# Fractional Sobolev spaces and interpolation in unbounded Lipschitz domains

I am not really familiar with the topic, thus I am looking for some references about the following problem.

Let $s>0$ be a positive real and let $p\in(1,+\infty)$. We define the Bessel Potential spaces on $\mathbb{R}^n$ via the Fourier transform $\mathcal{F}$ as follows. $$H^{s,p}(\mathbb{R}^n)=\{f\in L^p(\mathbb{R^n}):\mathcal{F}^{-1}(1+|\xi|^2)^{\frac{s}{2}}\mathcal{F}f\in L^p(\mathbb{R^n})\}.$$

In the case of $s\in\mathbb{Z}^+$ it holds that $H^{s,p}(\mathbb{R}^n)=W^{s,p}(\mathbb{R}^n)$ where the latter is the standard Sobolev space.

Moreover, the Bessel potential spaces can be obtained via complex interpolation from the standard Sobolev spaces of integer order. Therefore the spaces $H^{s,p}(\mathbb{R}^n)$ are also referred as Fractional Sobolev Space.

Let now $\Omega\subset \mathbb{R}^n$ an open subset (bounded or unbounded), we can define $$H^{s,p}(\Omega)=\{f\in L^p(\Omega): \exists g\in H^{s,p}(\mathbb{R}^n): g\big|_{\Omega}=f \}$$

with the norm $\|f\|_{H^{s,p}(\Omega)}=\inf\{\|g\|_{H^{s,p}(\mathbb{R}^n)}:g\big|_{\Omega}=f\}$.

Therefore, my question is. Let $k$ be in $\mathbb{Z}^+$. Are the spaces $H^{s,p}(\Omega)$ interpolation spaces between $W^{k,p}(\Omega)$ and $W^{k+1.p}(\Omega)$ if $k<s<k+1$ when $\Omega$ is an unbounded Lipschitz domain with noncompact boundary?

In particular, if $T$ is bounded linear operator $T: W^{k,p}(\Omega)\to W^{k,p}(\Omega)$ and $T:W^{k+1,p}(\Omega)\to W^{k+1,p}(\Omega)$, do we have the boundedness $T:H^{s,p}(\Omega)\to H^{s,p}(\Omega)$ for every $s\in[k,k+1]?$

Can you please suggest me some references?

You cannot expect this without imposing some condition at infinity. Otherwise, $\Omega$ could have a tentacle" which stretches to infinity and thins rapidly. In this fashion, you can have functions in $W^{k,p}(\Omega)$ which do not have an extension in $W^{k,p}(R^n)$.

• I expected that some regularity is needed. Can you suggest me some detailed references please?
– Pit
May 1, 2015 at 15:23

Here are two very good references:

R. A. Adams, Sobolev Spaces, Pure and Applied Mathematics, A Series of Monographs and Textbooks, 65 (Academic Press, New York, 1975).

H. Triebel, Interpolation Theory, Function Spaces, Di erential Operators (North-Holland, Amsterdam, 1978).

Best regards.