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

TL;DR

This paper introduces a new class of fractional Sobolev-Orlicz spaces extending classical spaces, studies their properties, and applies these to analyze PDEs involving the Riesz fractional gradient, including existence and stability results.

Contribution

It defines generalized Sobolev-Orlicz spaces associated with the Riesz fractional gradient, establishes their embedding properties, and analyzes PDEs depending on the fractional order.

Findings

Established continuous and compact embeddings of the new spaces.

Proved continuous dependence of the Riesz fractional gradient on the order s.

Demonstrated existence and uniqueness of PDE solutions involving these spaces.

Abstract

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