Does the equality $\|\nabla u\|^{p}_{L^{p}(\mathbb{R}^{n})}=[u]^{p}_{W^{s,p}(\Omega)}$ hold in the fractional Sobolev space?

Question

I have been reading the Hitchhiker’s guide to the fractional Sobolev spaces and there they define the fractional Sobolev space and its norm in the following way. $$W^{s,p}(\Omega):=\Biggl\{ u\in L^{p}(\Omega):(x,y)\mapsto\frac{|u(x)-u(y)|}{|x-y|^{\frac{n}{p}+s}}\in L^{p}(\Omega\times\Omega)\Biggl\},$$ and $$\|u\|^{p}_{W^{s,p}(\Omega)}:=\int_{\Omega}|u|^{p}dx+\int_{\Omega}\int_{\Omega}\frac{|u(x)-u(y)|^{p}}{|x-y|^{n+sp}}\,dx\,dy=\|u\|^{p}_{L^{p}(\Omega)}+[u]^{p}_{W^{s,p}(\Omega)}.$$ Where $[u]_{W^{s,p}(\Omega)}$ is the Gagliardo semi-norm. I understand the definition of both the norm and the space, my problem arises in the Proposition 2.2 specifically in the ecuation (2.6), where the next inequality is given. $$C_{1}(n,s,p)\|\nabla \overset{\backsim}{u}\|^{p}_{L^{p}(\mathbb{R}^{n})} \leq C_{1}(n,s,p)\| \overset{\backsim}{u}\|^{p}_{W^{1,p}(\Omega)}.$$ I asked a profesor and he gave me this $$C_{1}(n,s,p)\|\nabla \overset{\backsim}{u}\|^{p}_{L^{p}(\mathbb{R}^{n})} \leq C_{1}(n,s,p)\biggl(\|\nabla \overset{\backsim}{u}\|^{p}_{L^{p}(\mathbb{R}^{n})}+\| \overset{\backsim}{u}\|^{p}_{L^{p}(\mathbb{R}^{n})}\biggl)=C_{1}(n,s,p)\| \overset{\backsim}{u}\|^{p}_{W^{1,p}(\Omega)}.$$ Stating that the term $\|\nabla \overset{\backsim}{u}\|^{p}_{L^{p}(\mathbb{R}^{n})}+\| \overset{\backsim}{u}\|^{p}_{L^{p}(\mathbb{R}^{n})}$ is exactly the definition of the fractional Sobolev norm, but as stated above the definition would be $\|\overset{\backsim}{u}\|^{p}_{L^{p}(\Omega)}+[\overset{\backsim}{u}]^{p}_{W^{s,p}(\Omega)}$. Does that mean that the equality $\|\nabla \overset{\backsim}{u}\|^{p}_{L^{p}(\mathbb{R}^{n})}=[\overset{\backsim}{u}]^{p}_{W^{s,p}(\Omega)}$ holds? or is it that there is another definition for the norm in the fractional Sobolev space?

P.D. I tried to prove the equality and I could not achive anything.

Answer 1

The function $\tilde u$ is the extension of $u$, and the extension satisfies $\|\tilde u\|_{W^{1,p}(\mathbb R^n)} \le c \|u\|_{W^{1,p}(\Omega)}$, where $c$ depends on $\Omega$ (and possible $p$). This is the first paragraph of the proof of the proposition.

Also you did not reproduce the inequality in the paper, they state correctly (with different constants) $$ C_1(\dots) \|\nabla \tilde u\|_{L^p(\mathbb R^n)}^p \le C_2(\dots) \|u\|_{W^{1,p}(\Omega)}^p. $$ Also note that we are talking about the $W^{1,p}$ norm here, not the fractional norm of $W^{s,p}$.