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José Madrid

Publications and source records attributed to José Madrid.

36 records · Page 2Linked to original sources
arXiv2021

Decoupling for fractal subsets of the parabola

We consider decoupling for a fractal subset of the parabola. We reduce studying $l^{2}L^{p}$ decoupling for a fractal subset on the parabola $\{(t, t^2) : 0 \leq t \leq 1\}$ to studying $l^{2}L^{p/3}$ decoupling for the projection of this subset to the interval $[0, 1]$. This generalizes the decoupling theorem of Bourgain-Demeter in the case of the parabola. Due to the sparsity and fractal like structure, this allows us to improve upon Bourgain-Demeter's decoupling theorem for the parabola. In the case when $p/3$ is an even integer we derive theoretical and computational tools to explicitly compute the associated decoupling constant for this projection to $[0, 1]$. Our ideas are inspired by the recent work on ellipsephic sets by Biggs using nested efficient congruencing.

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arXiv2021

On classical inequalities for autocorrelations and autoconvolutions

In this paper we study an autocorrelation inequality proposed by Barnard and Steinerberger. The study of these problems is motivated by a classical problem in additive combinatorics. We establish the existence of extremizers to this inequality, for a general class of weights, including Gaussian functions (as studied by the second author and Ramos) and characteristic function (as originally studied by Barnard and Steinerberger). Moreover, via a discretization argument and numerical analysis, we find some almost optimal approximation for the best constant allowed in this inequality. We also discuss some other related problem about autoconvolutions.

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arXiv2021

Continuity of the gradient of the fractional maximal operator on $W^{1,1}(\mathbb{R}^d)$

We establish that the map $f\mapsto |\nabla \mathcal{M}_αf|$ is continuous from $W^{1,1}(\mathbb{R}^d)$ to $L^{q}(\mathbb{R}^d)$, where $α\in (0,d)$, $q=\frac{d}{d-α}$ and $\mathcal{M}_α$ denotes either the centered or non-centered fractional Hardy--Littlewood maximal operator. In particular, we cover the cases $d >1$ and $α\in (0,1)$ in full generality, for which results were only known for radial functions.

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arXiv2020

On optimal autocorrelation inequalities on the real line

We study autocorrelation inequalities, in the spirit of Barnard and Steinerberger's work. In particular, we obtain improvements on the sharp constants in some of the inequalities previously considered by these authors, and also prove existence of extremizers to these inequalities in certain specific settings. Our methods consist of relating the inequalities in question to other classical sharp inequalities in Fourier analysis, such as the sharp Hausdorff--Young inequality, and employing functional analysis as well as measure theory tools in connection to a suitable dual version of the problem to identify and impose conditions on extremizers.

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arXiv2020

Sharp inequalities for maximal operators on finite graphs, II

Let $M_{G}$ be the centered Hardy-Littlewood maximal operator on a finite graph $G$. We find $\underset{p\to \infty}{\lim}\|M_{G}\|_{p}^{p }$ when $G$ is the start graph ($S_n$) and the complete graph ($K_n$), and we fully describe $\|M_{S_n}\|_{p}$ and the corresponding extremizers for $p\in (1,2)$. We prove that $\underset{p\to \infty}{\lim}\|M_{S_n}\|_{p}^{p }=\frac{1+\sqrt{n}}{2}$ when $n\ge 25$. Also, we compute the best constant ${\bf C}_{S_n,2}$ such that for every $f:V\to \mathbb{R}$ we have $Var_{2}M_{S_n}f\le {\bf C}_{S_n,2} Var_{2}f$. We prove that ${\bf C}_{S_n,2}=\frac{(n^2-n-1)^{1/2}}{n}$ for all $n\geq 3$ and characterize the extremizers. Moreover, when $M$ is the Hardy-Littlewood maximal operator on $\mathbb{Z}$, we compute the best constant ${\bf C}_{p}$ such that $Var_{p}Mf\le {\bf C}_{p}\|f\|_{p}$ for $p\in (\frac{1}{2},1)$ and we describe the extremizers.

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arXiv2020

Sharp Inequalities for maximal operators on finite graphs

Let $G=(V,E)$ be a finite graph and $M_G$ be the centered Hardy-Littlewood maximal operator defined there. We find the optimal value $\bf{C}_{G,p}$ such that the inequality $$\text{Var}_{p}(M_{G}f)\leq {\textbf{C}}_{G,p}\text{Var}_{p}(f)$$ holds for every $f:V\to \mathbb{R},$ where $\text{Var}_p$ stands for the $p$-variation, when: (i) $G=K_n$ (complete graph) and $p\in [\frac{\log(4)}{\log(6)},\infty)$ or $G=K_4$ and $p\in (0,\infty)$; (ii) $G=S_n$ (star graph) and $1\ge p\ge \frac{1}{2}$; $p\in (0,\frac{1}{2})$ and $n\ge C(p)$ or $G=S_3$ and $p\in (1,\infty).$ We also find the value of the norm $\|M_{G}\|_{2}$ when: (i) $G=K_n$ and $n\ge 3$; (ii) $G=S_n$ and $n\ge 3.$

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arXiv2020

Sunrise strategy for the continuity of maximal operators

In this paper we address the $W^{1,1}$-continuity of several maximal operators at the gradient level. A key idea in our global strategy is the decomposition of a maximal operator, with the absence of strict local maxima in the disconnecting set, into "lateral" maximal operators with good monotonicity and convergence properties. This construction is inspired in the classical sunrise lemma in harmonic analysis. A model case for our sunrise strategy considers the uncentered Hardy-Littlewood maximal operator $\widetilde{M}$ acting on $W^{1,1}_{\rm rad}(\mathbb{R}^d)$, the subspace of $W^{1,1}(\mathbb{R}^d)$ consisting of radial functions. In dimension $d\geq 2$ it was recently established by H. Luiro that the map $f \mapsto \nabla \widetilde{M} f$ is bounded from $W^{1,1}_{\rm rad}(\mathbb{R}^d)$ to $L^1(\mathbb{R}^d)$, and we show that such map is also continuous. Further applications of the sunrise strategy in connection with the $W^{1,1}$-continuity problem include non-tangential maximal operators on $\mathbb{R}^d$ acting on radial functions when $d\geq 2$ and general functions when $d=1$, and the uncentered Hardy-Littlewood maximal operator on the sphere $\mathbb{S}^d$ acting on polar functions when $d\geq 2$ and general functions when $d=1$.

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arXiv2020

Improving estimates for discrete polynomial averages

For a polynomial $P$ mapping the integers into the integers, define an averaging operator $A_{N} f(x):=\frac{1}{N}\sum_{k=1}^N f(x+P(k))$ acting on functions on the integers. We prove sufficient conditions for the $\ell^{p}$-improving inequality \begin{equation*} \|A_N f\|_{\ell^q(\mathbb{Z})} \lesssim_{P,p,q} N^{-d(\frac{1}{p}-\frac{1}{q})} \|f\|_{\ell^p(\mathbb{Z})}, \qquad N \in\mathbb{N}, \end{equation*} where $1\leq p \leq q \leq \infty$. For a range of quadratic polynomials, the inequalities established are sharp, up to the boundary of the allowed pairs of $(p,q)$. For degree three and higher, the inequalities are close to being sharp. In the quadratic case, we appeal to discrete fractional integrals as studied by Stein and Wainger. In the higher degree case, we appeal to the Vinogradov Mean Value Theorem, recently established by Bourgain, Demeter, and Guth.

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arXiv2019

Regularity of the centered fractional maximal function on radial functions

We study the regularity properties of the centered fractional maximal function $M_β$. More precisely, we prove that the map $f \mapsto |\nabla M_βf|$ is bounded and continuous from $W^{1,1}(\mathbb{R}^d)$ to $L^q(\mathbb{R}^d)$ in the endpoint case $q=d/(d-β)$ if $f$ is radial function. For $d=1$, the radiality assumption can be removed. This corresponds to the counterparts of known results for the non-centered fractional maximal function. The main new idea consists in relating the centered and non-centered fractional maximal function at the derivative level.

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arXiv2019

A note about the $l^p$-improving property of the average operators

In this paper we give a short proof of the $\ell^p$-improving property of the average operator along the square integers and more general quadratic polynomials. Moreover we obtain a similar result for some higher degree polynomials. We also show an elementary proof of the $\ell^p$-improving property of the average operator along primes.

math.CA↗
arXiv2019

Directional maximal function along the primes

We study a two-dimensional discrete directional maximal operator along the set of the prime numbers. We show existence of a set of vectors, which are lattice points in a sufficiently large annulus, for which the $\ell^2$ norm of the associated maximal operator with supremum taken over all large scales grows with an epsilon power in the number of vectors. This paper is a follow-up to a prior work on the discrete directional maximal operator along the integers by the first and third author.

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arXiv2019

Endpoint Sobolev continuity of the fractional maximal function in higher dimensions

We establish continuity mapping properties of the non-centered fractional maximal operator $M_β$ in the endpoint input space $W^{1,1}(\mathbb{R}^d)$ for $d \geq 2$ in the cases for which its boundedness is known. More precisely, we prove that for $q=d/(d-β)$ the map $f \mapsto |\nabla M_βf|$ is continuous from $W^{1,1}(\mathbb{R}^d)$ to $L^{q}(\mathbb{R}^d)$ for $ 0 < β< 1$ if $f$ is radial and for $1 \leq β< d$ for general $f$. The results for $1\leq β< d$ extend to the centered counterpart $M_β^c$. Moreover, if $d=1$, we show that the conjectured boundedness of that map for $M_β^c$ implies its continuity.

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arXiv2017

The Variation of the Fractional Maximal Function of a Radial Function

In this paper we study the regularity of the non-centered fractional maximal operator $M_β$. As the main result, we prove that there exists $C(n,β)$ such that if $q=n/(n-β)$ and $f$ is a radial function, then $\|DM_βf\|_{L^{q}(\mathbb{R}^n)}\leq C(n,β)\|Df\|_{L^{1}(\mathbb{R}^n)}$. The corresponding result was previously known only if $n=1$ or $β=0$. Our proofs are almost free from one-dimensional arguments. Therefore, we believe that the new approach may be very useful when trying to extend the result for all $f\in W^{1,1}(\mathbb{R}^n)$.

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arXiv2017

Endpoint Sobolev and BV Continuity for maximal operators, II

In this paper we study some questions about the continuity of classical and fractional maximal operators in the Sobolev space $W^{1,1}$, in both continuous and discrete setting, giving a positive answer to two questions posed recently, one of them regarding the continuity of the map $f \mapsto \big(\widetilde M_βf\big)'$ from $W^{1,1}(\mathbb{R})$ to $L^q(\mathbb{R})$, for $q=\frac{1}{1-β}$. Here $\widetilde M_β$ denotes the non-centered fractional maximal operator on $\mathbb{R}$ with $β\in(0,1)$. The second one regarding the continuity of the discrete centered maximal operator in the space of functions of bounded variation BV$(\mathbb{Z})$, complementing some recent boundedness results.

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arXiv2017

Endpoint Sobolev and BV continuity for maximal operators

In this paper we investigate some questions related to the continuity of maximal operators in $W^{1,1}$ and $BV$ spaces, complementing some well-known boundedness results. Letting $\widetilde M$ be the one-dimensional uncentered Hardy-Littlewood maximal operator, we prove that the map $f \mapsto \big(\widetilde Mf\big)'$ is continuous from $W^{1,1}(\mathbb{R})$ to $L^1(\mathbb{R})$. In the discrete setting, we prove that $\widetilde M: BV(\mathbb{Z}) \to BV(\mathbb{Z})$ is also continuous. For the one-dimensional fractional Hardy-Littlewood maximal operator, we prove by means of counterexamples that the corresponding continuity statements do not hold, both in the continuous and discrete settings, and for the centered and uncentered versions.

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arXiv2016

Derivative bounds for fractional maximal functions

In this paper we study the regularity properties of fractional maximal operators acting on $BV$-functions. We establish new bounds for the derivative of the fractional maximal function, both in the continuous and in the discrete settings.

math.CA↗
arXiv2014

One-sided Band-limited Approximations of Some Radial Functions

We construct majorants and minorants of a Gaussian function in Euclidean space that have Fourier transforms supported in a box. The majorants that we construct are shown to be extremal and our minorants are shown to be asymptotically extremal as the sides of the box become uniformly large. We then adapt the Distribution and Gaussian Subordination methods of Carneiro-Littmann-Vaaler to the multidimensional setting to obtain majorants and minorants for a class of radial functions. Periodic analogues of the main results are proven and applications to Hilbert-type inequalities are given.

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