# Embedding of fractional Sobolev space into BMO

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?

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 $$p<\infty$$.
• Thank you. What is the critical $p(s)$ for the embedding? – user173196 Feb 23 at 9:43
• You are welcome, It is $p(s)=2/(1-s)$. – Raffaele Scandone Feb 23 at 9:46
• Isn't this strange, since the embedding into $BMO$ works with $\dot H^1$? – user173196 Feb 23 at 9:49