Caffarelli–Kohn–Nirenberg inequalities for Besov and Triebel–Lizorkin-type spaces

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
$$ ||f||_{\dot{k}_{v,\sigma}^{\alpha_1,r}}\leqslant c||f||_{\dot{K}_{u}^{\alpha_2,\delta}}^{1-\theta}||f||_{\dot{K}_{p}^{\alpha_3,\delta_1}A_\beta^s}^\theta $$
and
$$ ||f||_{\mathcal{E}_{p,2,u}^\sigma}\leqslant c||f||_{M_\mu^\delta}^{1-\theta}||f||_{\mathcal{N}_{q,\beta,v}}^\theta, $$
with some appropriate assumptions on the parameters, where $\dot{k}_{v,\sigma}^{\alpha_1,r}$ 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 $\dot{K}_p^{\alpha_3,\delta_1}A_\beta^s$ 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.