An antichain approach to a conjecture of Zygmund

Abstract

An antichain is a family of rectangles in which no member contains another. Given a family $\mathcal{E}$ of rectangles, let $h_{\mathcal{E}}$ be the sum of the indicator functions of its members. We show that there exist constants $c, C > 0$ such that for every sparse antichain $\mathcal{E}$ of dyadic rectangles in $\mathbb{R}^2$ one has $\int_E \exp(c h_{\mathcal{E}}) \leq C|E|$, where $E$ is the union of all the rectangles in $\mathcal{E}$. For general sparse families without the antichain condition, the estimate requires replacing $h_{\mathcal{E}}$ by $h_{\mathcal{E}}^{1/2}$, so antichains behave as if they lived in one dimension fewer. We give two applications. First, the dyadic Zygmund conjecture holds in dimension three: the maximal operator associated to dyadic rectangles with sidelengths $2^{m_1} \times 2^{m_2} \times 2^{\Phi(m_1,m_2)}$, where $\Phi$ is monotone increasing in each variable, is weak-type $L \log L$. This recovers a theorem of A. C\'ordoba. Second, the maximal operator of an arbitrary antichain of dyadic rectangles in the plane is bounded on $L^p$ with norm $O(p')$, which is the growth of the one-parameter maximal function. This bound is sharp, and removing the antichain condition forces a constant that grows like $(p')^2$ instead. The proofs proceed through bounds on $k$-fold intersections: for sparse antichains in the plane, the $k$-wise intersection sums grow at most geometrically in $k$, which we prove through an $L^2$ estimate for the Gram matrix of the normalized indicators of the family. We also show that, in every dimension, the analogous exponential estimate for antichains is equivalent to a $k$-wise intersection bound, and implies the corresponding case of Zygmund's conjecture.