Maximal weak Orlicz types and the strong maximal 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 two first-named authors, it was show that E has weak type (Φ, Φ) for a family of functions including Φ(t) = t \, log2+ε t, for every ε > 0. In this article, we prove that the weak Orlicz type of E cannot be lowered below L log² L, meaning that if E is of weak type (Φ, Φ), then Φ(s) ∈ o(s \, log² s). Our proof is based on interpolation. Namely, 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 L log² L would imply a p-operator constant for E smaller than the known optimum as p → 1⁺.