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.