This tag is devoted to any problem concerning fractional Sobolev spaces which are approximation of the classical Sobolev spaces
There are two version of Fractional sobolev spaces .
Definition1: (Via Galiardo semi-norm) Let $1\leq p\leq +\infty$, $0<s<1$ and $\Omega\subseteq \mathbb{R}^n$ an open set. The [fractional Sobolev space $W^{s,p}(\Omega)$][2] is defined to be
$$ W^{s,p}(\Omega) = \left\{ u\in L^p(\Omega) : \frac{|u(x)-u(y)|}{|x-y|^{\frac{n}{p} + s}} \in L^p(\Omega\times\Omega) \right\} $$
equipped with the norm
$$ \|u\|_{W^{s,p}(\Omega)} = \left( \int_\Omega |u|^p \; dx + \int_\Omega\int_\Omega \frac{|u(x)-u(y)|^p}{|x-y|^{n+ sp}} \; dx dy \right)^{1/p}. $$
Definition2:(Via Fourier Transform) For $s\in\mathbb{R}$, $1<p<\infty$, and $n\geq 1$, define the [Sobolev space $H^{s,p}(\mathbb{R}^{n})$][1] by $$H^{s,p}(\mathbb{R}^{n}):=\left\{f\in\mathcal{S}(\mathbb{R}^{n}) : \|(\langle{\xi}\rangle^{s}\widehat{f})^{\vee}\|_{L^{p}}<\infty\right\}$$, equipped with norm $$\|f\|_{H^{s,p}}=\|(\langle{\xi}\rangle^{s}\widehat{f})^{\vee}\|_{L^{p}}$$
Where, $$\langle{\xi}\rangle^{s} =(1+|\xi|^2)^s$$