Sobolev embeddings for Herz-type Triebel-Lizorkin spaces

TL;DR

This paper establishes Sobolev embeddings for Herz-type Triebel-Lizorkin spaces, extending classical results and including new embeddings involving weighted spaces, under specific parameter conditions.

Contribution

It provides new Sobolev embedding theorems for Herz-type Triebel-Lizorkin spaces and their weighted variants, broadening the understanding of these function spaces.

Findings

Proved Sobolev embeddings for Herz-type Triebel-Lizorkin spaces.

Derived new embeddings between Herz and Triebel-Lizorkin spaces.

Extended results to Triebel-Lizorkin spaces with power weights.

Abstract

In this paper we prove the Sobolev embeddings for Herz-type Triebel-Lizorkin spaces, \begin{equation*} \dot{K}_{q}^{\alpha_{2},r}F_{\theta }^{s_{2}}\hookrightarrow \dot{K}%_{s}^{\alpha_{1},p}F_{\beta }^{s_{1}} \end{equation*} where the parameters $\alpha_{1},\alpha_{2},s_{1},s_{2},s,q,r,p,\beta $ and $\theta $ satisfy some suitable conditions. An application we obtain new embeddings between Herz and Triebel-Lizorkin spaces. Moreover, we present the Sobolev embeddings for Triebel-Lizorkin spaces equipped with power weights. All these results cover the results on classical Triebel-Lizorkin spaces.