A few last words on pointwise multipliers of Calder\'on--Lozanovski\u{i} spaces

TL;DR

This paper provides a complete characterization of pointwise multipliers between Calderón–Lozanovski spaces, confirming a conjecture and offering new insights into their structure and applications.

Contribution

It offers a complete description of the multiplier space as another Calderón–Lozanovski space, advancing understanding and solving the factorization problem.

Findings

Multiplier space is another Calderón–Lozanovski space with a generalized Young conjugate.

Confirmed the conjecture by Kolwicz, Lesnik, and Maligranda.

Provided applications and outlined future research directions.

Abstract

We will provide a complete description of the space $M(X_F,X_G)$ of pointwise multipliers between two Calder\'on--Lozanovski\u{i} spaces $X_F$ and $X_G$ built upon a rearrangement invariant space $X$ and two Young functions $F$ and $G$. Meeting natural expectations, the space $M(X_F,X_G)$ turns out to be another Calder\'on--Lozanovski\u{i} space $X_{G \ominus F}$ with $G \ominus F$ being the appropriately understood generalized Young conjugate of $G$ with respect to $F$. Nevertheless, our argument is not a mere transplantation of existing techniques and requires a rather delicate analysis of the interplay between the space $X$ and functions $F$ and $G$. Furthermore, as an example to illustrate applications, we will solve the factorization problem for Calder\'on--Lozanovski\u{i} spaces. All this not only complements and improves earlier results (basically giving them the final touch), but also confirms the conjecture formulated by Kolwicz, Le\'snik and Maligranda in [Pointwise multipliers of Calder\'on--Lozanovski\u{i} spaces, Math. Nachr. 286 (2012), no. 8-9, 876--907]. We will close this work by formulating a number of open questions that outline a promising panorama for future research.