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

Abstract

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<\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 $L_{\mathrm{loc}}^{1}$. The usual Littlewood-Paley technique, Sobolev and Franke embeddings are the main tools of this paper. Some remarks on Hardy-Sobolev inequalities are given.