Zygmund Inequalities Research Articles

The Calderón–Zygmund inequality on a compact Riemannian manifold

For a compact closed n-dimensional manifold, we derive the Calderon-Zygmund inequality for the Hodge Laplacian, with constants depending only on bounds on the injectivity radius, volume and the curvature operator. We obtain the Poincare-Sobolev inequality for forms as a consequence.

Moment inequalities for B-valued random vectors with applications to the strong limit theorems

Analogues of the Marcinkiewicz–Zygmund, Rosenthal and Acosta inequalities for Banach space valued random vectors are proved. As an application some results on the complete convergence and the strong law of large numbers are obtained. Some well-known results are improved and extended.

Marcinkiewicz–Zygmund inequalities and the numerical approximation of singular integrals for exponential weights: methods, results and open problems, some new, some old

In this short article, we explore some methods, results and open problems dealing with weighted Marcinkiewicz–Zygmund inequalities as well as the numerical approximation of integrals for exponential weights on the real line and on finite intervals of the line. The problems posed are based primarily on ongoing work of the author and his collaborators and were presented at the November 2001 Oberwolfach workshop: ‘Numerical integration and Complexity’.

On extremal distributions and sharp $\bolds{L_p}$-bounds\\ for sums of multilinear forms

In this paper we present a study of the problem of approximating the expectations of functions of statistics in independent and dependent random variables in terms of the expectations of functions of the component random variables. We present results providing sharp analogues of the Burkholder--Rosenthal inequalities and related estimates for the expectations of functions of sums of dependent nonnegative r.v.'s and conditionally symmetric martingale differences with bounded conditional moments as well as for sums of multilinear forms. Among others, we obtain the following sharp inequalities: $E(\sum_{k=1}^n X_k)^t\le 2 \max (\sum_{k=1}^n EX_k^t, (\sum_{k=1}^n a_k)^t)$ for all nonnegative r.v.'s $X_1, \ldots, X_n$ with $E(X_k\mid X_1, \ldots, X_{k-1})\le a_k$, $EX_k^t<\infty$, $k=1, \ldots, n$, $1#x003C;t#x003C;2$; $E(\sum_{k=1}^n X_k)^t\le E\theta^t(1) \max (\sum_{k=1}^n b_k, (\sum_{k=1}^n a_k^s)^{t/s})$ for all nonnegative r.v.'s $X_1, \ldots, X_n$ with $E(X_k^s\mid X_1, \ldots, X_{k-1})\le a_k^s$, $E(X_k^t\mid X_1, \ldots, X_{k-1})\le b_k$, $k=1, \ldots, n$, $1#x003C;t#x003C;2$, $0#x003C;s\le t-1$ or $t\ge 2$, $0#x003C;s\le 1$, where $\theta(1)$ is a Poisson random variable with parameter 1. As applications, new decoupling inequalities for sums of multilinear forms are presented and sharp Khintchine--Marcinkiewicz--Zygmund inequalities for generalized moving averages are obtained. The results can also be used in the study of a wide class of nonlinear statistics connected to problems of long-range dependence and in an econometric setup, in particular, in stabilization policy problems and in the study of properties of moving average and autocorrelation processes. The results are based on the iteration of a series of key lemmas that capture the essential extremal properties of the moments of the statistics involved.

Sur une inégalité de Littlewood–Salem

We study uniform estimates for random functions of the form ∑ k=1 N X k f( n k x) where f∈A( T) , { X k } k⩾1 is a sequence of independent complex random variables such that EX k=0 and E|X k| 2<∞ , and { n k } k⩾1 is a strictly increasing sequence of positive integers. (the case of real sequence n k is also studied). Let σ N 2=∑ k=1 N E|X k| 2 . Under some condition on { X k }, we prove that ∫ Ω sup N⩾1 sup x∈ T ∑ k=1 NX kf(n kx) σ N 2 logn N d P<∞. This result generalizes to some extent Salem–Zygmund inequality. We apply this inequality to study the local regularities of random series of Rademacher type ∑ k=1 ∞ a k ε k f( n k x) and of non-stationnary gaussian type ∑ k=1 ∞ a k g k f( n k x).

On the best constant in Marcinkiewicz–Zygmund inequality

Let { X n , n⩾1} be a sequence of independent r.v.'s with EX n=0, C(p) be the best constant in the following Marcinkiewicz–Zygmund inequality: E ∑ k=1 n X k p⩽C(p)n p/2−1 ∑ k=1 n E|X k| p, p⩾2. In this paper we prove that [ C( p)] 1/ p grows like p as p→∞ and give an estimate C(p)⩽(3 2 ) pp p/2 .

Analogues of Khintchine, Marcinkiewicz–Zygmund and Rosenthal Inequalities for Symmetric Statistics

In this paper we prove the analogues of Khintchine, Marcinkiewicz–Zygmund and Rosenthal moment inequalities for symmetric statistics of second order in not identically distributed random variables.

A new weak dependence condition and applications to moment inequalities

The purpose of this paper is to propose a unifying weak dependence condition. Mixing sequences, functions of associated or Gaussian sequences, Bernoulli shifts as well as models with a Markovian representation are examples of the models considered. We establish Marcinkiewicz–Zygmund, Rosenthal and exponential inequalities for general sequences of centered random variables. Inequalities are stated in terms of the decay rate for the covariance of products of the initial random variables subject to the condition that the gap of time between both products tends to infinity. As applications of those notions, we obtain a version of the functional CLT and an invariance principle for the empirical process

On Converse Marcinkiewicz—Zygmund Inequalities in L p ,p&gt;1

We obtain converse Marcinkiewicz—Zygmund inequalities such as $$ \| P\nu \|_{L_{p}[-1,1]}\leq C\left( \sum_{j=1}^{n}\mu _{j}| P(t_{j})| ^{p}\right) ^{1/p} $$ for polynomials P of degree ≤ n-1 , under general conditions on the points {t j } n j=1 and on the function ν . The weights {μ j } n j=1 are appropriately chosen. We illustrate the results by applying them to extended Lagrange interpolation for exponential weights on [-1,1] .

Approximation of modified Lagrange interpolation in Orlicz spaces

In this paper, the authors give the Marcinkiewicz—Zygmund inequality based on the zeros of the first kind Chebyshev polynomials in Orlicz norm. As application, the degree of approximation by two kinds of modified Lagrange interpolatory polynomials in Orlicz spaces is studied.

A best constant for Zygmund’s conjugate function inequality

When the space L log + L L{\log ^ + }L is given the Hardy-Littlewood norm the best constant in the corresponding version of Zygmund’s conjugate function inequality is shown to be \[ K = 1 − 2 − 3 − 2 + 5 − 2 − 7 − 2 + ⋯ 1 − 2 + 3 − 2 + 5 − 2 + 7 − 2 + ⋯ . {\mathbf {K}} = \frac {{{1^{ - 2}} - {3^{ - 2}} + {5^{ - 2}} - {7^{ - 2}} + \cdots }}{{{1^{ - 2}} + {3^{ - 2}} + {5^{ - 2}} + {7^{ - 2}} + \cdots }}. \] This complements the recent result of Burgess Davis that the best constant in Kolmogorov’s inequality is K − 1 {{\mathbf {K}}^{ - 1}} .

Certain inequalities in various metrics for trigonometric polynomials and their derivatives

We establish for 0<p<1 the analog of the Bernstein- Zygmund inequality for the derivative of a trigonometric polynomial We prove weighted inequalities, exact in the sense of order, for trigonometric polynomials and their derivatives in various integral metrics with exponents 0<p, q≤∞.