Is fractional Sobolev space $H^s$ Hilbert?

Question

For $s \in (0,\infty)$ a fractional number, define $H^s(\Omega) = W^{s,2}(\Omega)$ on good domain $\Omega$.

Every textbook doesn't say that $H^s$ is Hilbert. Is it? I have only seen this fact when $\Omega = \mathbb{R}^n$.

Answer 1

Yes, it is a Hilbert space. On $\mathbb R^n$ this fact is easy to see because the Fourier transform.

On other domains, some form of integrated divided difference is used: $$\|f\|^2_{\dot H^s} = \iint_{\Omega\times \Omega} \frac{|f(x)-f(y)|^2}{|x-y|^{n+2s}}\,dx\,dy$$ (The above is the square of a seminorm; add $\int_\Omega |f|^2$ to make it the square of a norm.) The parallelogram law holds: $$ \|f+g\|^2_{\dot H^s} + \|f-g\|^2_{\dot H^s} = 2\|f\|^2_{\dot H^s} +2\|g\|^2_{\dot H^s} $$ because for each $x,y$, denoting $a=f(x)-f(y)$ and $b=g(x)-g(y)$, we have $$|a+b|^2+|a-b|^2=2|a|^2+2|b|^2$$ The polarization identity tells you what the inner product is: $$\iint_{\Omega\times \Omega} \operatorname{Re}\frac{(f(x)-f(y))(\overline{g(x)-g(y)})}{|x-y|^{n+2s}}\,dx\,dy$$ plus $\int_\Omega \operatorname{Re}\Big( f(x) \overline{g(x) }\Big)$ coming from the added $L^2$ norm.

Answer 2

The first question to answer is: what is your definition of $W^{s,2}(\Omega)$? I will disconsider this question try to help you with your problem.

Assume that $\Omega \subset \mathbb{R}^n$. Also, let's suppose that $s\in (0,1)$. Let $u\in L^2(\Omega)$ and define $$|u|_{2,s}=\int\int\left(\frac{|u(x)-u(y)|^2}{|x-y|^{n+2s}}\right)^{1/2}dxdy$$

Define $$W^{s,2}(\Omega)=\{u\in L^2(\Omega):\ |u|_{2,s}<\infty\}$$

Note that $|\cdot|_{2,s}$ is a semi norm in $W^{s,2}(\Omega)$, it is called Gagliardo semi norm. You can verify that $$\|u\|_{2,s}=\sqrt{\|u\|_2^2+|u|_{2,s}^2}$$

define a norm in $W_{s,2}(\Omega)$ ($\|u\|_2$ is the usual norm in $L^2(\Omega)$) and this is a Banach space with such norm.

Define in $W^{s,2}(\Omega)$ the following quantity $$[u,v]_{2,s}=(u,v)+\int\int\frac{u(x)-u(y)}{|x-y|^{n/2+s}}\frac{v(x)-v(y)}{|x-y|^{n/2+s}}dxdy$$

Now you can show that $[\cdot,\cdot]_{2,s}$ define a inner product in $W^{2,s}(\Omega)$ and this space is Hilbert with respect to this product.

Remark 1: I leave to you the task of tying up loose ends.

Remark 2: What is the best definition when $s\geq 1$, i.e. what is the definition which combines with the whole theory?

Remark 3: I don't know that is your definition for $W^{s,2}(\Omega)$, but it has to combine in somw way with the definition given here.