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Let $g_{\mu}(x) = \mu^{d/2}\exp(-\pi\mu|x|^2)$ for every $\mu > 0$. Prove that $$ \int_{\mathbb R^{d}}\left|(-\Delta)^{\frac{s}{2}} u\right|^{2} \geq \int_{\mathbb R^{4}}\left|(-\Delta)^{\frac{s}{2}} \sqrt{|u|^{2}*g_{\mu}}\right|^{2} $$ for every $u \in H^s(\mathbb R^d)$ with $0<s<1$.

Let $g_{\mu}(x) = \mu^{d/2}\exp(-\pi\mu|x|^2)$ for every $\mu > 0$. Prove that $$ \int_{\mathbb R^{d}}\left|(-\Delta)^{\frac{s}{2}} u\right|^{2} \geq \int_{\mathbb R^{d}}\left|(-\Delta)^{\frac{s}{2}} \sqrt{|u|^{2}*g_{\mu}}\right|^{2} $$ for every $u \in H^s(\mathbb R^d)$ with $0<s<1$.