The Zygmund conjecture and Rey's exponential integrability conjecture

We prove the Zygmund conjecture in all parameters: the maximal operator associated with products of cubes with side lengths $(s_1,\dots ,s_{m-1},\phi (s_1,\dots ,s_{m-1}))$, where $\phi$ is any positive coordinatewise nondecreasing function, satisfies the weak $L(\log L{)}^{m-2}$ estimate for every $m\ge 3$. 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 $1\mathrm{/}(m-1)$, for sparse incomparable families $\mathcal{G}$ of $m$-parameter dyadic rectangles, $m\ge 2$. The Zygmund conjecture, even in the continuous form above, follows by strengthening a reduction given by Rey.