New Characterizations of Musielak-Orlicz-Sobolev Spaces via Sharp Ball Averaging Functions

TL;DR

This paper introduces a new way to characterize Musielak-Orlicz-Sobolev spaces using sharp ball averaging functions, generalizing and improving previous results, especially for variable exponent Sobolev spaces.

Contribution

It provides a novel characterization of Musielak-Orlicz-Sobolev spaces that encompasses several classical and weighted Sobolev spaces, extending prior results with weaker assumptions.

Findings

Unified characterization of Musielak-Orlicz-Sobolev spaces

Improved results for variable exponent Sobolev spaces

Weaker integrability assumptions on functions

Abstract

In this article, the authors establish a new characterization of the Musielak--Orlicz--Sobolev space on $\mathbb{R}^n$, which includes the classical Orlicz--Sobolev space, the weighted Sobolev space and the variable exponent Sobolev space as special cases, in terms of sharp ball averaging functions. Even in a special case, namely, the variable exponent Sobolev space, the obtained result in this article improves the corresponding result obtained by P. H\"ast\"o and A. M. Ribeiro [Commun. Contemp. Math. 19 (2017), 1650022, 13 pp] via weakening the assumption $f\in L^1(\mathbb R^n)$ into $f\in L_{\mathrm{loc}}^1(\mathbb R^n)$, which was conjectured to be true by H\"ast\"o and Ribeiro in the aforementioned same article.