Lorentz Herz-type Besov-Triebel-Lizorkin spaces

TL;DR

This paper introduces a new family of Besov and Triebel-Lizorkin type function spaces, providing their characterizations, embeddings, decompositions, and examples, advancing the theoretical understanding of these spaces.

Contribution

It presents the first $ ext{φ}$-transform characterization, embeddings, and atomic decompositions for these new function spaces, along with illustrative examples.

Findings

Established $ ext{φ}$-transform characterization.

Proved Sobolev and Franke-Jewarth embeddings.

Developed atomic, molecular, and wavelet decompositions.

Abstract

In this paper, we introduce a new family of function spaces of Besov and Triebel-Lizorkin type. We present the $\varphi $-transform characterization of these spaces in the sense of Frazier and Jawerth and we prove their Sobolev and Franke-Jewarth embeddings. Also, we establish the smooth atomic, molecular and wavelet decomposition of these function spaces. Characterizations by ball means of differences are given. Finally, we investigate a series of examples which play an important role in the study of function spaces of Besov-Triebel-Lizorkin type.