On generalized Sobolev-Orlicz spaces associated to the Riesz fractional gradient

We introduce a new family of function spaces, the fractional generalized Sobolev-Orlicz spaces ${\mathrm{\Lambda }}_0^{s,A}(\mathrm{\Omega })$, where $A$ is a generalized $\mathrm{\Phi }$-function satisfying the $(\mathrm{I}\mathrm{n}\mathrm{c}{)}_p$ and $(\mathrm{D}\mathrm{e}\mathrm{c}{)}_q$ conditions for $1<p\le q<\mathrm{\infty }$, as an extension of the Lions-Calderón spaces (also known as Bessel potential spaces) ${\mathrm{\Lambda }}_0^{s,p}(\mathrm{\Omega })$ when $0<s<1$ to the generalized Orlicz framework. We obtain some continuous and compact embeddings for these spaces and study the continuous dependence of the Riesz fractional gradient $D^s$ with respect to $s\in [0,1]$ as $s\to \sigma \in [0,1]$. Finally, we apply these results to study the existence, uniqueness and continuous dependence of a family of partial differential equations depending on the Riesz fractional gradient as $s\to \sigma \in (0,1]$.