Jose M. Conde-Alonso

Universidad Autónoma de Madrid | UAM · Department of Mathematics

We investigate $L_p$-estimates for balanced averages of Fourier truncations in group algebras, in terms of differential operators acting on them. Our results extend a fundamental inequality of Naor for the hypercube (with profound consequences in metric geometry) to discrete groups. Different inequalities are established in terms of directional der...

In the line of previous work by Naor, we establish new forms of metric $\mathrm{X}_p$ inequalities in group algebras under very general assumptions. Our results' applicability goes beyond the previously known setting in two directions. In first place, we find continuous forms of the $\mathrm{X}_p$ inequality in the $n$-dimensional torus. Second, we...

In this paper we solve three problems in noncommutative harmonic analysis which are related to endpoint inequalities for singular integrals. In first place, we prove that an L2-form of Hörmander's kernel condition suffices for the weak type (1,1) of Calderón-Zygmund operators acting on matrix-valued functions. To that end, we introduce an improved...

We establish regularity conditions for $L_p$-boundedness of Fourier multipliers on the group von Neumann algebras of stratified Lie groups and high rank simple Lie groups, which give sharp canonical forms of the H\"ormander-Mikhlin criterion in terms of Lie derivatives of the symbol. As expected, such results for nilpotent groups require to work wi...

We establish a rather unexpected and simple criterion for the boundedness of Schur multipliers $S_M$ on Schatten $p$-classes which solves a conjecture proposed by Mikael de la Salle. Given $1 < p < \infty$, a simple form our main result reads for $\mathbf{R}^n \times \mathbf{R}^n$ matrices as follows $$\big\| S_M: S_p \to S_p \big\|_{\mathrm{cb}} \...

We present a general approach to sparse domination based on single-scale \(L^p\)-improving as a key assumption. The results are formulated in the setting of metric spaces of homogeneous type and avoid completely the use of dyadic-probabilistic techniques as well as of Christ–Hytönen–Kairema cubes. Among the applications of our general principle, we...

In this paper we solve three problems in noncommutative harmonic analysis which are related to endpoint inequalities for singular integrals. In first place, we prove that an $L_2$-form of H\"ormander's kernel condition suffices for the weak type (1,1) of Calder\'on-Zygmund operators acting on matrix-valued functions. To that end, we introduce an im...

Our first result is a noncommutative form of the Jessen-Marcinkiewicz-Zygmund theorem for the maximal limit of multiparametric martingales or ergodic means. It implies bilateral almost uniform convergence (a noncommutative analogue of almost everywhere convergence) with initial data in the expected Orlicz spaces. A key ingredient is the introductio...

We present a general approach to sparse domination based on single-scale $L^p$-improving as a key property. The results are formulated in the setting of metric spaces of homogeneous type and avoid completely the use of dyadic-probabilistic techniques as well as of Christ-Hyt\"onen-Kairema cubes. Among the applications of our general principle, we r...

Our first result is a noncommutative form of Jessen/Marcinkiewicz/Zygmund theorem for the maximal limit of multiparametric martingales or ergodic means. It implies bilateral almost uniform convergence with initial data in the expected Orlicz spaces. A key ingredient is the introduction of the $L_p$-norm of the $\limsup$ of a sequence of operators a...

Our first result is a noncommutative form of Jessen/Marcinkiewicz/Zygmund theorem for the maximal limit of multiparametric martingales or ergodic means. It implies bilateral almost uniform convergence (a noncommutative analogue of a.e. convergence) with initial data in the expected Orlicz spaces. A key ingredient is the introduction of the Lp-norm...

We study the dual space of the variable Lebesgue space $\Lp$ with unbounded exponent function $\pp$ and provide an answer to a question posed in~[fiorenza-cruzuribe2013]. Our approach is to decompose the dual into a topological direct sum of Banach spaces. The first component corresponds to the dual in the bounded exponent case, and the second is,...

Given a measure μ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu $$\end{document} of polynomial growth, we refine a deep result by David and Mattila to construct an...

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.

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

The usual one third trick allows to reduce problems involving general cubes to a countable family. Moreover, this covering lemma uses only dyadic cubes, which allows to use nice martingale properties in harmonic analysis problems. We consider alternatives to this technique in spaces equipped with nonhomogeneous measures. This entails additional dif...

Consider a totally irregular measure $\mu$ in $\mathbb{R}^{n+1}$, that is, the upper density $\limsup_{r\to0}\frac{\mu(B(x,r))}{(2r)^n}$ is positive $\mu$-a.e.\ in $\mathbb{R}^{n+1}$, and the lower density $\liminf_{r\to0}\frac{\mu(B(x,r))}{(2r)^n}$ vanishes $\mu$-a.e. in $\mathbb{R}^{n+1}$. We show that if $T_\mu f(x)=\int K(x,y)\,f(y)\,d\mu(y)$ i...

It is shown that if a Markov map $T$ on a noncommutative probability space $\mathcal{M}$ has a spectral gap on $L_2(\mathcal{M})$, then it also has one on $L_p(\mathcal{M})$ for $1<p<\infty$. For fixed $p$, the converse also holds if $T$ is factorizable. These results are also new for classical probability spaces.

Let T be a multilinear integral operator which is bounded on certain products of Lebesgue spaces on Rⁿ. We assume that its associated kernel satisfies some mild regularity condition which is weaker than the usual Hölder continuity of kernels of multilinear Calderón-Zygmund singular integral operators. In this paper, given a suitable multiple weight...

We prove that bilinear forms associated to the rough homogeneous singular integrals $T_\Omega$ on $\mathbb R^d$, where the angular part $\Omega \in L^q (S^{d-1})$ has vanishing average and $1<q\leq \infty$, and to Bochner-Riesz means at the critical index in $\mathbb R^d$ are dominated by sparse forms involving $(1,p)$ averages. This domination is...

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 Calder\'on-Zygmund operators and to answer a question posed by A. Lerner. Several applications to weighted estimates for both multilinear Calder\'on-Zygmund operators and square functions ar...

Given a probability space (Ω, Σ, μ), the Hardy space H1 (Ω) that is associated with the martingale square function does not admit a classical atomic decomposition when the underlying filtration is not regular. In this paper, we construct a decomposition of H1 (Ω) into "atomic blocks" in the spirit of Tolsa, which we will introduce for martingales....

Let $T$ be a multilinear integral operator which is bounded on certain products of Lebesgue spaces on $\mathbb R^n$. We assume that its associated kernel satisfies some mild regularity condition which is weaker than the usual H\"older continuity of those in the class of multilinear Calder\'on-Zygmund singular integral operators. In this paper, give...

We obtain a complete characterization of the weak-type (1,1) for Haar shift operators in terms of generalized Haar systems adapted to a Borel measure μ in the operator-valued setting. The main technical tool in our method is a noncommutative Calder\'on-Zygmund decomposition valid for arbitrary Borel measures.

Given a probability space $(\Omega,\Sigma,\mu)$, the Hardy space $\mathrm{H}_1(\Omega)$ which is associated to the martingale square function does not admit any atomic decomposition when the underlying filtration is not regular. In this paper we adapt Tolsa's ideas for nondoubling measures to the present context and construct a decomposition of $\m...

In this paper we introduce a class of BMO spaces which interpolate with $L_p$ and are sufficiently large to serve as endpoints for new singular integral operators. More precisely, let $(\Omega, \Sigma, \mu)$ be a $\sigma$-finite measure space. Consider two filtrations of $\Sigma$ by successive refinement of two atomic $\sigma$-algebras $\Sigma_\mat...

We construct a family of n+1 dyadic filtrations in R^n, so that every Euclidean ball B is contained in some cube Q of our family satisfying diam(Q) \le c_n diam(B) for some dimensional constant c_n. Our dyadic covering is optimal on the number of filtrations and improves previous results of Christ and Garnett/Jones by extending a construction of Me...

We present an e-learning system based on online forms that allows teachers to easily organise competitions in a classroom. This system is used in a preliminary study to evaluate whether cooperative competition is positive or not in education, and to identify which are the characteristics this kind of activity should have to be no harmful for studen...

In this paper, we present a folksonomy-based approach for implicit user intent extraction during a Web search process. We present a number of result re-ranking techniques based on this representation that can be applied to any Web search engine. We perform a user experiment the results of which indicate that this type of representation is better at...

Resumen. En este trabajo investigamos la adaptación de técnicas implícitas de extracción del contexto del usuario durante la realización de una búsqueda Web. Presentamos varios modelos de obtención del contexto que utilizan la información aportada por servicios Web de anotación de recursos (e.g. Delicious) en vez de la representación textual del do...

We analyse the effects of competition in education. We identify the benefits and drawbacks of forcing students to compete themselves during their learning process, and investigate a number of features a competitive learning activity should have in order to motivate students, and improve their academic performance. More specifically, by using a simp...