I was reading Fractional Sobolev space topic.
$s\in (0,1)$, $p\in [1,\infty)$,
$$ W^{s,p}(\Omega)=\left\{u\in L^p(\Omega)|\frac{|u(x)-u(y)|}{|x-y|^{n/p+s}}\in L^p(\Omega\times \Omega)\right\} $$.
But I do know why there is extra term of $n/p$ come in power. Natural extension of integer Sobolev space should be $$ W^{s,p}(\Omega)=\left\{u\in L^p(\Omega)|\frac{|u(x)-u(y)|}{|x-y|^{s}}\in L^p(\Omega\times \Omega)\right\} $$.
Please any one through some light on such definition of Sobolev space.
Edit: I though may be this because of integrability possible iff $(n/p+s)p>n$. But I am not sure.
Any Help will be appreciated.
If e.g. $u\in L^1$ with compact support for simplicity in $\Omega=\mathbb R^n$, then $$\iint_{\mathbb R^{2n}} \frac{|u(x)-u(y)|}{|x-y|^s} \ dxdy \le\int_{\operatorname{supp} u}|u(x)|\int_{\operatorname{supp} u}\frac{1}{|x-y|^s}dydx + \int_{\operatorname{supp} u}|u(y)|\int_{\operatorname{supp} u}\frac{1}{|x-y|^s}dxdy \le C\|u\|_{L^1} $$ for any $s<n$, nevermind $s<1$, as $$\int_{\text{bounded set}} \frac{dz}{|z|^s} < \infty. $$ Consequently, the condition $ \frac{|u(x)-u(y)|}{|x-y|^s}\in L^1(\mathbb R^{2n})$ fails to ask for any new regularity properties on $L^1$ functions.
You may also look to the Hitchhiker's guide to fractional sobolev spaces for the computation that $\left\| \frac{|u(x)-u(y)|}{|x-y|^{n/2+s}}\right\|_{L^2(\mathbb R^{2n},dxdy)} = C_{n,s} \| |\xi|^s \hat f(\xi)\|_{L^2(\mathbb R^n,d\xi)} $, which explains (in the case $p=2$) the appearance of the $n/2$.