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Is there a way to represent a Sobolev space as the image of a fractional integral operator over an $L^p$ Lebesgue space? Yes, as it was comment, there is an answer for that in the book "Singular Integrals and Differentiability Properties of Functions" by Elias M. Stein (Princeton, 1970), Section V.3, pp. 130ff.

But, what i really want to know is if there exist an operator $D_k$ such that $W^{k,p}=\{f\in L^p: D_k f \in L^p\}$. And what will really help me if this operator can be defined for any real (positive) number $k$, in order to extend the definition of Sobolev space $W^{k,p}$ to any real $k$, in a fancy way. Is this possible?

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  • $\begingroup$ Maybe you could say a little on the background / motivation for your question? $\endgroup$
    – Stefan Kohl ♦
    Commented Nov 4, 2013 at 21:03
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    $\begingroup$ Check "Bessel potential spaces" in the book "Singular Integrals and Differentiability Properties of Functions" by Elias M. Stein (Princeton, 1970), Section V.3, pp. 130ff. $\endgroup$
    – Pedro Lauridsen Ribeiro
    Commented Nov 4, 2013 at 22:56
  • $\begingroup$ The idea is to obtain a charaterization of Sobolev spaces $W^{s,p}$, where $s$ could be any positive real number, in terms of a Fractional integral operator (as the image of such operator over $L^p$), or as the pre-image of a suitable Fractional derivative operator, it this way: if $D_s$ is the desired derivative operator of order $s$, then $W^{s,p}=\{f \in L^p: D_s f \in L^p\}$. $\endgroup$
    – Anibal Chicco
    Commented Nov 5, 2013 at 14:30
  • $\begingroup$ I know that this is possible is the context of Laguerre and Hermite expansions, but I don't know it for the classical setting. I will check that book Pedro $\endgroup$
    – Anibal Chicco
    Commented Nov 5, 2013 at 14:34
  • $\begingroup$ Aha! I have regained my account! So the answer is yes, and I can send you a preprint if you are interested to the development. I perfectly well understand what you are saying, and the notion is quite interesting. $\endgroup$
    – Daniel Spector
    Commented Nov 15, 2013 at 20:09

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Let $p\in (1,+\infty)$ and $s\in \mathbb R$. For $\xi \in \mathbb R^n$, we define $ \langle\xi \rangle=(1+\vert \xi\vert^2)^{1/2} $ and accordingly the Fourier multiplier $\langle D \rangle^s$ as $$ (\widehat{\langle D \rangle^s u})(\xi)=\langle\xi \rangle^s\hat u(\xi). $$ Then we have $ W^{s,p}(\mathbb R^n)=\{u\in \mathscr S'(\mathbb R^n),\langle D \rangle^s u\in L^p(\mathbb R^n) \}, $ where $ \mathscr S'(\mathbb R^n)$ stands for the temperate distributions. The main point is to prove the continuity of classical singular integrals on $L^p$.

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