This might look too an elementary question, but I am confined and is not able to find a textbook which answers the following question.

I have a function $f:{\mathbb R}\rightarrow{\mathbb R}$, such that $f\in L^3({\mathbb R})$ and $$\int\int\frac{|f(y)-f(x)|^3}{|y-x|^4}dydx<\infty.$$ May I conclude that $f\in W^{1,3}({\mathbb R})$ ?

This is a limit case of Sobolev-Slobodeckij space, as $4=1\cdot3+1$. Obviously, the same integral but with exponent $s\cdot3+1$ with $s<1$ is valid, hence $f\in W^{s,3}({\mathbb R})$