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Guillermo Rey

Publications and source records attributed to Guillermo Rey.

14 recordsLinked to original sources
arXiv2026

An antichain approach to a conjecture of Zygmund

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.

math.CA↗
arXiv2026

Pairwise meets of antichains in $\mathbb{Z}^d$

The meet of two points in $\mathbb{Z}^d$ is their coordinatewise minimum. We show that every finite antichain $A$ in $\mathbb{Z}^d$ has at least $c_d |A|^{d/(d-1)}$ distinct pairwise meets, where $c_d > 0$ depends only on $d$, and that the exponent $d/(d-1)$ is best possible. As a corollary we obtain an isoperimetric inequality for downsets: every finite downset $D$ in $\mathbb{Z}^d_{\geq 0}$ satisfies $|D| \geq c_d |\max(D)|^{d/(d-1)}$, where $\max(D)$ is its set of maximal elements. By prime factorization the meet bound also yields a lower bound on greatest common divisors: a primitive set of $N$ integers supported on at most $d$ primes has at least $c_d N^{d/(d-1)}$ distinct pairwise gcds.

math.CO↗
arXiv2026

Cancellative sparse domination

We present a general sparse domination principle which respects the cancellative structure of the functions under study. We obtain sparse domination results in general measure spaces, including general martingale settings in one and two parameters, and in the Euclidean setting. In the one-parameter martingale setting, we obtain a sparse characterization of the $H^1$ norm. The proofs make critical use of precise level-set estimates for generalized versions of medians. Our results imply new, quantitatively sharp, weighted results for martingales and Calder\'on-Zygmund operators acting on $H^p$ spaces.

math.CA↗
arXiv2025

Merryfield's inequality for multiparameter martingales

We extend an inequality of Merryfield, valid in the continuous setting, to discrete multiparameter martingales. As a consequence, we obtain the $L^p$ comparison of the maximal function with the square function: \begin{align*} E[(Sf)^p] \lesssim E[(f^*)^p] \end{align*} for regular multiparameter filtrations and $0 < p < \infty$.

math.PR↗
arXiv2023

Weak-type maximal function estimates on the infinite-dimensional torus

We prove necessary and sufficient conditions for the weak-$L^p$ boundedness, for $p \in (1,\infty)$, of a maximal operator on the infinite-dimensional torus. In the endpoint case $p=1$ we obtain the same weak-type inequality enjoyed by the strong maximal function in dimension two. Our results are quantitatively sharp.

math.CA↗
arXiv2022

Greedy approximation algorithms for sparse collections

We describe a greedy algorithm that approximates the Carleson constant of a collection of general sets. The approximation has a logarithmic loss in a general setting, but is optimal up to a constant with only mild geometric assumptions. The constructive nature of the algorithm gives additional information about the almost-disjoint structure of sparse collections. As applications, we give three results for collections of axis-parallel rectangles in every dimension. The first is a constructive proof of the equivalence between Carleson and sparse collections, first shown by Hänninen. The second is a structure theorem proving that every collection $\mathcal{E}$ can be partitioned into $\mathcal{O}(N)$ sparse subfamilies where $N$ is the Carleson constant of $\mathcal{E}$. We also give examples showing that such a decomposition is impossible when the geometric assumptions are dropped. The third application is a characterization of the Carleson constant involving only $L^{1,\infty}$ estimates.

math.CA↗
arXiv2020

Another counterexample to Zygmund's conjecture

We present a simple dyadic construction that yields a new counterexample to Zygmund's conjecture. Our result recovers Soria's classical result in dimension three, through a different construction, and gives new ones in all other dimensions $d > 3$.

math.CA↗
arXiv2018

Sparse domination and the strong maximal function

We study the problem of dominating the dyadic strong maximal function by $(1, 1)$-type sparse forms based on rectangles with sides parallel to the axes, and show that such domination is impossible. Our proof relies on an explicit construction of a pair of maximally separated point sets with respect to an appropriately defined notion of distance

math.CA↗
arXiv2016

On the embedding of $A_1$ into $A_\infty$

We give a quantitative embedding of the Muckenhoupt class $A_1$ into $A_\infty$. In particular, we show how $ε$ depends on $[w]_{A_1}$ in the inequality which characterizes $A_\infty$ weights: \[ \frac{w(E)}{w(Q)} \leq \biggl( \frac{|E|}{|Q|} \biggr)^ε, \] where $Q$ is any dyadic cube and $E$ is any subset of $Q$. This embedding yields a sharp reverse-Hölder inequality as an easy corollary.

math.CA↗
arXiv2015

Borderline Weak Type Estimates for Singular Integrals and Square Functions

For any Calderón-Zygmund operator $ T$, any weight $ w$, and $ α>1$, the operator $ T$ is bounded as a map from $ L ^{1} (M _{ L \log\log L (\log\log\log L) ^{α} } w )$ into weak-$L^1(w)$. The interest in questions of this type goes back to the beginnings of the weighted theory, with prior results, due to Coifman-Fefferman, Pérez, and Hytönen-Pérez, on the $ L (\log L) ^{ε}$ scale. Also, for square functions $ S f$, and weights $ w \in A_p$, the norm of $ S$ from $ L ^p (w)$ to weak-$L^p (w)$, $ 2\leq p < \infty $, is bounded by $ [w] _{A_p}^{1/2} (1+\log [w] _{A_ \infty }) ^{1/2} $, which is a sharp estimate.

math.CA↗
arXiv2014

A pointwise estimate for positive dyadic shifts and some applications

We prove a pointwise estimate for positive dyadic shifts of complexity $m$ which is linear in the complexity. This can be used to give a pointwise estimate for Calderón-Zygmund operators and to answer a question posed by A. Lerner. Several applications to weighted estimates for both multilinear Calderón-Zygmund operators and square functions are discussed.

math.CA↗