I'm reading Pseudodifferential Operators by M. E. Taylor, where the author talks about $H^s(\Omega)$ for $s\in\mathbb{R}$ and $\Omega\subset\mathbb{R}^n$ an open set (for example, in the statement of Gårding's inequality) without ever defining it. Indeed, he has only defined such Sobolev spaces for $\mathbb{R}^n$ and compact manifolds. In both of these cases, one has an $s$-order pseudodifferential operator $\Lambda^s$ (with principal symbol $\langle\xi\rangle^s$) which induces an isomorphism $H^s\to L^2$. This could be taken as the definition of $H^s$. However, I don't know how to do the same for general open sets in Euclidean space. Some thoughts:

  1. On p.51, the author remarks that this is done by altering $\Lambda$ so that it is properly supported. However, I'm unsure what he means by this.
  2. Perhaps one could use functional calculus on the standard Laplacian $\Delta$. There are several problems with this approach: (a) I would need $\Delta^{s/2}$ to be defined on the space of distributions (so that the definition would be like: a distribution $u$ belongs to $H^s$ if $\Delta^{s/2}u\in L^2$), but functional calculus only defines it on a subspace of $L^2$. (b) Is $\Delta^{s/2}$ really a pseudodifferential operator with the correct symbol?

So what is the correct definition in this context? Any help will be appreciated!

  1. Typical definitions of Sobolev spaces

For a general open subset $\Omega$ (without regularity assumptions on its boundary), the Sobolev-spaces $H^s(\Omega)$ are first defined for $s\in \mathbb{N}$ (in the obvious way: derivatives up to order $s$ shall be in $L^2$) and for general $s\in \mathbb{R}$ via interpolation/duality.

However, if $\partial \Omega$ is sufficiently regular there is an easier way: Let's assume for simplicity that $\partial \Omega \in C^\infty$, then one typically defines $H^s(\Omega)$ as the space of distributions on $\Omega$ that admit an extension to $\mathbb{R}^d$ that lies in $H^s(\Omega)$. Equivalently $H^s(\Omega)=rH^s(\mathbb{R}^d)\subset\mathcal{D}'(\Omega)$, where $r:\mathcal{D}'(\mathbb{R}^d)\rightarrow \mathcal{D}'(\Omega)$ is the restriction operator. This yields the same spaces as in the first paragraph.

As a reference on these things I can recommend Taylor's PDE book, which has a whole chapter on various definitions of Sobolev spaces. (Also for $\mathbb{R}^d$ being replaced by a closed manifold).

  1. Elliptic scales

Now, regarding the comment on properly supported $\psi$do's $\Lambda^s$ you can consider Lemma 7.1 in Shubin's $\psi$do book. Indeed, this states that on an arbitrary manifold $X$ (in particular you could take $X=\Omega$) that there exists a scale of properly supported operators $\Lambda^s\in \Psi^s_{\mathrm{cl}}(X)$ (subscript denoting classicality) with positive principal symbols. Shubin then defines local Sobolev spaces by $H^s_\mathrm{loc}(X)=\{u\in \mathcal{D}'(X): \Lambda^su\in L^2_{\mathrm{loc}}(X)\}$ and proves this to be equivalent with some other definitions.

The point is, that for a general (non-compact) manifold this is as good as it gets: There is no notion of $H^s(X)$ without specifying the behaviour of its functions at infinity. If $X$ happens to be an open subset of $\mathbb{R}^d$ or a closed manifold, the behaviour at infinity (or rather at the boundary) is specified by requiring functions to be extendible across $\partial X$ and we are in the setting of the first few paragraphs.

What if $X$ has a Riemannian metric $g$? I suppose that in this case one could define $H^s(X,g)$ for $s\in \mathbb{N}$ by requiring its functions to satisfy $X_1\dots X_k f \in L^2(M,g)$ for any vector fields $X_1,\dots,X_k$ $(0\le k \le s)$ which satisfy $\vert X_i \vert_g\in L^\infty(X)$. For non-integer $s$ then via interpolation\duality.

If $(X,g)$ happens to be complete (like $\mathbb{R}^d$), then Gaffney showed that the Laplacian $1+\Delta_g$ has a unique self-adjoint realisation in $L^2(X,g)$ and I suppose one could call its domain $\tilde H^2(X,g)$. The same is true for its powers and thus we can define $\tilde H^s(X,g)$ for $s\in 2\mathbb{N}$ and extend to general $s$ by interpolation/duality. I would not be surprised (but have not checked it), if indeed $H^s(X,g)=\tilde H^s(X,g)$ in that case.

  1. Complex powers

You were interested in whether you can define Sobolev spaces on $\Omega$ via powers of the Laplacian. It makes more sense to take powers of $P=1+\Delta$ (in analogy with $\mathbb{R}^d$) and indeed there is a nice theory that tells you that this is possible, at least if you are on a closed manifold. So suppose that $\Omega$ lives inside a closed Riemannian manifold $(M,g)$ (and $\partial \Omega \in C^\infty)$, then $P^z$ is defined for all $z\in \mathbb{C}$ and is a classical $\psi$do of order $\mathrm{Re}(z)$ with the obvious algebraic properties. (This is due to Seeley, but you can find a nice account on it in Shubin's book).

Now you might want to define $H^s(\Omega)=\{f:P^s f\in L^2(\Omega,g)\}$ and at least for $s\in \mathbb{N}$ this gives the same as defined in the beginning, i.e. $H^s(\Omega) = r H^s(M)$. A sufficient criterion for the two spaces to agree is that $P^s$ satisfies the so called transmission condition at $\partial \Omega$: This is Definition 18.2.13 in Hörmander and says that $rP^se_0(C^\infty(\bar \Omega)) \subset C^\infty(\bar \Omega)$, where $e_0$ denotes extension by zero. Now for positive integers-powers $P^s$ is a differential operator and clearly satisfies the condition. For non-integer powers this might fail, as is mentioned at the beginning of page 184 here. This is all I can say about it at the moment.

  • $\begingroup$ Thank yopu for your informative answer! About 1: Are you referring to Taylor's trilogy entitled Partial Differential Equations? Actually, I've read carefully the material on Sobolev spaces in the first volume, and I think there was no mention of Sobolev spaces on general open domains, except in the overly brief treatment (with no proofs whatsoever) in Section 4.7... $\endgroup$ – Colescu Aug 19 '20 at 12:43
  • $\begingroup$ Yes, that is what I was referring to. Indeed, his treatment of `rough domains' is brief. For reading his $\psi$do book I would recommend to just assume the boundary $\partial \Omega$ to be smooth. If you are interested in whether a specific result holds for rougher domains, it might be worth asking a separate question. In that case there are indeed things that can go wrong (cf. exercises in Section 4.7) and one has to carefully check on which properties higher level results depend on. $\endgroup$ – Jan Bohr Aug 19 '20 at 13:52

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