# Equivalent norms of fractional Sobolev spaces on bounded Lipschitz domain

Let $$s>0$$, $$1 and let $$\Omega\subset\mathbb R^n$$ be a bounded Lipschitz domain. Set $$H^{s,p}(\Omega)=\{u\in L^p(\Omega):\exists\tilde u\in L^p(\mathbb R^n),\tilde u|_{\Omega}=u,(I-\Delta)^\frac s2\tilde u\in L^p(\mathbb R^n)\}$$ and $$\|u\|_{W^{s,p}(\Omega)}=\inf\limits_{\tilde u}\|(I-\Delta)^\frac s2\tilde u\|_{L^p(\mathbb R^n)}$$ be the Bessel potential spaces.

Question: If $$s>1$$, how can we show from the definition that $$\|u\|_{W^{s,p}(\Omega)}\approx\|u\|_{W^{s-1,p}(\Omega)}+\sum_{i=1}^n\|\partial_iu\|_{W^{s-1,p}(\Omega)}$$?

And how does the existence of extension operator guarantee this fact?

Indeed for each $$\partial_iu$$ by definition we can only find a $$W^{s-1,p}$$-extension $$\tilde v_i$$, but $$\tilde v_i$$ may not be the gradient of some function.

If $$s$$ is integer, I know this is true because of the classical characterization of Sobolev norm is just the L^p-norm of the derivatives inside $$\Omega$$, which does not require the extension.