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

Abstract

We introduce a new family of function spaces, the fractional generalized Sobolev-Orlicz spaces ${\Lambda }_0^{s,A}(\Omega )$, where $A$ is a generalized $\Phi$-function satisfying the $(\mathrm{Inc}{)}_p$ and $(\mathrm{Dec}{)}_q$ conditions for $1<p\le q<\infty$, as an extension of the Lions-Calderón spaces (also known as Bessel potential spaces) ${\Lambda }_0^{s,p}(\Omega )$ 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 $s\to \sigma \in (0,1]$.

26 pages