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 $K\subset R^n$?
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 $I(f)<\infty$.
