arXiv subjects

María Cristina Pereyra

Publications and source records attributed to María Cristina Pereyra.

6 recordsLinked to original sources

arXiv2026

Further evidence towards the Fourier Entropy-Influence conjecture

The Fourier Entropy-Influence (FEI) conjecture states that the Fourier entropy of Boolean functions is uniformly bounded by their total influence. It has been verified for canonical examples such as disjoint tribes and for some classes of Boolean functions such as symmetric functions and read-$k$ decision trees (with a constant that depends linearly on $k$). In this note we present new classes of Boolean functions that verify the FEI conjecture. The key element is an inequality controlling the difference between the entropy of a function $f$ and the average of the entropies of $f^{\pm}$, the sub-functions obtained by setting $x_m=\pm1$ for some $m$, by the $m$-influence of $f$. If this key inequality were to hold for Boolean functions, then the full FEI conjecture would follow by induction. We introduce the notion of a stopping binary tree and observe that functions that satisfy the key inequality at the branching nodes of the tree and the FEI conjecture at the stopping nodes will satisfy the FEI conjecture. We identify some classes of functions that fit this framework: the $δ$-tribes functions, the monotone Boolean functions with the tribe separation property, and the Boolean functions with the semi-separation property, and, and, along the way, demonstrate some results that we hope the experts in this fascinating field might find useful.

math.CO↗

arXiv2025

Las funciones booleans y el lema de Bonami

In this expository article, we study the relation between the boolean functions and the hypercontractivity theorems of Aline Bonami. We focus on the social choice theory, and present some of the most important results in the area, such as the Friedgut-Kalai-Naor (FKN) and the Kahn-Kalai-Linial (KKL) theorems, and the famous Fourier Entropy/Influence conjecture. -- En este artículo expositivo estudiamos la relación entre las funciones booleanas y los teoremas de hipercontractividad de Aline Bonami. Nos concentramos en la teoría de la elección social, y presentamos algunos de los resultados más importantes en el área como los teoremas de Friedgut-Kalai-Naor (FKN) y de Kahn-Kalai-Linial (KKL), y la famosa conjetura Entropíıa de Fourier/Influencia.

math.CA↗

arXiv2025

Weighted Inequalities for $t$-Haar multipliers

In this paper, we provide necessary and sufficient conditions on a triple of weights $(u,v,w)$ so that the $t$-Haar multipliers $T^t_{w,σ}$, $t\in \R$, %defined in \cite{P} when $σ=1$, are uniformly (on the choice of signs $σ$) bounded from $L^2(u)$ into $L^2(v)$. These dyadic operators have symbols $s(x,I)=σ_I\,(w(x)/\langle w\rangle_I)^t$ which are functions of the space variable $x\in\R$ and the frequency variable $I\in \mathcal{D}$, making them dyadic analogues of pseudo-differential operators. Here $\mathcal{D}$ denotes the dyadic intervals, $σ_I=\pm1$, and $\langle w\rangle_I$ denotes the integral average of $w$ on $I$. When $w\equiv 1$ we have the martingale transform and our conditions recover the known two-weight necessary and sufficient conditions of Nazarov, Treil and Volberg. %We will discuss some relations between the three weights inequality for these operators given the inequality for other dyadic operators. We also show how these conditions are simplified when $u=v$. In particular, the martingale one-weight and the $t$-Haar multiplier unsigned and unweighted (corresponding to $σ_I\equiv 1$ and $u=v\equiv 1$) known results are recovered or improved. We also obtain necessary and sufficient testing conditions of Sawyer type for the two-weight boundedness of a single variable Haar multiplier similar to those known for the martingale transform.

math.CA↗

arXiv2025

Two-weight estimates for the square function and $t$-Haar multipliers

We present necessary and sufficient conditions on triples of weights $(u,v,w)$ for the boundedness of the dyadic weighted square function $S_w$ from $L^2(u)$ into $L^2(v)$. We use this characterization to obtain necessary and sufficient conditions for the boundedness of the $t$-Haar multipliers from $L^2(u)$ into $L^2(v)$ in terms of boundedness of the dyadic weighted square function.

math.CA↗

arXiv2018

Dyadic harmonic analysis and weighted inequalities: the sparse revolution

We will introduce the basics of dyadic harmonic analysis and how it can be used to obtain weighted estimates for classical Calderón-Zygmund singular integral operators and their commutators. Harmonic analysts have used dyadic models for many years as a first step towards the understanding of more complex continuous operators. In 2000 Stefanie Petermichl discovered a representation formula for the venerable Hilbert transform as an average (over grids) of dyadic shift operators, allowing her to reduce arguments to finding estimates for these simpler dyadic models. For the next decade the technique used to get sharp weighted inequalities was the Bellman function method introduced by Nazarov, Treil, and Volberg, paired with sharp extrapolation by Dragičević et al. Other methods where introduced by Hytönen, Lerner, Cruz-Uribe, Martell, Pérez, Lacey, Reguera, Sawyer, Uriarte-Tuero, involving stopping time and median oscillation arguments, precursors of the very successful domination by positive sparse operators methodology. The culmination of this work was Tuomas Hytönen's 2012 proof of the $A_2$ conjecture based on a representation formula for any Calderón-Zygmund operator as an average of appropriate dyadic operators. Since then domination by sparse dyadic operators has taken central stage and has found applications well beyond Hytönen's $A_p$ theorem. We will survey this remarkable progression and more in these lecture notes.

math.CA↗

arXiv2013

Weighted estimates for dyadic paraproducts and t-Haar multipiers with complexity (m,n)

We extend the definitions of dyadic paraproduct and t-Haar multipliers to dyadic operators that depend on the complexity (m,n), for m and n positive integers. We will use the ideas developed by Nazarov and Volberg to prove that the weighted L^2(w)-norm of a paraproduct with complexity (m,n) associated to a function b\in BMO, depends linearly on the A_2-characteristic of the weight w, linearly on the BMO-norm of b, and polynomially in the complexity. This argument provides a new proof of the linear bound for the dyadic paraproduct (the one with complexity (0,0)). Also we prove that the L^2-norm of a t-Haar multiplier for any t and weight w depends on the square root of the C_{2t}-characteristic of w times the square root of the A_2-characteristic of w^{2t} and polynomially in the complexity (m,n), recovering a result of Beznosova for the (0,0)-complexity case.

math.CA↗