For $1\leq p <+\infty$, $0<s<1$ and $\Omega\subset R^n$ domain, the fractional Sobolev space $W^{s,p}$ is defined as


$$W^{s,p}(\Omega):=\big\{f \in L^p(\Omega)\colon \int\_{\Omega} \int\_{\Omega} \frac{|f(x)-f(y)|^p}{|x-y|^{s p + n}}dx dy<+\infty .\big\}$$


I wonder if this definition makes sense for $s=0$ for bounded $\Omega$, in particular, can one describe functions $f$ such that


$$\int\_{K} \int\_{K} \frac{|f(x)-f(y)|^p}{|x-y|^{n}}dx dy<+\infty$$


for any compact $K\subset R^n$?



Fractional Sobolev spaces of order 0
Let $n=1$, $p=2$ for a partial answer: characterizing $1$-periodic functions $f\in L^1(0,1)$ such that $I(f):=\int\_0^1 h^{-1}[\int\_0^1(f(x+h)-f(x))^2\ dx]\ dh<\infty$.


Let $\hat f(n):=\int\_0^1 e^{2\pi inx}f(x)\ dx$ denote the Fourier coefficients of $f$. Then $$I(f)=\sum\_{n\in\mathbb Z}|\hat f(n)|^2\int\_0^1|e^{2\pi inh}-1|^2h^{-1}\ dh.$$But $\int\_0^1|e^{2\pi inh}-1|^2h^{-1}\ dh=\int\_0^1(2-2\cos(2\pi nh))h^{-1}\ dh$ is essentially (between two positive constants times) $\log(1+n^2)$.


Thus $I(f)<\infty$ iff $f\in L^2(0,1)$ with $\sum|\hat f(n)|^2\log|n|<\infty$. From this you can easily define an example of an $f\notin\cup\_{0<s<1}W^{s,2}$ such that $I(f)<\infty$.


