I know that this is possible is the context of Laguerre and Hermite expansions, but I don't know it for the classical setting. I will check that book Pedro

The idea is to obtain a charaterization of Sobolev spaces $W^{s,p}$, where $s$ could be any positive real number, in terms of a Fractional integral operator (as the image of such operator over $L^p$), or as the pre-image of a suitable Fractional derivative operator, it this way: if $D_s$ is the desired derivative operator of order $s$, then $W^{s,p}=\{f \in L^p: D_s f \in L^p\}$.