Papers and preprints

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. Córdoba. Second, the maximal operator of an arbitrary antichain of dyadic rectangles in the plane is bounded on $L^p$ with norm $\mathcal{O}(p')$, 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.

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.

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 Calderón-Zygmund operators acting on $H^p$ spaces.

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 for regular multiparameter filtrations and $0 < p < \infty$.

Sharp restricted weak-type estimates for sparse operators

We find the exact Bellman function associated to the level-sets of sparse operators acting on characteristic functions.

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.

Weighted BMO estimates for singular integrals and endpoint extrapolation in Banach function spaces

In this paper we prove sharp weighted BMO estimates for singular integrals, and we show how such estimates can be extrapolated to Banach function spaces.

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 finite 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.

Another counterexample to Zygmund's conjecture

We present a simple dyadic construction that yields a new counterexample to Zygmund's conjecture in differentiation of integrals. Our result recovers Soria's classical results in dimensions three and four, through a different construction, and gives new ones in all other dimensions

@Article{Rey2020,
  Title   = {Another counterexample to {Z}ygmund's connjecture},
  Author  = {Guillermo Rey},
  Journal = {Proc. Amer. Math. Soc.},
  Year    = {2020},
  Pages   = {5269--5275},
  Volume  = "148"
}
        

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

@article{BCOR2019,
title = "Sparse domination and the strong maximal function",
journal = "Advances in Mathematics",
volume = "345",
pages = "1 - 26",
year = "2019",
issn = "0001-8708",
doi = "https://doi.org/10.1016/j.aim.2019.01.007",
url = "http://www.sciencedirect.com/science/article/pii/S0001870819300325",
author = "Alex Barron and José M. Conde-Alonso and Yumeng Ou and Guillermo Rey",
keywords = "Biparameter, Multiparameter, Sparse domination, Maximal functions",
abstract = "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."
}
        

Borderline weak type estimates for singular integrals and square functions

For any Calder&oactue;on-Zygmund operator $T$, any weight $ w$, and $\alpha >1$, the operator $T$ is bounded as a map from $ L ^{1} (M _{ L \log\log L (\log\log\log L) ^{\alpha } } 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) ^{\epsilon }$ 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}$.

@article {DSLR,
author    = {Domingo-Salazar, Carlos and Lacey, Michael and Rey, Guillermo},
title     = {Borderline weak-type estimates for singular integrals and square functions},
journal   = {Bulletin of the London Mathematical Society},
volume    = {48},
number    = {1},
publisher = {Oxford University Press},
issn      = {1469-2120},
url       = {http://dx.doi.org/10.1112/blms/bdv090},
doi       = {10.1112/blms/bdv090},
pages     = {63--73},
year      = {2016},
}
        

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

We prove the following endpoint embedding between the $A_1$ and $A_\infty$ Muckenhoupt classes: \[ \frac{w(E)}{w(P)} \leq [w]_{A_1^d} \biggl( \frac{|E|}{|P|} \biggr)^{[w]_{A_1^d}^{-1}} \] for all cubes $P$ in $\mathbb{R}^d$ and all subsets $E \subseteq P$. To prove this result we compute the exact Bellman function associated with the problem (in any dimension) and construct explicit examples to prove sharpness.

@Article{Rey2016,
  Title   = {On the embedding of ${A}_1$ into ${A}_\infty$},
  Author  = {Guillermo Rey},
  Journal = {Proc. Amer. Math. Soc.},
  Year    = {2016},
  Pages   = {4455--4470},
  number  = "10"
}
        

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.

@ARTICLE
{2014arXiv1409.4351C,
author = {{Conde-Alonso}, J.~M. and {Rey}, G.},
title = "{A pointwise estimate for positive dyadic shifts and some applications}",
journal = {ArXiv e-prints},
archivePrefix = "arXiv",
eprint = {1409.4351},
primaryClass = "math.CA",
keywords = {Mathematics - Classical Analysis and ODEs},
year = 2014,
month = sep,
adsurl = {http://adsabs.harvard.edu/abs/2014arXiv1409.4351C},
adsnote = {Provided by the SAO/NASA Astrophysics Data System}
}
        

Extremizers and sharp weak-type estimates for positive dyadic shifts

We find the exact Bellman function for the weak $L^1$ norm of local positive dyadic shifts. We also describe a sequence of functions, self-similar in nature, which in the limit extremize the local weak-type $(1,1)$ inequality.

@article {MR3161110,
AUTHOR = {Rey, Guillermo and Reznikov, Alexander},
TITLE = {Extremizers and sharp weak-type estimates for positive dyadic
       shifts},
JOURNAL = {Adv. Math.},
FJOURNAL = {Advances in Mathematics},
VOLUME = {254},
YEAR = {2014},
PAGES = {664--681},
ISSN = {0001-8708},
MRCLASS = {42B25},
MRNUMBER = {3161110},
MRREVIEWER = {B. Muckenhoupt},
DOI = {10.1016/j.aim.2013.12.030},
URL = {http://dx.doi.org/10.1016/j.aim.2013.12.030},
}