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Let $\Omega \subset \mathbb R^n$ be a Lipschitz domain, let $s \in \mathbb R$. Let $W^s(\Omega)$ denote the Sobolev-Slobodeckij space on $\Omega$, and let $H^s(\Omega)$ denote the Bessel-potential spaces on $\Omega$, each for the parameter $p = 2$.

Both spaces are often refered to simply as Sobolev spaces but are not the same in general. Under what set of additional conditions (which may be the empty set), do these spaces coincide, i.e. $W^s = H^s$?

I have not found a resource that explains the situation.

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2 Answers 2

On $\mathbb R^n$ they are the same: $W^s=B^s_{2,2}=F^s_{2,2}=H^s$, all of which can be found in Theory of Function spaces by Triebel. (Also in Stein's Singular integrals..., page 155.) Triebel considers only smooth domains in the chapter on function spaces on domains. However, I think that having the result for $\mathbb R^n$ is sufficient, since Lipschitz domains admit bounded extension operators for any reasonable function spaces.

For example, given $u=H^s(\Omega)$ one can write $u=\varphi*\mathscr J_s$ where $\varphi\in L^p(\mathbb R^n)$ and $\mathscr J_s$ is the Bessel potential.. The point is that $u=\varphi*\mathscr J_s$ makes sense also as a function on $\mathbb R^n$, which therefore belongs to $W^s(\mathbb R^n)$. The restriction to $\Omega$ gives a function in $W^s(\mathbb R^n)$.

To go in the opposite direction, one needs a Whitney-type extension $W^s(\Omega)\to W^s(\mathbb R^n)$. Unfortunately, I don't have a reference for such a result when $s$ is fractional.

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One reference that deals with less than smooth (e.g. Lipschitz) domains is McLean's Strongly elliptic systems and boundary integral equations.

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