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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.

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1 Answer 1

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The result is mentioned in Triebel's Function Spaces and Wavelets on Domains Proposition 4.21. Also see Theorem 1.1 in my paper.

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