I have to prove the following inequality $$\Vert fg\Vert_{H^s(\mathbb{R}^3)}\leq C \Vert f\Vert_{H^\sigma(\mathbb{R}^3)}\Vert g\Vert_{H^{s+1}(\mathbb{R}^3)}$$ with $s\geq 0$, $\sigma\geq\max\{\frac{1}{2},s\}$ and $(s,\sigma)\neq (\frac{1}{2},\frac{1}{2})$.
I've tried to use the following generalization of Leibnitz rule but without success $$\Vert J^s(fg)\Vert_p\leq \Vert J^{s+a}(f)\Vert_{p_1}\Vert J^{-a}(g)\Vert_{q_1}+\Vert J^{-b}f\Vert_{p_2}\Vert J^{s+b}(g)\Vert_{q_2}$$ where $J^s=(1-\Delta)^{\frac{s}{2}}$ and $\frac{1}{p}=\frac{1}{p_i}+\frac{1}{q_i}$ and $s,a,b\geq 0$. Any help?