Sobolev embeddings for Herz-type Triebel-Lizorkin spaces

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.