# Basic results of fractional Orlicz-Sobolev space and applications to non-local problems

## TL;DR

This paper explores the properties of fractional Orlicz-Sobolev spaces and introduces a fractional M-Laplacian operator, demonstrating its application in proving the existence of solutions to non-local problems.

## Contribution

It defines and analyzes the fractional Orlicz-Sobolev space $W^{s,M}$ and introduces a fractional M-Laplacian operator, extending non-local analysis in Orlicz spaces.

## Key findings

- Qualitative properties of $W^{s,M}$ established
- Existence of weak solutions for non-local problems proved
- Introduction of a fractional M-Laplacian operator

## Abstract

In this paper, we study the interplay between Orlicz-Sobolev spaces $L^{M}$ and $W^{1,M}$ and fractional Sobolev spaces $W^{s,p}$. More precisely, we give some qualitative properties of the new fractional Orlicz-Sobolev space $W^{s,M}$, where $s\in (0,1)$ and $M$ is an $N-$function. We also study a related non-local operator, which is a fractional version of the nonhomogeneous $M$-Laplace operator. As an application, we prove existence of weak solution for a non-local problem involving the new fractional $M-$Laplacian operator.