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José Madrid

Publications and source records attributed to José Madrid.

At least 19 recordsLinked to original sources
arXiv2026

Translational tilings by axis-parallel polygonal sets

The periodic tiling conjecture asserts that if a bounded set $Ω\subset \mathbb R^d$ of finite measure tiles $\mathbb R^d$ by translations, then it admits at least one fully periodic tiling. This conjecture was proved to hold in $\mathbb R$ by Lagarias and Wang, and was recently disproved by Greenfeld and Tao in sufficiently high dimensions. Its discrete analogue in $\mathbb Z^2$ was proved by Bhattacharya, whereas the conjecture in $\mathbb R^2$ remains open. In this paper, we analyze the structure of tilings by axis-parallel polygonal sets: bounded open sets in $\mathbb R^2$ whose boundary is a finite union of line segments parallel to the axes. We prove that every such tiling that is topologically minimal is weakly periodic. Moreover, we show that if an axis-parallel polygonal set $Ω$ tiles $\mathbb{R}^2$ then it admits a quasi-periodic tiling (i.e., a finite union of translated lattices). The proofs combine geometric decompositions of tilings with rational approximation and discrete tiling theory. As a consequence, we obtain that the translational tiling problem for axis-parallel polygonal sets with real algebraic vertex coordinates is decidable.

math.DS↗
arXiv2026

On $L^p$ norms of maximal operators

We study the exact $L^p$ norms for a number of maximal operators, which are prominent in harmonic analysis, semigroup theory, and probability. We prove that a strongly continuous positive symmetric sub-Markovian semigroup $(T_t)_{t\ge0}$ on a nonatomic $σ$-finite measure space $(X,\mathcal F,μ)$ has maximal $L^p$ norm $p'=p/(p-1)$ for every $1 0$ and some measurable set $E$ of positive measure. The lower bound follows by approximating finite dyadic conditional expectations at suitable semigroup times; the upper bound is Stein's maximal inequality in its sub-Markovian form. Important examples of such maximal operators we discuss include maximal functions of various semigroups (heat, Poisson, Ornstein--Uhlenbeck, Schrödinger, Dunkl, Laguerre, Jacobi) as well as the centered Hardy-Littlewood maximal operator on $\mathbb{R}.$ In particular, the heat and Poisson maximal operators on every complete connected Riemannian manifold of positive dimension without boundary have exact $L^p$ norm $p'$, with no curvature or stochastic completeness assumptions. Combining our approach with previous work of the first and last authors we also prove that $p'$ is the limit, as $d\to \infty,$ of the $L^p$ norms $B(p,d)$ of the centered Hardy-Littlewood maximal operators on $\mathbb{R}^d.$ On the other hand, we show that the norm of this maximal operator is strictly larger than $p',$ in dimensions $d\ge2.$

math.CA↗
arXiv2026

Sharp endpoint extension inequalities for the moment curve on finite fields II: an extremal property of the uniform distribution

We identify the optimal constant and describe all maximizers for the Fourier endpoint extension inequality associated with the moment curve over finite fields. This confirms a conjecture proposed in the authors' earlier work. The proof proceeds by showing that the problem is equivalent to a purely probabilistic extremal statement: among all probability distributions on a finite set, the uniform distribution uniquely maximizes the expected number of distinct rearrangements of an i.i.d. sample.

math.CA↗
arXiv2026

The norm of the centered Hardy--Littlewood maximal operator

We determine the exact $L^p(\mathbb{R})$ norm of the centered Hardy--Littlewood maximal operator for every $1<p<\infty$, proving that it equals $p/(p-1)$. This settles the sharp strong-type problem for the centered maximal operator on the real line. For every $p$ in this range, the norm is strictly larger than the constant associated with the critical power $|x|^{-1/p}$, thereby disproving the conjecture of Dror, Ganguli, and Strichartz that this profile determines the unrestricted norm.

math.CA↗
arXiv2026

Convex order, log-convexity and moments of averages of random variables

We establish a convex order comparison for products of averages of exchangeable nonnegative random variables. As a consequence, we solve a conjecture of Lamkin and Tkocz (2022) on the log-convexity of moments of sample means, as well as its pointwise version, in a more general form involving arbitrary integer partitions.

math.PR↗
arXiv2026

On the frequency function of Hardy-Littlewood maximal functions

We study the frequency function (introduced by Temur) in both the discrete and continuous settings. More precisely, we extend the definition of the frequency function to the higher-dimensional continuous setting and to the uncentered Hardy-Littlewood maximal function. We analyze the asymptotic behavior of the frequency function and the density of its small values for functions in $\ell^1(\mathbb{Z)}$ and $L^1(\mathbb{R}^d)$ answering some questions posed by Temur. Finally, we study the size of the frequency function for functions in $\ell^p(\mathbb{Z})$ with $p>1$, showing that this case differs significantly from the case $p=1$.

math.CA↗
arXiv2026

Sharp higher order regularity of discrete maximal functions

We derive sharp $\ell^p(\mathbb{Z})$ bounds for the $k$th derivative of the discrete uncentered maximal operator applied to characteristic functions $f:\mathbb{Z}\to\{0,1\}$ in the cases $k=0,1,2$. When $k=1,2$ these are the first sharp bounds for derivatives of a Hardy-Littlewood maximal function in continuous or discrete settings when $1<p<\infty$. We also establish several lower bounds for $k\geq 3$ and for general functions $f:\mathbb{Z}\to\mathbb{R}$.

math.CA↗
arXiv2026

The smoothest average and some extremal problems for polynomials

We study the problem of finding the "smoothest'' local average of a function $f \in \ell^2(\mathbb{Z})$ when we consider its convolution with suitable kernels $u$. The measurement of smoothness is as follows: Given a positive integer $k$, we aim to minimize the constant \begin{equation*} \sup_{0 \neq f \in \ell^2(\mathbb{Z})} \frac{\|\nabla^{k}(u\ast f)\|_{\ell^2(\mathbb{Z})}}{\|f\|_{\ell^2(\mathbb{Z})}} \end{equation*} among all symmetric kernels $u : \{-n,\dots,n\} \to \mathbb{R}$ with normalization $\sum_{j=-n}^{n}u(j) = 1$. We are also interested in finding the kernel for which the least constant is attained. For $k=1$ and $k=2$, the sharp constants and optimal kernels were obtained by Kravitz-Steinerberger, and Richardson. In this paper, we provide alternative proofs for $k\in \{1,2\}$ by using complex analysis tools. Moreover, we establish the case $k=3$, and also the cases $k\in \{4,6\}$ when the kernels are restricted to have non-negative Fourier transform. These are the first results in the literature for $k>2$. Finally, we deduce a general relation between the sharp constants and optimal kernels corresponding to $\nabla^k$ and $\nabla^{2k}$.

math.CA↗
arXiv2026

Sharp variational inequalities for the Hardy-Littlewood maximal operator on finite undirected graphs

We study sharp $p$-variational inequalities for the Hardy-Littlewood maximal operator on complete graphs, answering in the affirmative a question by Feng Liu and Qingying Xue. We also use computational assistance to find sharp constants in $1$-variational inequalities for all connected graphs on at most five vertices and pose a conjecture on the corresponding sharp constants for path graphs. Finally, we construct finite graphs with arbitrarily large $p$-variational constants.

math.CA↗
arXiv2025

On suprema of convolutions on discrete cubes

We find the optimal constant $C$ such that \begin{equation*} \|f_1*f_2*\dots*f_{k}\|_{\infty}\geq C\prod_{i=1}^{k}\|f_i\|_1 \end{equation*} for functions $f_i:\{0,1\}^d\to\mathbb{R}$. As applications, we derive bounds for Sidon sets on hypercubes, and, we also obtain bounds for the continuous analogue problem.

math.CA↗
arXiv2025

Optimal Young's convolutions inequality and its reverse form on the hypercube

We establish sharp forms of Young's convolution inequality and its reverse on the discrete hypercube $\{0,1\}^d$ in the diagonal case $p=q$. As applications, we derive bounds for additive energies and sumsets. We also investigate the non-diagonal regime $p\neq q$, providing necessary conditions for the inequality to hold, along with partial results in the case $r = 2$.

math.CA↗
arXiv2024

Sharp Poincare-Wirtinger inequalities on complete graphs

Let $K_n=(V,E)$ be the complete graph with $n\geq 3$ vertices (here $V$ and $E$ denote the set of vertices and edges of $K_n$ respectively). We find the optimal value ${\bf{C}}_{n,p}$ such that the inequality $$\|f-m_f\|_p\le {\bf C}_{n,p}{\rm Var}_{p}f$$ holds for every $f:V\to \mathbb{R},$ where ${\rm Var}_p$ stands for the $p$-variation, and $m_f$ stands for the average value of $f$, for all $p\in[1,3+δ^1_n)\cup (3+δ^2_n,+\infty)$, for $δ^1_n=\frac{1}{2n^2\log(n)}+O(1/n^3)$ and $δ^2_n=\frac{2}{n}+O(1/n^2).$ Moreover, we characterize all the maximizer functions in that case. The behavior of the maximizers is different in each of the intervals $(1,2)$, $(2,3+δ^{1}_n)$ and $(3+δ^{2}_n,\infty).$

math.CA↗
arXiv2023

An uncountable ergodic Roth theorem and applications

We establish an uncountable amenable ergodic Roth theorem, in which the acting group is not assumed to be countable and the space need not be separable. This generalizes a previous result of Bergelson, McCutcheon and Zhang, and complements a result of Zorin-Kranich. We establish the following two additional results: First, a combinatorial application about triangular patterns in certain subsets of the Cartesian square of arbitrary amenable groups, extending a result of Bergelson, McCutcheon and Zhang for countable amenable groups. Second, a new uniformity aspect in the double recurrence theorem for $Γ$-systems for arbitrary uniformly amenable groups $Γ$. Our uncountable Roth theorem is crucial in the proof of both of these results.

math.DS↗
arXiv2023

Additive energies on discrete cubes

We prove that for $d\geq 0$ and $k\geq 2$, for any subset $A$ of a discrete cube $\{0,1\}^d$, the $k-$higher energy of $A$ (the number of $2k-$tuples $(a_1,a_2,\dots,a_{2k})$ in $A^{2k}$ with $a_1-a_2=a_3-a_4=\dots=a_{2k-1}-a_{2k}$) is at most $|A|^{\log_{2}(2^k+2)}$, and $\log_{2}(2^k+2)$ is the best possible exponent. We also show that if $d\geq 0$ and $2\leq k\leq 10$, for any subset $A$ of a discrete cube $\{0,1\}^d$, the $k-$additive energy of $A$ (the number of $2k-$tuples $(a_1,a_2,\dots,a_{2k})$ in $A^{2k}$ with $a_1+a_2+\dots+a_k=a_{k+1}+a_{k+2}+\dots+a_{2k}$) is at most $|A|^{\log_2{ \binom{2k}{k}}}$, and $\log_2{ \binom{2k}{k}}$ is the best possible exponent. We discuss the analogous problems for the sets $\{0,1,\dots,n\}^d$ for $n\geq 2$.

math.CO↗
arXiv2023

On sharp isoperimetric inequalities on the hypercube

We prove the sharp isoperimetric inequality $$ \mathbb{E} \,h_{A}^{\log_{2}(3/2)} \geq μ(A)^{*} (\log_{2}(1/μ(A)^{*}))^{\log_{2}(3/2)} $$ for all sets $A \subseteq \{0,1\}^n$, where $μ$ denotes the uniform probability measure, $μ(A)^{*}=\min\{μ(A), 1-μ(A)\}$, $h_A$ is supported on $A$ and to each vertex $x$ assigns the number of neighbour vertices in the complement of $A$. The inequality becomes equality for any subcube. Moreover, we provide lower bounds on $\mathbb{E} h_{A}^β$ in terms of $μ(A)$ for all $β\in [1/2,1]$, improving, and in some cases tightening, previously known results. In particular, we obtain the sharp inequality $\mathbb{E}h_{A}^{0.53}\geq 2 μ(A)(1-μ(A))$ for all sets with $μ(A)\geq 1/2$, which allows us to refine a recent result of Kahn and Park on isoperimetric inequalities about partitioning the hypercube. Furthermore, we derive Talagrand's isoperimetric inequalities for functions with values in a Banach space having finite cotype: for all $f :\{-1,1\}^{n} \to X$, $\|f\|_{\infty}\leq 1$, and any $p \in [1,2]$ we have $$ \|Df\|_{p} \gtrsim \frac{1}{q^{3/2}C_{q}(X)} \|f\|_{2}^{2/p}\left(\log \frac{e\|f\|_{2}}{\|f\|_{1}}\right)^{1/q}, $$ where $\| Df\|_{p}^{p} = \mathbb{E} \| \sum_{1\leq j \leq n} x'_{j} D_{j} f(x)\|^{p}$, $x'$ is independent copy of $x$, and $C_{q}(X)$ is the cotype $q$ constant of $X$. Different proofs of the recently resolved Talagrand's conjecture will be presented.

math.CA↗
arXiv2023

A new proof of the description of the convex hull of space curves with totally positive torsion

We give new proofs of the description convex hulls of space curves $γ: [a,b] \mapsto \mathbb{R}^{d}$ having totally positive torsion. These are curves such that all the leading principal minors of $d\times d$ matrix $(γ', γ'', \ldots, γ^{(d)})$ are positive. In particular, we recover parametric representation of the boundary of the convex hull, different formulas for its surface area and the volume of the convex hull, and the solution to a general moment problem corresponding to $γ$.

math.PR↗
arXiv2022

Comparison of Ising Models Under Change of Apriori Measure

We study comparison of correlation functions for ferromagnetic generalized Ising models with two different apriori measures. One purpose of this note is to publicize some unpublished 45 year old work of Daniel Wells on the issue. We then prove results for the apriori measures associated to one component of D-vectors uniformly distributed on the unit sphere and also the case of spin S (2S +1 equally spaced values symmetric about 0 and with equal weights) that improves some 50 year old bounds of Griffiths on transition temperatures.

math-ph↗