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Denote by $D^{\alpha}=(-\triangle)^{\frac{\alpha}{2}}$,then we have $$\|D^{\alpha}(f\cdot g)\|\leq g)\| \leq C(\|D^{\alpha+s}(f)\|_{p_1}\|D^{-s}(g)\|_{q_1}+\|D^{\alpha+t}(f)\|_{p_2}\|D^{-t}(f)\|_{q_2})$$ where $\alpha$,s,t are positive real numbers,and $\frac{1}{p}=\frac{1}{p_{i}}+\frac{1}{q_{i}}$ with $i=1,2$. The proof can be seen in [Exact smoothing properties of Schrödinger semigroups]（ http://www.jstor.org/stable/10.2307/25098514.）

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Denote by $J^{\alpha}=(1-\triangle)^{\frac{\alpha}{2}}$,and by $D^{\alpha}=(-\triangle)^{\frac{\alpha}{2}}$,then we have $$\|J^{\alpha}(fg)\|{p}\leq C(\|J^{\alpha+s}(f)\|{p_1}\|J^{-s}(g)\|_{q_1}+\|J^{\alpha+t}(f)\|_{p_2}\|J^{-t}(f)\|_{q_2})$$,where $\|D^{\alpha}(f\cdot g)\|\leq C(\|D^{\alpha+s}(f)\|_{p_1}\|D^{-s}(g)\|_{q_1}+\|D^{\alpha+t}(f)\|_{p_2}\|D^{-t}(f)\|_{q_2})$$where 1{p}\leq C(\|D^{\alpha+s}(f)\|{p_1}\|D^{-s}(g)\|_{q_1}+\|D^{\alpha+t}(f)\|_{p_2}\|D^{-t}(f)\|_{q_2})$$ \alpha$,s,t are positive real numbers,and $\frac{1}{p}=\frac{1}{p_{i}}+\frac{1}{q_{i}}$ with $i=1,2$. The proof can be seen in [Exact smoothing properties of Schrödinger semigroups]（ http://www.jstor.org/stable/10.2307/25098514.）

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