2026/07/28 by Guillermo Rey
Mathematics · #math.CA
An antichain is a family of rectangles in which no member contains another. Given a family E of rectangles, let hE 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 E of dyadic rectangles in ℝ2 one has ∫E exp(c hE) ≤ C|E|, where E is the union of all the rectangles in E. For general sparse families without the antichain condition, the estimate requires replacing hE by hE1/2, 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 2m1 × 2m2 × 2Φ(m1,m2), where Φ is monotone increasing in each variable, is weak-type L log L. This recovers a theorem of A. Córdoba. Second, the maximal operator of an arbitrary antichain of dyadic rectangles in the plane is bounded on Lp 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 L2 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.