J. M. Aldaz

Universidad Autónoma de Madrid | UAM · Department of Mathematics

We continue the study initiated by H. S. Shapiro on Fischer decompositions of entire functions, showing that such decomposition exist in a weak sense (we do not prove uniqueness) under hypotheses regarding the order of the entire function f to be expressed as f=P·q+r\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage...

We explore some variants of the Boman covering lemma, and their relationship to the boundedness properties of the maximal operator. Let $1 < p < \infty$ and let $q$ be its conjugate exponent. We prove that the strong type $(q,q)$ of the uncentered maximal operator, by itself, implies certain generalizations of the Boman covering lemma for the expon...

We prove that in a metric measure space $X$, if for some $p \in (1,\infty)$ there are uniform bounds (independent of the measure) for the weak type $(p,p)$ of the centered maximal operator, then $X$ satisfies a certain geometric condition, the Besicovitch intersection property, which in turn implies the uniform weak type $(1,1)$ of the centered ope...

We study generalizations of the classical Bernstein operators on the polynomial spaces \(\mathbb {P}_{n}[a,b]\), where instead of fixing \(\mathbf {1}\) and x, we reproduce exactly \(\mathbf {1}\) and a polynomial \(f_1\), strictly increasing on [a, b]. We prove that for sufficiently large n, there always exist generalized Bernstein operators fixin...

We explore the consequences for the boundedness properties of averaging and maximal averaging operators, of the following local comparabiliity condition for measures: Intersecting balls of the same radius have comparable sizes. Since in geometrically doubling spaces this property yields the same results as doubling, we study under which circumstanc...

In \cite{NaTa} Naor and Tao extended to the metric setting the $O(d \log d)$ bounds given by Stein and Str\"omberg for Lebesgue measure in $\mathbb{R}^d$, deriving these bounds first from a localization result, and second, from a random Vitali lemma. Here we show that the Stein-Str\"omberg original argument can also be adapted to the metric setting...

The Cauchy-Schwarz (C-S) inequality is one of the most famous inequalities in mathematics. In this survey article, we rst give a brief history of the inequality. Afterward, we present the C-S inequality for inner product spaces. Focusing on operator inequalities, we then review some signicant recent developments of the C-S inequality and its revers...

In this note we describe some recent advances in the area of maximal function inequalities. We also study the behaviour of the centered Hardy-Littlewood maximal operator associated to certain families of doubling, radial decreasing measures, and acting on radial functions. In fact, we precisely determine when the weak type $(1,1)$ bounds are unifor...

We present some identities related to the Cauchy-Schwarz inequality in complex inner product spaces. A new proof of the basic result on the subject of Strengthened Cauchy-Schwarz inequalities is derived using these identities. Also, an analogous version of this result is given for Strengthened H\"older inequalities.

We prove that variances of non-negative random variables have the following monotonicity property: For all $0 < r < s \le 1$, and all $0 \le X \in L^2$, we have $\operatorname{Var}(X^r)^{1/r} \le \operatorname{Var}(X^s)^{1/s}$. We also discuss the real valued case.

We present sharp bounds for $\sum_{i=1}^n \alpha_i x_i - \prod_{i=1}^n x_i^{\alpha_i}$ in terms of the variance of the vector $(x_1^{1/2},...,x_n^{1/2})$.

We complement a recent result of S. Furuichi, by showing that the differences $\sum_{i=1}^n \alpha_i x_i - \prod_{i=1}^n x_i^{\alpha_i}$ associated to distinct sequences of weights are comparable, with constants that depend on the smallest and largest quotients of the weights.

We extend Dragomir's refinement of Jensen's inequality from the dicrete to the general case, identifying the equality conditions.

We obtain sharp bounds for the modulus of continuity of the uncentered maximal function in terms of the modulus of continuity of the given function, via integral formulas. Some of the results deduced from these formulas are the following: The best constants for Lipschitz and H\"older functions on proper subintervals of $\mathbb{R}$ are $\operatorna...

Let Un ⊂ Cn[a, b] be an extended Chebyshev space of dimension n + 1. Suppose that f0 ∈ Un is strictly positive and f1 ∈ Un has the property that f1/f0 is strictly increasing. We search for conditions ensuring the existence of points t0, …, tn ∈ [a, b] and positive coefficients α0, …, αn such that for all f ∈ C[a, b], the operator Bn:C[a, b] → Un de...

We show that a certain optimality property of the classical Bernstein operator also holds, when suitably reinterpreted, for generalized Bernstein operators on extended Chebyshev systems.

We complement a recent result of S. Furuichi, by showing that the differences $\sum_{i=1}^n \alpha_i x_i - \prod_{i=1}^n x_i^{\alpha_i}$ associated to distinct sequences of weights are comparable, with constants that depend on the smallest and largest quotients of the weights. Comment: Improvements on the bounds after becoming aware of the paper by...

As shown in [A1], the lowest constants appearing in the weak type (1,1) inequalities satisfied by the centered Hardy-Littlewood maximal operator associated to certain finite radial measures, grow exponentially fast with the dimension. Here we extend this result to a wider class of radial measures and to some values of p>1. Furthermore, we improve t...

We study the Hardy-Littlewood maximal operator defined via an unconditional norm, acting on block decreasing functions. We show that the uncentered maximal operator maps block decreasing functions of special bounded variation to functions with integrable distributional derivatives, thus improving their regularity. In the special case of the maximal...

We show that the lowest constant appearing in the weak type (1,1) inequality satisfied by the centered Hardy-Littlewood maximal operator on radial integrable functions is 1.

We present some identities related to the Cauchy-Schwarz inequality in complex inner product spaces. A new proof of the basic result on the subject of strengthened Cauchy-Schwarz inequalities is derived using these identities. Also, an analogous version of this result is given for strengthened Hölder inequalities.

In this note we present a refinement of the AM-GM inequality, and then we estimate in a special case the typical size of the improvement. Comment: To appear, Journal of Mathematical Inequalities

We explore the concentration properties of the ratio between the geometric mean and the arithmetic mean, showing that for certain sequences of weights one does obtain concentration, around a value that depends on the sequence. Comment: 10 pages

We present a refinement, by selfimprovement, of the arithmetic geometric inequality. Comment: 3 pages

We present a stability version of Hölder’s inequality, incorporating an extra term that measures the deviation from equality. Applications are given. 1. Introduction. In the field of geometric inequalities, the expression Bonnesen type is used after Bonnesen classical refinement of the isoperimetric inequality (cf., for instance, [Os1], [Os2]), whe...

We study the existence and shape preserving properties of a generalized Bernstein operator $B_{n}$ fixing a strictly positive function $f_{0}$, and a second function $f_{1}$ such that $f_{1}/f_{0}$ is strictly increasing, within the framework of extended Chebyshev spaces $U_{n}$. The first main result gives an inductive criterion for existence: sup...

Let $L$ be a linear differential operator with constant coefficients of order $n$ and complex eigenvalues $\lambda_{0},...,\lambda_{n}$. Assume that the set $U_{n}$ of all solutions of the equation $Lf=0$ is closed under complex conjugation. If the length of the interval $[ a,b] $ is smaller than $\pi /M_{n}$, where $M_{n}:=\max \left\{| \text{Im}%...

Let $M_d$ be the centered Hardy-Littlewood maximal function associated to cubes in $\mathbb{R}^d$ with Lebesgue measure, and let $c_d$ denote the lowest constant appearing in the weak type (1,1) inequality satisfied by $M_d$. We show that $c_d \to \infty$ as $d\to \infty$, thus answering, for the case of cubes, a long standing open question of E. M...

A product of doubling measures on the real line can be defined in such a way that another doubling measure on the line is obtained. It follows that doubling measures on the line form a semiring.

Let LN+1 be a linear differential operator of order N+1 with constant coefficients and real eigenvalues λ1,…,λN+1, let E(ΛN+1) be the space of all C∞-solutions of LN+1 on the real line. We show that for N⩾2 and n=2,…,N, there is a recurrence relation from suitable subspaces En to En+1 involving real-analytic functions, and with EN+1=E(ΛN+1) if and...

We show that the smallest constants appearing in the weak type (1,1) inequalities satisfied by the centred Hardy–Littlewood maximal function associated to some finite radial measures, such as the standard gaussian measure, grow exponentially fast with the dimension.

We characterize the space BV(I) of functions of bounded variation on an arbitrary interval I⊂R, in terms of a uniform boundedness condition satisfied by the local uncentered maximal operator MR from BV(I) into the Sobolev space W1,1(I). By restriction, the corresponding characterization holds for W1,1(I). We also show that if U is open in Rd, d>1,...

We prove that if $f:I\subset \Bbb R\to \Bbb R$ is of bounded variation, then the noncentered maximal function $Mf$ is absolutely continuous, and its derivative satisfies the sharp inequality $\|DMf\|_1\le |Df|(I)$. This allows us obtain, under less regularity, versions of classical inequalities involving derivatives. Comment: To appear, TAMS, 21 pa...

We show that the best constants appearing in the weak type (1,1) inequalities satisfied by the centered Hardy-Littlewood maximal function associated to some finite radial measures, such as the standard gaussian measure, grow exponentially fast with the dimension. Comment: 7 pages, to appear in the Bull. London Math. Soc

We show that there is a measure $\mu$, defined on the hyperbolic plane and with polynomial growth, such that the centered maximal operator associated to $\mu$ does not satisfy weak type $(1,1)$ bounds.

We show that in the study of certain convolution operators, functions can be replaced by measures without changing the size of the constants appearing in weak type (1,1) inequalities. As an application, we prove that the best constants for the centered Hardy-Littlewood maximal operator associated to parallelotopes do not decrease with the dimension...

We show that the Lebesgue differentiation theorem does not hold in Ρ with the “product” Lebesgue measure.

We show that given any Borel measure onR, every Lipschitz function is μ-a.e. differentiable with respect to μ.

We develop a new approach to the measure extension problem, based on nonstandard analysis. The class of thick topological spaces, which includes all locally compact and all K-analytic spaces, is introduced in this paper, and measure extension results of the following type are obtained: If (X, ) is a regular, Lindelöf, and thick space, ⊂σ[] is a σ-a...

We study the behaviour of the n-dimensional centered Hardy-Littlewood maximal operator associated to the family of cubes with sides parallel to the axes, improving the previously known lower bounds for the best constants c n that appear in the weak type (1,1) inequalities.

We answer questions of A. Carbery, M. Trinidad Menarguez and F. Soria by proving, firstly, that for the centred Hardy-Littlewood maximal function on the real line, the best constant C for the weak type(1, 1)inequality is strictly larger than 3/2, and secondly, that C is strictly less than 2 (known to be the best constant in the noncentred case).

We show that in a countably metacompact space, if a Baire measure admits a Borel extension, then it admits a regular Borel extension. We also prove that under the special axiom Clubsuit sign there is a Dowker space which is quasi-Mařík but not Mařík, answering a question of H. Ohta and K. Tamano, and under P(c), that there is a Mařík Dowker space,...

We show that both the rationals (a σ-compact metric space) and the irrationals (a polish space) admit a topology, strictly finer than the euclidean at every point, such that the resulting spaces are homeomorphic to the original ones. An example with the same property is also given, where the metric space is complete and locally compact. However, if...

We present a characterization of the completed Borel measure spaces for which every measurable function, with values in a separable Frechet space, is the almost everywhere limit of a sequence of continuous functions. From this characterization one can easily obtain results that have appeared recently in the literature, in a more general form. We al...

Loeb measures have been utilized to represent Radon and r-smooth measures on topological spaces via the standard part map. The purpose of this paper is to show how to extend these results to a nontopological setting.

We show that several ''good'' properties of the standard part map on regular Hausdorff spaces do not hold for arbitrary Hausdorff spaces.

The purpose of this note is to show that neither a Loeb measure nor the image of a Loeb measure have to be compact, thus answering in the negative two questions of D. Ross.

The purpose of this note is to show that neither a Loeb measure nor the image of a Loeb measure have to be compact, thus answering in the negative two questions of D. Ross.

In this paper it is shown that the construction of measures on standard spaces via Loeb measures and the standard part map does not depend on the full structure of the internal algebra being used. A characterization of universal Loeb measurability is given for completely regular Hausdorff spaces, and the behavior of this property under various topo...

We explore, in a fairly elementary fashion, a variety of topics in the Theory of Numbers, presenting along the way some conjectures and open problems. 1. Introducci´

We use a refinement of Hölder's inequality for 1 < p < 1 to obtain the corre- sponding refinement when r 2 (0,1). This in turn allows us to sharpen the reverse triangle inequality on the nonnegative functions in Lr, for r 2 (0,1).