Is it true that $$\Vert u \Vert_{BMO(\mathbb R^2)} \lesssim_{s} \Vert u \Vert_{\dot H^s(\mathbb R^2)},$$ for $s \in (0,1)$, where $\dot H^s(\mathbb R)$ is the homogeneous fractional Sobolev space?
1 Answer
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For every $s\in(0,1)$, $H^s(\mathbb{R}^2)$ fails to embed into $L^p(\mathbb{R}^2)$, for some $p:=p(s)$ large enough. Hence your inequality can not be true, as BMO functions are locally $L^p$ (with a continuous embedding on bounded domains) for every $p<\infty$.

$\begingroup$ Thank you. What is the critical $p(s)$ for the embedding? $\endgroup$ Commented Feb 23, 2021 at 9:43

$\begingroup$ You are welcome, It is $p(s)=2/(1s)$. $\endgroup$ Commented Feb 23, 2021 at 9:46

$\begingroup$ Isn't this strange, since the embedding into $BMO$ works with $\dot H^1$? $\endgroup$ Commented Feb 23, 2021 at 9:49

$\begingroup$ There is nothing strange, it's the classical Sobolev embedding. $\endgroup$ Commented Feb 23, 2021 at 9:59