A revised condition for harmonic analysis in generalized Orlicz spaces on unbounded domains

TL;DR

This paper corrects a flaw in the decay condition for harmonic analysis in generalized Orlicz spaces and introduces new results, including a broader density theorem for smooth functions on unbounded domains.

Contribution

It identifies and corrects a flaw in the inverse function decay condition (A2) and provides new results on the density of smooth functions in generalized Orlicz spaces.

Findings

Corrected the decay condition (A2) for harmonic analysis

Established a more general density result for smooth functions

Enhanced understanding of conditions in generalized Orlicz spaces

Abstract

Conditions for harmonic analysis in generalized Orlicz spaces have been studied over the past decade. One approach involves the generalized inverse of so-called weak $\Phi$-functions. It featured prominently in the monograph Orlicz Spaces and Generalized Orlicz Spaces [P. Harjulehto and P. H\"ast\"o, Lecture Notes in Mathematics, vol. 2236, Springer, Cham, 2019]. While generally successful, the inverse function formulation of the decay condition (A2) in the monograph contains a flaw, which we explain and correct in this note. We also present some new results related to the conditions, including a more general result for the density of smooth functions.