Characterization of generalized Orlicz spaces

TL;DR

This paper introduces a new smoothed difference quotient to characterize generalized Orlicz-Sobolev spaces, extending classical concepts to more flexible function spaces, including Orlicz and variable exponent spaces.

Contribution

It proposes a novel smoothed difference quotient method for characterizing generalized Orlicz-Sobolev spaces, which was not possible with traditional difference quotients.

Findings

The smoothed difference quotient successfully characterizes generalized Orlicz-Sobolev spaces.

Results are new even for classical Orlicz and variable exponent spaces.

The approach extends fractional smoothness concepts to broader function spaces.

Abstract

The norm in classical Sobolev spaces can be expressed as a difference quotient. This expression can be used to generalize the space to the fractional smoothness case. Because the difference quotient is based on shifting the function, it cannot be used in generalized Orlicz spaces. In its place, we introduce a smoothed difference quotient and show that it can be used to characterize the generalized Orlicz-Sobolev space. Our results are new even in Orlicz spaces and variable exponent spaces.