MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|cite|improve this question

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.

share|cite|improve this answer


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 :

share|cite|improve this answer
Thanks for the reply. You are right about the compact support being there, as well as the right analogue for the fractional case. I guess the point is to find the right definition of the fractional laplacian, which will be tied to this in a very critical way. – Daniel Spector Feb 14 '13 at 16:52

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

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