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,σ}^{α_{1},r}}\leq c\big\|f\big\|_{\dot{K}_{u}^{α_{2},δ}}^{1-θ}\big\|f\big\|_{\dot{K}_{p}^{α_{3},δ_{1}}A_{β}^{s}}^{θ}, \end{equation*} and \begin{equation*} \big\|f\big\|_{\mathcal{E}_{p,2,u}^{σ}}\leq c\big\|f\big\|_{\mathcal{M}_{μ}^{δ}}^{1-θ}\big\|f\big\|_{\mathcal{N}_{q,β,v}^{s}}^{θ}, \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 , 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.

arXiv comment36 pages