A revised condition for harmonic analysis in generalized Orlicz spaces on unbounded domains
Petteri Harjulehto, Peter Hästö, Artur Słabuszewski
TL;DR
This work identifies a flaw in the inverse-function formulation of the decay condition $A2$ for unbounded generalized Orlicz spaces and introduces a corrected $A2$ that uses $\varphi^{-1}$ with a max argument $\max\{\tau, h(x)+h(y)\}$. The authors prove invariance of $A2$ under equivalence of weak $\Phi$-functions, establish that $\varphi^*$ inherits $A2$, and connect several equivalent formulations, including a max-$A2$ variant and the old $A2$ under certain assumptions. These results clarify the proper framework for harmonic analysis in nonstandard growth spaces and have practical implications for density of smooth functions in $W^{1,\varphi}$ on (un)bounded domains. The findings also align and extend recent work on density, improving the robustness of the theory in unbounded settings.
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 $Φ$-functions. It featured prominently in the monograph Orlicz Spaces and Generalized Orlicz Spaces [P. Harjulehto and P. Hästö, 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.