization of these spaces in the sense of Frazier and Jawerth and we prove their Sobolev, Franke and Jawerth embeddings. Our embeddings extend and improve Sobolev, Franke and Jawerth embeddings of mixed-norm Besov and Triebel-Lizorkin spaces.