# References on duality of fractional order Sobolev spaces

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 $$0) and $$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.

• 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. :) Commented Jun 4, 2022 at 4:35
• @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 $\dot{H}^1$? Commented Jun 6, 2022 at 10:31
• 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. Commented Jun 6, 2022 at 10:53

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 $$T=J^su$$.