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

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.