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Suppose that $\Omega\subset\mathbb{R}^N$ is a bounded smooth domain. It is common to see the definition of the Sobolev space $W^{k,p}(\Omega)$ as the completion of $C^{k}(\overline{\Omega})$ with the norm $\|u\|_{k,p}=\left(\sum_{|\alpha|\leq k}D^\alpha u\right)^{1/p}$ (here $k$ is a positive integer).

My question is: For $k\in (0,1)$, is $W^{k,p}(\Omega)$ the completion of $C^{0,k}(\overline{\Omega})$ with the norm: $$\|u\|_{k,p}=\left(\int_\Omega|u|^p+\int_\Omega\int_\Omega\frac{|u(x)-u(y)|^p}{|x-y|^{N+kp}}\right)^{1/p}$$

Remark: Here $C^{0,k}(\overline{\Omega})$ denotes the usual Holder space.

Remark 2: Usually $W^{k,p}(\Omega)$ for $k\in (0,1)$ is defined by $$W^{k,p}(\Omega)=\left\{u\in L^p(\Omega):\ \frac{|u(x)-u(y)|^p}{|x-y|^{\frac{N}{p}+k}}\in L^p(\Omega\times\Omega)\right\}$$

and $W^{k,p}(\Omega)$ is complete under the norm $\|\cdot\|_{k,p}$ defned above.

Thank you

Tomás
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No, because $C^{0,k}$ is not necessarily contained in $W^{k,p}$ when $k<1$. Consider the Hardy-Littlewood trigonometric series $$f(t)=\sum_{n=1}^\infty e^{i n\log n}n^{-\frac12-\alpha} e^{inx},\quad 0<\alpha<1 \tag1$$ It is true, though nontrivial (page 197 in Trigonometric Series, Volume I by Zygmund) that $f\in C^{0,\alpha}$ (on any bounded interval, since it's periodic). On the other hand, a trigonometric series is in $W^{\alpha,2}$ if and only if its coefficients $c_n$ satisfy $$\sum_{n\in\mathbb Z}|n|^{2\alpha}|c_n|^2<\infty\tag2$$ For (1) we get $$\sum_{n=1}^\infty n^{2\alpha} n^{-1-2\alpha }=\infty$$ Thus, $C^{0,\alpha}\not\subset W^{\alpha,2}$.

But you can generalize the definition anyway, because $C^\infty$ functions are dense in function spaces with finite integrability exponents (e.g., Proposition 14.5 in First Course in Sobolev Spaces by Leoni). The definition of $W^{k,p}$ for integer $k$ could as well have $C^\infty$ there instead of $C^k$. And then for fractional $k$ it's the same thing.

user90090
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  • Sorry, but I made a big mistake in the formulation of my question. I will edit it, please take a look. – Tomás Aug 15 '13 at 00:31
  • @Tomás OK, edited. – user90090 Aug 15 '13 at 16:52
  • Ok I understood. I asked this question because I was trying to find the "biggest" space for which its completion with the norm $|\cdot|_{k,p}$ is $W^{k,p}(\Omega)$ for $k\in (0,1)$. (+1) – Tomás Aug 15 '13 at 17:26
  • @Tomás You can probably use $C^{0,\alpha}$ with some $k<\alpha<1$; the choice of $\alpha$ will require digging into the Besov space literature. – user90090 Aug 15 '13 at 17:37
  • But do you think that such $\alpha$ exist, or it does converge to $k$? – Tomás Aug 16 '13 at 12:38
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    @Tomás A definitive answer requires more function spaces expertise than I have. But I can observe that $f\in C^{0,\alpha}$ implies $|\hat f(\xi)|=O(|\xi|^{-\alpha})$. It follows that $\int_{\mathbb R^n} |\xi|^{2k}|\hat f(\xi)|^2<\infty$ when $k<\alpha-n/2$. On the other hand, I think that Morrey embedding $W^{k,p}\subset C^{k-n/p}$ is sharp. So it seems that $\alpha>k+n/p$ is the threshold for containment $C^{0,\alpha}\subset W^{k,p}$. – user90090 Aug 16 '13 at 19:46