Adrián González Pérez

Maximal interpolation in von Neumann algebras

The strong maximal function is a well-known object in classical harmonic analysis whose study goes back to the pioneering work of Jessen, Marcinkiewicz and Zygmund. It was proved independently by Cordoba-Feffermann and by de Guzman that, in two variables, the strong maximal function is of weak Orlicz type (Φ,Φ), where Φ(s)=s log(s). In this talk we will introduce a von Neumann analogue of the strong maximal function whose optimal weak Orlicz type is not yet known. In previous work with Jose Conde and Javier Parcet, we proved that such a maximal operator is of weak Orlicz type s log^2+ε(s), for every ε > 0. In this talk we will present a recent result that implies that weak Orlicz type cannot be improved below O(s log²(s)). This is built on recent results of Léonard Cadilhac and Éric Ricard and it is joint work with Javier Parcet and Jorge Pérez García.