I have two questions concerning Fractional Sobolev Spaces. Let

$W^{s,p}(\Omega) := \{u\in L^p(\Omega) |\frac{|u(x)-u(y)|}{|x-y|^{\frac{n}{p}+s}} \in L^p(\Omega \times \Omega)\}$


$W^{s,\infty}(\Omega) := \{u\in L^\infty(\Omega) |\frac{|u(x)-u(y)|}{|x-y|^s} \in L^\infty(\Omega \times \Omega)\}$

following the Hitchhiker's Guide to Sobolev Space (https://arxiv.org/abs/1104.4345). On page 59 in there, there is a remark that the latter space boils down to be the Hölder Space $C^{0,s}(\Omega)$, but why does this hold? I know that it would hold either if all functions in there would be continuous or if approximating functions in this space by continuous functions works, but I couldn't show any of these properties.

The second question is about whether like in the case for the normal Sobolev Spaces, do we have that $W^{s,p}(\Omega) \subseteq W^{s,q}(\Omega)$ when $\Omega$ is bounded and $p\geq q$? I tried to show but failed.

Thanks in advance.


2 Answers 2


For the first question is natural and legitimize since form integration theory we know the following convergence of Lebesgue space


Where $L^p(\Omega)$ are endowed with Lebesgue measure. Similarly Doing this on $\Omega\times\Omega$ with Lebesgue measure $dxdy $ we have


Here with the slight different that we have the factor $\frac{n}{p}\to 0$ as $p\to \infty$. with get

$$\lim_{p\to\infty}\left(\iint_{\Omega\times\Omega}\frac{|u(x)-u(y)|^p}{|x-y|^{n+sp}}dxdy\right)^{1/p} =\lim_{p\to\infty}\left(\iint_{\Omega\times\Omega}\left(\frac{|u(x)-u(y)|}{|x-y|^{\color{red}{\frac{n}{p}}+s}}\right)^pdxdy\right)^{1/p}=\sup_{x\neq y}\frac{|u(x)-u(y)|}{|x-y|^s} $$

This holds true for sufficiently smooth functions.


For the second question, look at this paper. https://hal.archives-ouvertes.fr/hal-01162231/document


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