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 $W^{k,p}(R^n)$.

  • $\begingroup$ I expected that some regularity is needed. Can you suggest me some detailed references please? $\endgroup$ – Pit May 1 '15 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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.