Weighted endpoint fractional Leibniz rule

Abstract

We prove that the weighted endpoint fractional Leibniz rules

$$\begin{aligned} \Vert D^s (fg)\Vert _{L^{p}_{w}({{\mathbb {R}}^{n}})}\lesssim & {} \Vert D^s f\Vert _{L^{1}_{w_1}({{\mathbb {R}}^{n}})} \Vert g\Vert _{L^{p_2}_{w_2}({{\mathbb {R}}^{n}})} + \Vert f\Vert _{L^{1}_{w_1}({{\mathbb {R}}^{n}})}\Vert D^s g\Vert _{L^{p_2}_{w_2}({{\mathbb {R}}^{n}})}, \\ \Vert J^s (fg)\Vert _{L^{p}_{w}({{\mathbb {R}}^{n}})}\lesssim & {} \Vert J^s f\Vert _{L^{1}_{w_1}({{\mathbb {R}}^{n}})} \Vert g\Vert _{L^{p_2}_{w_2}({{\mathbb {R}}^{n}})} + \Vert f\Vert _{L^{1}_{w_1}({{\mathbb {R}}^{n}})}\Vert J^s g\Vert _{L^{p_2}_{w_2}({{\mathbb {R}}^{n}})} \end{aligned}$$

hold for $1\le p_2< \infty , 1/2\le p<\infty$ with $1/p=1+1/p_2$ and $s>\max \left\{ n(\frac{\tau (w)}{p}-1),0\right\}$ or $s\in 2{\mathbb {N}},$ provided that $w_1\in A_{1}({{\mathbb {R}}^{n}})$ and $w_2\in A_{p_2}({{\mathbb {R}}^{n}})$. Our result complements and extends some existing results.