I am reading a paper and the in the first pages it is written what follows.
Let $s\in (0, 1), p>1$, $n> sp$ and $\Omega$ be a bounded domain in $\mathbb{R}^N$ with Lipschitz boundary. Let $$ [u]_{s,p} = \left(\int_{\mathbb{R}^{2n}} \frac{|u(x) - u(y)|^p}{|x - y|^{n+sp}}\, dx dy\right)^{1/p} $$ be the Gagliardo seminorm of the measurable function $u : \mathbb{R}^n \to \mathbb{R}$, and let $$ W^{s,p}(\mathbb{R}^n) = \{u \in L^p(\mathbb{R}^n) : [u]_{s,p} < \infty\} $$ be the fractional Sobolev space endowed with the norm $$ \|u\|_{s, p} = \big(|u|_p^p + [u]_{s,p}^p\big)^{1/p}, $$ where $|\cdot|_p$ is the norm in $L^p(\mathbb{R}^n)$. Consider the closed linear subspace $$ X_p^s(\Omega) = \{u \in W^{s,p}(\mathbb{R}^n) : u = 0 \text{ a.e. in } \mathbb{R}^n \setminus \Omega\}, $$ equivalently renormed by setting $\|\cdot\|_{s, p} = [\cdot]_{s,p}$.
Could someone please help me in proving that $\|\cdot\|_{s, p}$ and $[\cdot]_{s,p}$ are equivalent norms in $X^s_p (\Omega)$? Actually, I am interested in proving only that there exists $c>0$ such that $$\|u\|_{s, p} \le c [u]_{s, p}$$ since I have proved on my own the inequality in the other direction.
Also, which role plays the fact that $\Omega$ is a bounded domain?
Thank you in advance.
Indeed, one of the direction is trivial since $a\leq a+b$ if $b\geq 0$. For the other direction, by Sobolev's embedding, $\|u\|_{L^q(\Omega)} \leq C\,[u]_{W^{s,p}(\Omega)}$ for $q = \frac{np}{n -sp} > p$, and since the domain is bounded $\|u\|_{L^p(\Omega)} \leq C_\Omega\,\|u\|_{L^q(\Omega)}$ by Hölder's inequality.