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J. M. Aldaz

Publications and source records attributed to J. M. Aldaz.

At least 37 records · Page 2Linked to original sources
arXiv2016

Local comparability of measures, averaging and maximal averaging operators

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 circumstances it is equivalent to the latter condition, and when it is more general. We also study the concrete case of the standard gaussian measure, where this property fails, but nevertheles averaging operators are uniformly bounded, with respect to the radius, in $L^1$. However, such bounds grow exponentially fast with the dimension, very much unlike the case of Lebesgue measure.

math.CA↗
arXiv2013

On high dimensional maximal operators

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 uniform in the dimension.

math.CA↗
arXiv2013

Strengthened Cauchy-Schwarz and Hölder inequalities

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.

math.NA↗
arXiv2012

A monotonicity property of variances

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.

math.PR↗
arXiv2010

Comparison of differences between arithmetic and geometric means

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

math.CA↗
arXiv2010

Optimal bounds on the modulus of continuity of the uncentered Hardy-Littlewood maximal function

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 Hölder functions on proper subintervals of $\mathbb{R}$ are $\operatorname{Lip}_α( Mf) \le (1 + α)^{-1}\operatorname{Lip}_α( f)$, $α\in (0,1]$. On $\mathbb{R}$, the best bound for Lipschitz functions is $ \operatorname{Lip} ( Mf) \le (\sqrt2 -1)\operatorname{Lip}( f).$ In higher dimensions, we determine the asymptotic behavior, as $d\to\infty$, of the norm of the maximal operator associated to cross-polytopes, euclidean balls and cubes, that is, $\ell_p$ balls for $p = 1, 2, \infty$. We do this for arbitrary moduli of continuity. In the specific case of Lipschitz and Hölder functions, the operator norm of the maximal operator is uniformly bounded by $2^{-α/q}$, where $q$ is the conjugate exponent of $p=1,2$, and as $d\to\infty$ the norms approach this bound. When $p=\infty$, best constants are the same as when $p = 1$.

math.CA↗
arXiv2010

The weak type $(1,1)$ bounds for the maximal function associated to cubes grow to infinity with the dimension

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. Stein and J. O. Strömberg.

math.CA↗
arXiv2010

Behavior of weak type bounds for high dimensional maximal operators defined by certain radial measures

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 the previously known bounds for p=1. Roughly speaking, whenever p\in (1, 1.03], if μis defined by a radial, radially decreasing density satisfying some mild growth conditions, then the best constants c_{p,d,μ} in the weak type (p,p) inequalities satisfy c_{p,d,μ} \ge 1.005^d for all d sufficiently large. We also show that exponential increase of the best constants occurs for certain families of doubling measures, and for arbitrarily high values of p. [A1] Aldaz, J.M. Dimension dependency of the weak type (1,1) bounds for maximal functions associated to finite radial measures. Bull. Lond. Math. Soc. 39 (2007) 203-208. Also available at the Mathematics ArXiv.

math.CA↗
arXiv2010

Optimality of generalized Bernstein operators

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.

math.CA↗
arXiv2009

Regularity of the Hardy-Littlewood maximal operator on block decreasing functions

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 operator defined by the l_infty-norm, that is, by averaging over cubes, the result extends to block decreasing functions of bounded variation, not necessarily special.

math.CA↗
arXiv2009

Shape preserving properties of generalized Bernstein operators on Extended Chebyshev spaces

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: suppose there exists a Bernstein operator $B_{n}:C[a,b]\to U_{n}$ with strictly increasing nodes, fixing $f_{0}, f_{1}\in U_{n}$. If $U_{n}\subset U_{n + 1}$ and $U_{n + 1}$ has a non-negative Bernstein basis, then there exists a Bernstein operator $B_{n+1}:C[a,b]\to U_{n+1}$ with strictly increasing nodes, fixing $f_{0}$ and $f_{1}.$ In particular, if $% f_{0},f_{1},...,f_{n}$ is a basis of $U_{n}$ such that the linear span of $% f_{0},..,f_{k}$ is an extended Chebyshev space over $[ a,b] $ for each $k=0,...,n$, then there exists a Bernstein operator $B_{n}$ with increasing nodes fixing $f_{0}$ and $f_{1}.$ The second main result says that under the above assumptions the following inequalities hold B_{n}f\geq B_{n+1}f\geq f for all $(f_{0},f_{1})$-convex functions $f\in C[ a,b] .$ Furthermore, $B_{n}f$ is $(f_{0},f_{1})$-convex for all $(f_{0},f_{1})$% -convex functions $f\in C[ a,b] .$ In the specific case of exponential polynomials we give alternative proofs of shape preserving properties by computing derivatives of the generalized Bernstein polynomials.

math.CA↗