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Let me expand my comment into an answer. Let $s>\frac12$ and pick any $r\in(\frac12,s)$. Interpolating between Sobolev spaces gives $\|u\|_{H^r_P}\lesssim\|u\|_{H^0_P}^\delta\|u\|_{H^s_P}^{1-\delta}$ for some $\delta>0$. As was shown to you in the MSE answer to the question, $\|u\|_{L^\infty}\lesssim\|u\|_{H^r_P}$. Putting these estimates together ...

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If $A$ is $n\times n$, then $$\frac1{\sqrt n}\|A\|_F\le\|A\|_1\le\sqrt n\,\|A\|_F,\qquad \|A\|_2\le\|A\|_F\le\sqrt n\,\|A\|_2.$$ More generally, if $A$ is $n\times m$, then $$\frac1{\sqrt m}\|A\|_F\le\|A\|_1\le\sqrt n\,\|A\|_F,\qquad \|A\|_2\le\|A\|_F\le\min(\sqrt n\,,\sqrt m)\,\|A\|_2.$$ To see that these inequalities are sharp, take respectively the ...

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