Existence of an extension operator $E: W_0^{s,p}(\Omega)\rightarrow W^{s,p}(\mathbb{R}^d)$?

Question

If $\Omega$ is an open Lipschitz domain with bounded boundary then their exists a continuous operator $E: W^{s,p}(\Omega) \rightarrow W^{s,p}(\mathbb{R}^d)$ where $s\in (0,1)$ such that $Eu=u$ on $\Omega$.

A similar result holds for integer order Sobolev spaces.

But in case of integer order sobolev spaces one always has the existence of a continous extension operator $E: W_0^{k,p}(\Omega) \rightarrow W^{k,p}(\mathbb{R}^d)$ regardless of the regularity of $\Omega$ e.g. the zero extension.

My question is, if $\Omega$ is only open, do we also have the existence of a continuous extension operator

$$E: W_0^{s,p}(\Omega) \rightarrow W^{s,p}(\mathbb{R}^d) \text{ where } s\in(0,1)$$

If yes, can someone suggest literature where this is mentioned?

Honestly, I don't want to bother trying to prove it myself, or rather, I would like to have at least assurance that this is true first.

By the way the space $W^{s,p}_0(\Omega)$ is defined as the closure of $C_0^\infty(\Omega)$ wrt. to the norm $\|u\|_{W^{s,p}}^p:=|\cdot|_{s,p,\Omega}^p+\|\cdot\|_{L^p(\Omega)}^p$, where $|u|_{s,p,\Omega}^p:=\int_{\Omega \times \Omega} \frac{|u(x)-u(y)|^p}{|x-y|^{d+sp}}d(x,y)$ is the Gagliardo seminorm. If $\Omega=\mathbb{R}^d$ the definition is similar. But one can also use the definition via Fourrier transform they both equivalent in this case.

My thought on this: As in the integer-order case one might consider the extension by zero of $u \in W^{s,p}_0(\Omega)$ and try to show that this extension satsifies the above requirement. Yet, we have that

$$\|Eu\|_{W^{s,p}(\mathbb{R}^d)}^p=\|u\|_{W^{s,p}(\Omega)}^p+2\int_U \left(\int_{\mathbb{R}^d\setminus U} \frac{1}{|x-y|^{d+sp}} dy \right)|u(x)|^pdx\quad (1)$$

If this could be further estimated by $C\|u\|_{W^{s,p}}$ for some constant $C$ then the extension by zero would be continuous from $W^{s,p}_0(\Omega)$ to $W^{s,p}(\mathbb{R}^d)$.

Answer 1

This may be an overkill : you can use the closed graph theorem. If $(u_n)_n$ converges to $u$ in $W^{s,p}_0(\Omega)$ and $(E(u_n))_n$ converges to $v$ in $W^{s,p}(\mathbf{R}^d)$, then both convergences imply convergence in $L^p(\Omega)$ and $E(u_n)=u_n$ in $\Omega$, so you have $v=u=E(u)$ inside $\Omega$, and also $v=E(u)$ outside $\Omega$ because both are vanishing ($E(u)$ by definition and $v$ as $L^p(\mathbf{R}^d\setminus \Omega)$ limit of zero functions ...). So $E$ is continuous, since both spaces are Banach.

Answer 2

This is a somewhat delicate topic and I think what has been proposed so far did not capture fully what is going on. (Probably I am missing something, too, and I also did not fully answer OP's question, so maybe this is just a very extended comment.)

From OPs last formula identity for $\|Eu\|_{W^{s,p}(\mathbb{R}^d)}$, a short calculation shows that zero extension is in fact a bounded linear operator $$W^{s,p}(\Omega) \cap L^p(\Omega;\operatorname{dist}_{\partial\Omega}^{-s}) \to W^{s,p}(\mathbb{R}^d).$$

However, in general, the space defined in OP $$W^{s,p}_0(\Omega) := \overline{C_0^\infty(\Omega)}^{\|\cdot\|_{W^{s,p}(\Omega)}}$$ is not embedded into the weighted space $W^{s,p}(\Omega) \cap L^p(\Omega;\operatorname{dist}_{\partial\Omega}^{-s})$. In fact, this requires some boundary regularity. It is unclear to me whether zero extension is still a bounded linear operator on $W^{s,p}_0(\Omega)$ if that space does not embed into the weighted space.

There is a bunch of related literature about the relation of the above mentioned fractional Sobolev spaces with zero boundary conditions, referring to $(s,p)$ [or fractional] Hardy inequalities, with sufficient conditions. See for instance On density of compactly supported smooth functions in fractional Sobolev spaces by Dyda and Kijaczko (in particular Section 5) and the references there.

Note that it is classical that $W^{1,p}(\Omega) \cap L^p(\Omega;\operatorname{dist}_{\partial\Omega}^{-1})$ is embedded into $W^{1,p}_0(\Omega)$ (with the analogous definition as above) irrespective of boundary regularity. The other way around, again, requires boundary regularity. I would expect the same to be the case for the present case of fractional Sobolev spaces, but I do not know a striking reference right now.

Answer 3

This is "trivial" in the sense that $W^{s,p}_0(\Omega)$ can be regarded as a subspace of $W^{s,p}(\mathbb R^n)$ where the functions are vanishing on $\Omega^c$. And you don't need to take any special construct to build a $E$ for that.