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

Douadi Drihem

Published 2018-08-24Version 1

We present some Caffarelli-Kohn-Nirenberg-type inequalities on Herz-type Besov-Triebel-Lizorkin 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*} with some appropriate assumptions on the parameters, where $\dot{k}_{v,\sigma }^{\alpha _{1},r}$ is the Herz-type Bessel potential spaces. To do these, we study when distributions belonging to these spaces can be interpreted as functions in $L_{\mathrm{loc}}^{1}$. Our main tools is the usual Littlewood-Paley technique, Sobolev and Franke embeddings, and interpolation theory. Our results improve their results in some sense.