recently I have been studying the article named "Hitchhiker’s guide to the fractional Sobolev spaces" (I leave the external link here: https://arxiv.org/pdf/1104.4345.pdf) and as the definition of a fractional Sobolev space the next definition was given: $$ W^{s,p}(\Omega):=\Biggl\{ u\in L^{p}(\Omega):\frac{|u(x)-u(y)|}{|x-y|^{\frac{n}{p}+s}}\in L^{p}(\Omega\times\Omega)\Biggl\}$$ Where $\Omega\subseteq\mathbb{R}$ is an open set, also we have that $p\in [1,\infty]$, $s\in (0,1)$ and $n$ is the dimension of $\Omega$. Here the weak fractional derivative is: $$ \frac{|u(x)-u(y)|}{|x-y|^{\frac{n}{p}+s}}$$ My problem is that I can't understand why this is the correct definition for a fractional derivative (In fact I think that I don´t understand the definition at all).
The few things that I think I understand are the following:
The first thought in my mind is that this definition is a way to ensure some way of regularity for the functions $u\in L^{p}(\Omega)$. From what I understand is that the term $|x-y|$ can give me an indeterminate value in $x=y$, so to eliminate this problem you have to ask certain criteria to $u$, and those criteria are given in the term $\frac{n}{p}+s$ but there is where I start to get lost, as I don't understand the role of $n$ and why is this term so specific.
Any help would be really appreciated, Thank you very much in advance.