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I would like to ask you for any good references regarding fractional order Sobolev spaces. I know Hitchhiker's guide to the fractional Sobolev spaces is a very popular one, and I found it to be quite a nice introduction. However, I would like a more comprehensive study. In particular, I am struggling with the following question:

Let $s>0$ (in my case, it is enough that $0<s<1$) and $1\leq p <\infty$. Define the Sobolev space $W^{s,p}$ as the completion of $\mathcal{S}(\mathbb{R}^n)$ with the norm $$\|u\|_{W^{s,p}}=\|(I-\Delta)^{s/2}u\|_{L^p},$$ where $(I-\Delta)^{s/2}$ is the operator with symbol $(1+|\xi|^2)^{s/2}$. I know this may not be the customary approach, I'm following the potential space approach by Jerison and Kenig. As far as I know, alternative definitions turn out to be equivalent.

Define now $W^{-s,p'}$ as the dual of $W^{s,p}$. Clearly, if $$f=(I-\Delta^2)^{s/2}v,$$ for some $v\in L^{p'}$, then $f\in W^{-s,p'}$.

My question is, is the converse statement true? Does every $f\in W^{-s,p'}$ take the form above? For integer $s$, the answer is positive, see Adams and Fournier, Theorem 3.9.

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    $\begingroup$ If I recall correctly, the Bessel potential spaces you wrote are not equivalent to the fractional Sobolev spaces in the Hitchhiker article unless $p=2$ or $s \in Z$. The Bessel potential spaces are sometimes written $H^{s,p}$, and they can be obtained from the $W^{k,p}$ via complex interpolation in $s$. The ones defined in terms of Gagliardo seminorms are actually the Besov spaces $B^s_{p,p}$, obtained via real interpolation. The Bessel potential spaces are the same as the Triebel-Lizorkin spaces $F^s_{p,2}$. Also, the answer to your question is yes, but I don’t have a reference handy. :) $\endgroup$
    – sharpend
    Commented Jun 4, 2022 at 4:35
  • $\begingroup$ @sharpend Thank you! :) I have been looking more into Bessel potential spaces and I'm quite satisfied with what I have found. Do you know whether the homogeneous potential space, defined by the Riesz potential with symbol $|\xi|$, with $p=2$ is equivalent to the homogeneous Sobolev space $\dot{H}^1$? $\endgroup$
    – Manuel Cañizares
    Commented Jun 6, 2022 at 10:31
  • $\begingroup$ Actually, it would make all the sense in the world, considering that we can use Plancherel theorem in $p=2$ and how the Fourier transform interacts with derivatives, but I often get surprised by some functional analytic detail that passes unnoticed over my head. $\endgroup$
    – Manuel Cañizares
    Commented Jun 6, 2022 at 10:53

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The answer is positive. If $T$ is a continuous functional on $W^{s,p}$ then $TJ^{-s}$ is a continuous functional on $L^p$, which can be represented by an element $u\in L^{p'}$: thus for all $f\in L^p$ we have $$ TJ^{-s}(f)=\int uf dx. $$ You also have $\|TJ^{-s}\|=\|u\|$. Now for all $g\in W^{s,p}$ we have $g=J^{-s}f$ for some $f\in L^p$ and $$ T(g)=T(J^{-s}f)=u(f)=J^su(J^{-s}f)=J^su(g) $$ where I switched to distributional notation. This proves $T=J^su$.

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