Maximal weak Orlicz types and the strong maximal 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 $\mathcal{E}=({\mathbf{E}}_n\otimes {\mathbf{E}}_m{)}_{n,m}$ is not yet known. In a recent work of Jose Conde and the two first-named authors, it was show that $\mathcal{E}$ has weak type $(\mathrm{\Phi },\mathrm{\Phi })$ for a family of functions including $\mathrm{\Phi }(t)=t{\log }^{2+\epsilon }t$, for every $\epsilon >0$. In this article, we prove that the weak Orlicz type of $\mathcal{E}$ cannot be lowered below $L{\log }^{2}L$, meaning that if $\mathcal{E}$ is of weak type $(\mathrm{\Phi },\mathrm{\Phi })$, then $\mathrm{\Phi }(s)\in ̸o(s{\log }^{2}s)$. 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 $L{\log }^{2}L$ would imply a $p$-operator constant for $\mathcal{E}$ smaller than the known optimum as $p\to 1^{+}$.