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

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.