Caffarelli-Kohn-Nirenberg inequalities on Besov and Triebel-Lizorkin-type spaces

We present some Caffarelli-Kohn-Nirenberg-type inequalities on Herz-type Besov-Triebel-Lizorkin spaces, Besov-Morrey spaces and Triebel-Lizorkin-Morrey spaces. More Precisely, we investigate the inequalities \begin{equation*} \big|f\big|{\dot{k}{v,\sigma }^{\alpha_{1},r}}\leq c\big|f\big|{\dot{K}{u}^{\alpha_{2},\delta }}^{1-\theta }\big|f\big|{\dot{K}{p}^{\alpha_{3},\delta_{1}}A_{\beta }^{s}}^{\theta }, \end{equation*} and \begin{equation*} \big|f\big|{$\mathcal{E}${p,2,u}^{\sigma }}\leq c\big|f\big|{$\mathcal{M}${\mu }^{\delta }}^{1-\theta }\big|f\big|{$\mathcal{N}${q,\beta ,v}^{s}}^{\theta }, \end{equation*} with some appropriate assumptions on the parameters, where $\dot{k}{v,\sigma }^{\alpha{1},r}$ is the Herz-type Bessel potential spaces, which are just the Sobolev spaces if ${\alpha }_1=0,1<r=v<\mathrm{\infty }$ and $% \sigma \in $\mathbb{N}${0}$,and$\dot{K}{p}^{\alpha_{3},\delta_{1}}A_{\beta }^{s}$ are Besov or Triebel-Lizorkin spaces if ${\alpha }_3=0$ and${\delta }_1=p$. To do these, we study when distributions belonging to these spaces can be interpreted as functions in $L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1$. The usual Littlewood-Paley technique, Sobolev and Franke embeddings are the main tools of this paper. Some remarks on Hardy-Sobolev inequalities are given.