On Orlicz classes defined in terms of associated weight functions

TL;DR

This paper explores Orlicz classes defined via associated weight functions, linking properties of N-functions to weight sequences, and establishing connections with dual sequences to deepen understanding of ultradifferentiable function classes.

Contribution

It characterizes properties of N-functions through weight sequences and introduces a framework connecting these to dual sequences, enabling new constructions and insights.

Findings

Characterization of N-function properties via weight sequences

Construction of examples and counterexamples using weight sequences

Establishment of a link between the complementary N-function and dual sequences

Abstract

N-functions and their growth and regularity properties are crucial in order to introduce and study Orlicz classes and Orlicz spaces. We consider N-functions which are given in terms of so-called associated weight functions. These functions are frequently appearing in the theory of ultradifferentiable function classes and in this setting additional information is available since associated weight functions are defined in terms of a given weight sequence. We express and characterize several known properties for N-functions purely in terms of weight sequences which allows to construct (counter-)examples. Moreover, we study how for abstractly given N-functions this framework becomes meaningful and finally we establish a connection between the complementary N-function and the recently introduced notion of the so-called dual sequence.