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arXiv · 2609.23895

The Zygmund conjecture and Rey's exponential integrability conjecture

Henri Martikainen
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Abstract

We prove the Zygmund conjecture in all parameters: the maximal operator associated with products of cubes with side lengths $(s_1,\ldots,s_{m-1},ϕ(s_1,\ldots,s_{m-1}))$, where $ϕ$ is any positive coordinatewise nondecreasing function, satisfies the weak $L(\log L)^{m-2}$ estimate for every $m\geq3$. The estimate is new for $m>3$. Our approach is to prove Rey's recent conjecture on the exponential integrability of the overlap function $h_{\mathcal G}=\sum_{I\in\mathcal G}\mathbf{1}_I$, with exponent $1/(m-1)$, for sparse incomparable families $\mathcal G$ of $m$-parameter dyadic rectangles, $m\geq2$. The Zygmund conjecture, even in the continuous form above, follows by strengthening a reduction given by Rey.

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Henri Martikainen. 2026-09-20. The Zygmund conjecture and Rey's exponential integrability conjecture. https://arxiv.org/abs/2609.23895

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