# Pointwise multipliers of Musielak--Orlicz spaces and factorization

## TL;DR

This paper characterizes pointwise multipliers between Musielak-Orlicz spaces as another Musielak-Orlicz space defined via a generalized Legendre transform, and explores factorization properties highlighting differences from Orlicz spaces.

## Contribution

It provides a new characterization of pointwise multipliers in Musielak-Orlicz spaces using a generalized Legendre transform and discusses factorization issues distinct from classical Orlicz spaces.

## Key findings

- Pointwise multipliers form a Musielak-Orlicz space defined by a generalized Legendre transform.
- Characterization of multipliers between Nakano spaces as a special case.
- Differences in factorization properties between Musielak-Orlicz and Orlicz spaces.

## Abstract

We prove that the space of pointwise multipliers between two distinct Musielak--Orlicz spaces is another Musielak-Orlicz space and the function defining it is given by an appropriately generalized Legendre transform. In particular, we obtain characterization of pointwise multipliers between Nakano spaces. We also discuss factorization problem for Musielak-Orlicz spaces and exhibit some differences between Orlicz and Musielak-Orlicz cases.