Let $s>0$, $1<p<\infty$ 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.



Equivalent norms of fractional Sobolev spaces on bounded Lipschitz domain
The result is mentioned in Triebel's *Function Spaces and Wavelets on Domains* Proposition 4.21. Also see Theorem 1.4 for my paper <https://arxiv.org/abs/2110.14477>


