I am interested in literature/results characterizing the dual of the fractional Sobolev space $W^{s,p}(\Omega)$, where $\Omega \subset \mathbb{R}^N$ is open, bounded, and smooth, $0< s<1$, and $1< p<\infty$.

For example, we can characterize the dual of $W^{1,2}(\Omega)$ as follows. If $f \in W^{1,2}(\Omega)$ then there exist $f_i \in L^2(\Omega)$ such that

$\langle f,v \rangle = \int_\Omega f_0v+\sum_{i=1}^N f_iv_{x_i}\;dx$,

and so we write $f = f_0 - \sum_{i=1}^N (f_i)_{x_i}$.

A similar characterization holds for the dual of $W^{1,p}(\Omega)$, where instead the $f_i$ are in $L^p(\Omega)$ (though I do not have a reference for this - the $W^{1,2}$ case can be bound in Evan's PDE book).

My question is do we have such a representation $f=f_0 + (-\Delta)^sf_1$, for $f_0,f_1 \in L^p(\Omega)$, or something like this?

I am interested in literature/results characterizing the dual of the fractional Sobolev space $W^{s,p}(\Omega)$, where $\Omega \subset \mathbb{R}^N$ is open, bounded, and smooth, $0< s<1$, and $1< p<\infty$.

For example, we can characterize the dual of $W_0^{1,2}(\Omega)$ as follows. If $f \in W^{1,2}(\Omega)$ then there exist $f_i \in L^2(\Omega)$ such that (for $v \in W^{1,2}_0(\Omega)$)

$\langle f,v \rangle = \int_\Omega f_0v+\sum_{i=1}^N f_iv_{x_i}\;dx$,

and so we write $f = f_0 - \sum_{i=1}^N (f_i)_{x_i}$.

A similar characterization holds for the dual of $W_0^{1,p}(\Omega)$, where instead the $f_i$ are in $L^p(\Omega)$ (though I do not have a reference for this - the $W_0^{1,2}$ case can be bound in Evan's PDE book).

My question is do we have such a representation $f=f_0 + (-\Delta)^\frac{s}{2} f_1$, for $f_0,f_1 \in L^p(\Omega)$, or something like this?