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

Let $\mathbf {E}_n: \mathcal {M} \to \mathcal {M}_n$ and $\mathbf {E}_m: \mathcal {N} \to \mathcal {N}_m$ be two sequences of conditional expectations on finite von Neumann algebras. The optimal weak Orlicz type of the associated strong maximal operator $\mathscr {E} = (\mathbf {E}_n\otimes \mathbf {E}_m)_{n,m}$ is not yet known. In a recent work of Jose Conde and the first two authors, it was shown that $\mathscr {E}$ has weak type $(\varPhi , \varPhi )$ for a family of functions including $\varPhi (t) = t \, \log ^{2+\varepsilon } t$ for every $\varepsilon \gt 0$. We prove that the weak Orlicz type of $\mathscr {E}$ cannot be lowered below $L \,\log ^2 L$, meaning that if $\mathscr {E}$ is of weak type $(\varPhi , \varPhi )$, then $\varPhi (s) \not \in o(s \, \log ^2 s)$. 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 $L \,\log ^2 L$ would imply a $p$-operator constant for $\mathscr {E}$ smaller than the known optimum as $p \to 1^{+}$.