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I'm not able to prove the following inequality: Fix $s>0$

$$\|fg\|_{H^s}\lesssim \|fJ^sg\|_{L^2}+\|gJ^sf\|_{L^2},$$ where $\widehat{J^sf}(\xi)=(1+|\xi|^2)^{s/2}\hat{f}(\xi)$ (Bessel potential).

Actually, I used the inequality $(1+|\xi|^2)^{s/2}\lesssim (1+|\xi-y|^2)^{s/2}+(1+|y|^2)^{s/2}$ (obvious), to "spread" the weight in the convolution, but I had problems to finish.

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