How do I show that the definition of Sobolev–Slobodeckij spaces $W^{s,p}(\Omega)$ coincides with the classical one when $s$ is integer.

Question

For any nonnegative real number $s$ and $1<p<\infty$, the Sobolev–Slobodeckij space $W^{s,p}(\Omega)$ is defined as in the following link:

https://en.wikipedia.org/wiki/Sobolev_space#Sobolev%E2%80%93Slobodeckij_spaces

Here, we assume for brevity that $\Omega$ is either a bounded region in some Euclidean space with smooth boundaries or a compact Riemannian manifold.

I think that when $s$ is an integer, such Sobolev–Slobodeckij spaces $W^{s,p}(\Omega)$ must coincide with the "classical Sobolev space" defined in terms of weak derivatives. Of course, the norms from each definition must be equivalent as well.

However, I cannot find a reference explicitly dealing with this issue.

Answer 1

I cannot click on your link because I am traveling but I suppose you are referring to the definition $$ |u|_{W^{s,p}(\Omega)}^p = \int_{\Omega^{2}} \frac{|u(x)-u(y)|^p}{|x-y|^{d+sp}} \,\mathrm d x\,\mathrm d y \\ \|u\|_{W^{s,p}(\Omega)}^p = \|u\|_{L^p}^p + |u|_{W^{s,p}(\Omega)}^p $$ with this definition, the limit $s\to 1$ does not give the $W^{1,p}$ norm, but you can get it by multiplying the norm by the right factor.

I invite you to read the paper "How to recognize constant functions. Connections with Sobolev spaces" by H. Brezis. In particular, Equation (44) there shows that $$ p\,(1-s)\,|u|_{W^{s,p}(\Omega)}^p \to C_{d,p}\,|u|_{W^{1,p}(\Omega)}^p $$ when $s\to 1$. If you do not put the factor $1-s$, then $|u|_{W^{s,p}(\Omega)}^p$ converges to $$ \int_{\Omega^2} \frac{|u(x)-u(y)|^p}{|x-y|^{d+p}} \,\mathrm d x\,\mathrm d y $$ but the only functions for which the above quantity is bounded are the constant functions (that's Corollary $1$ of the same paper).

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Going deeper.

• There are other equivalent definitions of the fractional norms. Actually, they can be defined as real interpolation spaces of Sobolev spaces. Then there is a general result by M. Milmann "Notes on limits of Sobolev spaces and the continuity of interpolation scales" about the factor to put in front of the interpolated spaces when you reach the boundary spaces.
• The constant $C_{d,p}$ can be computed more exactly as $C_{d,p} = \frac{2\,\omega_{d+p}}{\omega_{p+1}}$ where $\omega_d = \frac{2\,\pi^{d/2}}{\Gamma(d/2)}$. In my recent paper arxiv.org:2210.03013, I had fun finding the constant to get a norm that is compatible on all the sides $p\to\infty$, $s\to 0$, $s\to 1$, $p=2$, then you can define (Remark 2.1 of the mentioned paper, in dimension $2d$) the seminorm $$ |u|_{W^{s,p}(\Omega)}^p = C_{d,s,p} \int_{\!\Omega^2} \frac{|u(x)-u(y)|^p}{|x-y|^{d+sp}} \,\mathrm d x\,\mathrm d y $$ with $C_{d,s,p} = \frac{p\,|\omega_{-2s}|}{2\,\omega_{d+sp}} \left(\frac{\pi\,\omega_{p+1}}{s^{p/2-1}}\right)^s$.
• One could use second order differences instead of first order differences in the definition of the seminorm, that is replace $u(x+z)-u(x)$ by $u(x+z)+u(x-z)-2\,u(x)$. Then the limiting norm in the limit $s\to 1$ is the Besov space $B^s_{p,p}$. And indeed, for $s$ not an integer, $B^s_{p,p} = W^{s,p}$, but not anymore for integers. Except in the case $p=2$. In this case, by the Remark 2.5 of my above mentioned paper, one finds in dimension $d$ $$ |u|_{B^1_{2,2}}^2 = \frac{2\ln(4)\,\omega_d}{d} \,\|\nabla u\|_{L^2}^2 $$ where the first seminorm is the one with the double difference, that is $$ |u|_{B^1_{2,2}}^2 = \int_{\!\Omega^2} \frac{|u(x+z)+u(x-z)-2u(x)|^2}{|z|^{d+2s}} \,\mathrm d x\,\mathrm d z. $$