scholarly journals Nilsystems and ergodic averages along primes

2019 ◽  
Vol 40 (10) ◽  
pp. 2769-2777
Author(s):  
TANJA EISNER
Keyword(s):  
Continuous Functions ◽  
Ergodic Averages ◽  
Von Mangoldt Function ◽  
Almost Everywhere ◽  
Correlation Result

A celebrated result by Bourgain and Wierdl states that ergodic averages along primes converge almost everywhere for $L^{p}$-functions, $p>1$, with a polynomial version by Wierdl and Nair. Using an anti-correlation result for the von Mangoldt function due to Green and Tao, we observe everywhere convergence of such averages for nilsystems and continuous functions.

scholarly journals Restriction of Families of Conversion Operators in Lp Spaces

2021 ◽  
Vol 27 (3) ◽  
pp. 449-460
Author(s):  
A. D. Nakhman
Keyword(s):  
Fourier Coefficients ◽  
Continuous Functions ◽  
Parameter Family ◽  
Maximal Operators ◽  
Limiting Behavior ◽  
Lp Spaces ◽  
Almost Everywhere ◽  
Spaces Of Continuous Functions

We study a one-parameter family of convolutional operators acting in Lebesgue Lp spaces. The case of integral kernels given by the Fourier coefficients is considered. It is established that the condition of the coefficients being quasiconvex ensures the boundedness of the corresponding maximal operators. The limiting behavior of families in the metrics of spaces of continuous functions and Lp, p ≥ 1, classes is studied, and their convergence is obtained almost everywhere. The ways of possible generalizations and distributions are indicated.

Weak-type inequalities and convergence of the ergodic averages in variable Lebesgue spaces with weights

2009 ◽  
Vol 139 (4) ◽  
pp. 673-683 ◽  
Author(s):  
M. Isabel Aguilar Cañestro ◽  
Pedro Ortega Salvador
Keyword(s):  
Lebesgue Space ◽  
Weak Type ◽  
Suitable Condition ◽  
Variable Exponents ◽  
Ergodic Averages ◽  
Lebesgue Spaces ◽  
Variable Lebesgue Spaces ◽  
Almost Everywhere ◽  
Variable Lebesgue Space ◽  
Measure Preserving Transformation

We characterize the weighted weak-type inequalities with variable exponents for the maximal operator associated with an ergodic, invertible, measure-preserving transformation and prove the almost everywhere convergence of the ergodic averages for all functions in a variable Lebesgue space with a weight verifying a suitable condition.

scholarly journals On the almost everywhere convergence of the ergodic averages

1990 ◽  
Vol 10 (1) ◽  
pp. 141-149
Author(s):  
F. J. Martín-Reyes ◽  
A. De La Torre
Keyword(s):  
Measure Space ◽  
Uniform Boundedness ◽  
Finite Measure ◽  
Necessary And Sufficient Condition ◽  
Ergodic Averages ◽  
Measurable Transformation ◽  
Almost Everywhere ◽  
Finite Measure Space ◽  
Finite Invariant Measure ◽  
Necessary And Sufficient

AbstractLet (X, ν) be a finite measure space and let T: X → X be a measurable transformation. In this paper we prove that the averages converge a.e. for every f in Lp(dν), 1 < p < ∞, if and only if there exists a measure γ equivalent to ν such that the averages apply uniformly Lp(dν) into weak-Lp(dγ). As a corollary, we get that uniform boundedness of the averages in Lp(dν) implies a.e. convergence of the averages (a result recently obtained by Assani). In order to do this, we first study measures v equivalent to a finite invariant measure μ, and we prove that supn≥0An(dν/dμ)−1/(p−1) a.e. is a necessary and sufficient condition for the averages to converge a.e. for every f in Lp(dν).

Ergodic Averages for Weight Functions Moved by Non-Linear Transformations on ℝn

1995 ◽  
Vol 47 (4) ◽  
pp. 852-876
Author(s):  
David I. McIntosh
Keyword(s):  
Sufficient Conditions ◽  
Convergent Sequence ◽  
Finite Measure ◽  
Weight Functions ◽  
Ergodic Averages ◽  
Linear Transformations ◽  
Borel Sets ◽  
Almost Everywhere ◽  
Image Position ◽  
Bounded Sets

AbstractLet ℝ+ denote the non-negative half of the real line, and let λ denote Lebesgue measure on the Borel sets of ℝn. A function φ: ℝn → ℝ+ is called a weight function if ʃℝn φ dλ = 1. Let (X, ℱ, μ) be a non-atomic, finite measure space, let ƒ: X → ℝ+, and suppose { Tν}ν∊ℝn is an ergodic, aperiodic ℝn-flow on X. We consider the weighted ergodic averages where is a sequence of weight functions. Sufficient as well as necessary and sufficient conditions for the pointwise, almost-everywhere convergence of are developed for a particular class of weight functions φk. Specifically, let {τk: ℝn → ℝn} be a sequence of measurable, non-singular maps with measurable, non-singular inverses such that the Radon-Nikodym derivatives dλ oτk /dλ and dλ oτk-1 / dλ are L∞ (ℝn), and such that τk and τ-1 map bounded sets to bounded sets. We examine convergence for the sequence where θk is an a.e.-convergent sequence of weight functions which are dominated by a fixed L1(ℝn) function with bounded support.

Convergence and divergence of ergodic averages

1994 ◽  
Vol 14 (3) ◽  
pp. 515-535 ◽  
Author(s):  
Roger L. Jones ◽  
Mate Wierdl
Keyword(s):  
Ergodic Averages ◽  
Almost Everywhere Convergence ◽  
Large Classes ◽  
Almost Everywhere ◽  
Cesaro Averages ◽  
Convergence And Divergence ◽  
General Method

AbstractIn this paper we consider almost everywhere convergence and divergence properties of various ergodic averages. A general method is given which can be used to construct averages for which a.e. convergence fails, and to show divergence (and in some cases ‘strong sweeping out’) for large classes of ergodic averages. We also show that there are sequences with the gaps between successive terms converging to zero, but such that the Cesaro averages obtained by sampling a flow along these sequences of times converge a.e. for all f∈L1(X).

scholarly journals On Fréchet theorem in the set of measure preserving functions over the unit interval

1990 ◽  
Vol 13 (2) ◽  
pp. 373-378 ◽  
Author(s):  
So-Hsiang Chou ◽  
Truc T. Nguyen
Keyword(s):  
Piecewise Linear ◽  
Continuous Functions ◽  
Unit Interval ◽  
Almost Everywhere

In this paper, we study the Fréchet theorem in the set of measure preserving functions over the unit interval and show that any measure preserving function on[0,1]can be approximated by a sequence of measure preserving piecewise linear continuous functions almost everywhere. Some application is included.

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