For $s\in\mathbb{R}$ and $1<p<\infty$, one can define the fractional Sobolev space $W^{s,p}(\mathbb{R}^n)$. Every element $f$ in $W^{s,p}(\mathbb{R}^d)$ is a tempered distribution such that the inverse Fourier transform of $$\widehat{f_s}(\xi):=(1+|\xi|^2)^{\frac{s}{2}}\hat{f}(\xi)$$ is an element of $L^p(\mathbb{R}^d)$ and we define $$\|f\|_{W^{s,p}}:=\|f_s\|_{L^p}.$$ Note that the function $(1+|\xi|^2)^{\frac{s}{2}}$ has at most polynomial order growth, thus $(1+|\xi|^2)^{\frac{s}{2}}\hat{f}(\xi)$ is in fact a well defined tempered distribution and so is its inverse Fourier transform. I'm looking for refrences which deal with the dual space of $W^{s,p}(\mathbb{R}^n)$. I have seen huge literatures on the case $p=2$, in which case the space $W^{s,2}(\mathbb{R}^n)$ is a Hilbert space with inner product $$\langle f,g\rangle_{W^{s,2}}=\frac{1}{2\pi}\int_{\mathbb{R}^d}\hat{f}(\xi)\overline{\hat{g}(\xi)}(1+|\xi|^2)^sd\xi.$$ And the dual space of $W^{s,2}(\mathbb{R}^n)$ is precisely $W^{-s,2}(\mathbb{R}^d)$.
Unlike the case $p=2$, I didn't find any resources which discuss the duality when $p\neq 2$. I've seen some resources mentioned that the dual of $W^{s,p}(\mathbb{R}^n)$ is $W^{-s,q}(\mathbb{R}^n)$, where $\frac{1}{p}+\frac{1}{q}=1$. Intuitively this makes sense from the duality of $L^p$ spaces. But I didn't see any proof of this.
Could someone provide me some references related to this quesiton? Or maybe explain to me in details if applicable? Thanks in advance!
P.S. I saw some resources which define the Sobolev spaces of negative powers to be the dual of Soboleve spaces of positive powers, i.e., they define $$W^{-s,q}(\mathbb{R}^n):=(W^{s,p}(\mathbb{R}^n))'$$ when $s>0$ and $\frac{1}{p}+\frac{1}{q}=1$. I would like to see whether this coincides with the definition I provided above.
So a good reference for that are the books Theory of function spaces I and II by H. Triebel.
In particular, the space that you define here is the space $F^s_{p,2}$ (see for example the last Theorem in section 1.5.1 in Theory of function spaces II) often denoted as $H^{s,p}$ and called the Bessel-Sobolev space which can be seen as complex interpolation of Sobolev spaces (see also Is $B^s_{p,p} = W^{s,p}$?), or also defined with a Littlewood-Paley decomposition. And you can find the fact that $(F^s_{p,q})' = F^{-s}_{p',q'}$ in Theory of function spaces II, section 2.11.2 (and it is also indicated in Theory of function spaces II, section 1.5.2)