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

Abstract

Based on mixed-norm Herz spaces, we introduce the classes of mixed-norm Herz-type Besov and Triebel-Lizorkin spaces. We establish their $φ$-transform characterization in the sense of Frazier and Jawerth and prove Sobolev, Franke and Jawerth embedding theorems for these spaces. The obtained Sobolev embeddings extend and improve several known embedding results for mixed-norm Besov and Triebel-Lizorkin spaces. As a consequence of our results, the Franke and Jawerth embeddings for mixed-norm Besov and Triebel-Lizorkin spaces are established.