Characterizing the Dual of $W_0^{s,p}$

Question

I am interested in literature/results characterizing the dual of the fractional Sobolev space $W^{s,p}(\Omega)$, where $\Omega \subset \mathbb{R}^N$ is open, bounded, and smooth, $0< s<1$, and $1< p<\infty$.

For example, we can characterize the dual of $W_0^{1,2}(\Omega)$ as follows. If $f \in W^{1,2}(\Omega)$ then there exist $f_i \in L^2(\Omega)$ such that (for $v \in W^{1,2}_0(\Omega)$)

$\langle f,v \rangle = \int_\Omega f_0v+\sum_{i=1}^N f_iv_{x_i}\;dx$,

and so we write $f = f_0 - \sum_{i=1}^N (f_i)_{x_i}$.

A similar characterization holds for the dual of $W_0^{1,p}(\Omega)$, where instead the $f_i$ are in $L^p(\Omega)$ (though I do not have a reference for this - the $W_0^{1,2}$ case can be bound in Evan's PDE book).

My question is do we have such a representation $f=f_0 + (-\Delta)^\frac{s}{2} f_1$, for $f_0,f_1 \in L^p(\Omega)$, or something like this?

Answer 1

Hi,

I guess what is done in the Evan's PDE book is for $W^{1,2}_0(\Omega)$ functions, right ? If you look at all $W^{1,2}(\Omega)$, $\mathscr{D}(\Omega)$ (test functions) is not a dense subspace. Hence you may not inject the dual of $W^{1,2}(\Omega)$ in the space of distributions.

Your formula $f=f_0 -\sum_{i=1}^N \partial_{x_i} f_i$ may then be misleading, no ?

Nevertheless, identifying $W^{1,p}(\Omega)$ with a " pair " of $L^p(\Omega)$ spaces, you may prove that its dual consists of a " pair " of $L^{p'}(\Omega)$ spaces (for finite values of $p$ of course). This approach is detailed in the book of Adams [*], p.62.

For the fractionnal case, I don't know about their duals. But I guess that your expected formula (which should be $f=f_0+(-\Delta)^{s/2} f_1$, no ?) would make sense only in the distribution framework (if not, I don't understand what is your definition of the fractionnal laplacian) : such a description (if it does exist) should hold only for elements of $W^{s,p}_0(\Omega)'$ .

[*] See here : http://www.amazon.com/Sobolev-Spaces-Second-Applied-Mathematics/dp/0120441438

Answer 2

In the case $p=2$, there is a pretty elegant formalization to be found in negative Sobolev spaces. It is mentioned briefly here, and less briefly here.

The idea is that for any real number, the space $H^s(\mathbb R^n)$ with norm $$ \| f \|_{H^2} = \|(1+|\xi|^2)^{s/2} \hat f(\xi) \| _{L^2} $$ Where $\hat f$ is the Fourier transform of $f$, which may be a tempered distribution. The dual of $H^s(\mathbb R^n)$ is $H^{-s}(\mathbb R^n)$ which is the is the space of tempered distributions $f$ such that there is a an $L^2$ function $f_0$ such that $\hat f = (1+|\xi|^2)^{-s/2}\hat f_0$. Which is at least something like $f = f_0 +(-\Delta)^{s/2} f_0$.

I don't know about the case $p\neq 2$, but would be really interested to see if anyone else does.