Maximal weak Orlicz types and the strong maximal operator on von Neumann algebras

Let En:M→Mn and Em:N→Nm be two sequences of conditional expectations on finite von Neumann algebras. The optimal weak Orlicz type of the associated strong maximal operator E=(En⊗Em)n,m is not yet known. In a recent work of Jose Conde and the first two authors, it was shown that E has weak type (Φ,Φ) for a family of functions including Φ(t)=tlog2+εt for every ε>0. We prove that the weak Orlicz type of E cannot be lowered below Llog2L, meaning that if E is of weak type (Φ,Φ), then Φ(s)/∈o(slog2s). Our proof is based on interpolation. We use recent techniques of Cadilhac/Ricard to formulate a Marcinkiewicz-type theorem for maximal weak Orlicz types. Then, we show that a weak Orlicz type lower than Llog2L would imply a p-operator constant for E smaller than the known optimum as p→1+.