Caffarelli-Kohn-Nirenberg Inequalities for Besov and Triebel-Lizorkin-Type Spaces

Abstract

We present some Caffarelli-Kohn-Nirenberg-type inequalities for Herz-type Besov-Triebel-Lizorkin spaces, Besov-Morrey and Triebel-Lizorkin-Morrey spaces. More precisely, we investigate the inequalities

\[ \|f\|_{\dot{k}^{\alpha_1, r}_{v, \sigma}} \leq c \|f\|_{\dot{K}^{1-\theta}_{\alpha_2, \delta}} \|f\|_{\dot{K}^{\theta}_{\alpha_3, \delta_1} A^s_\beta} \]

and

\[ \|f\|_{\mathcal{E}^{\sigma}_{p, 2, u}} \leq c \|f\|^{1-\theta}_{\dot{M}^\delta_\mu} \|f\|^{\theta}_{\mathcal{N}^{s}_{q, \beta, v}} \]

with some appropriate assumptions on the parameters, where \(\dot{k}^{\alpha_1, r}_{v, \sigma}\) are 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 \(\dot{K}^{\alpha_3, \delta_1}_{p} A^s_\beta\) are Besov or Triebel-Lizorkin spaces if \(\alpha_3 = 0\) and \(\delta_1 = p\). The usual Littlewood-Paley technique, Sobolev and Franke embeddings are the main tools of this paper. Some remarks on Hardy-Sobolev inequalities are given.