scholarly journals Pointwise convergence of certain continuous-time double ergodic averages

2021 ◽  
pp. 1-11
Author(s):  
MICHAEL CHRIST ◽  
POLONA DURCIK ◽  
VJEKOSLAV KOVAČ ◽  
JORIS ROOS
Keyword(s):  
Recent Work ◽  
Hilbert Transform ◽  
Continuous Time ◽  
Pointwise Convergence ◽  
Probability Space ◽  
Singular Integrals ◽  
Ergodic Averages ◽  
Almost Everywhere Convergence ◽  
Almost Everywhere ◽  
Multilinear Singular Integrals

Abstract We prove almost everywhere convergence of continuous-time quadratic averages with respect to two commuting $\mathbb {R}$ -actions, coming from a single jointly measurable measure-preserving $\mathbb {R}^2$ -action on a probability space. The key ingredient of the proof comes from recent work on multilinear singular integrals; more specifically, from the study of a curved model for the triangular Hilbert transform.

scholarly journals Norm variation of ergodic averages with respect to two commuting transformations

2017 ◽  
Vol 39 (3) ◽  
pp. 658-688 ◽  
Author(s):  
POLONA DURCIK ◽  
VJEKOSLAV KOVAČ ◽  
KRISTINA ANA ŠKREB ◽  
CHRISTOPH THIELE
Keyword(s):  
Harmonic Analysis ◽  
Hilbert Transform ◽  
Singular Integrals ◽  
Quantitative Result ◽  
Square Function ◽  
Two Dimensional ◽  
Ergodic Averages ◽  
Multilinear Singular Integrals

We study double ergodic averages with respect to two general commuting transformations and establish a sharp quantitative result on their convergence in the norm. We approach the problem via real harmonic analysis, using recently developed methods for bounding multilinear singular integrals with certain entangled structure. A byproduct of our proof is a bound for a two-dimensional bilinear square function related to the so-called triangular Hilbert transform.

Some Pointwise Convergence Results in Lp(μ), 1 < p < ∞

1977 ◽  
Vol 20 (3) ◽  
pp. 277-284 ◽  
Author(s):  
Richard Duncan
Keyword(s):  
Probability Theory ◽  
Pointwise Convergence ◽  
Function Spaces ◽  
Almost Everywhere Convergence ◽  
Measurable Functions ◽  
Almost Everywhere ◽  
Convergence Results ◽  
Convergence In Norm

The theory of almost everywhere convergence has its roots in the poineering work of A. Kolmogorov, and today it constitutes one of the most captivating and challenging chapters in modern probability theory and analysis. Whereas some modes of convergence for sequences of measurable functions, e.g. convergence in norm, can be readily obtained by an intelligent exploitation of the various properties of the function spaces involved, a.e. convergence invariably requires a rather high, and sometimes surprising, degree of mathematical virtuosity.

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).

Uniform estimates for families of singular integrals and double Fourier series

1986 ◽  
Vol 41 (1) ◽  
pp. 1-12 ◽  
Author(s):  
Elena Prestini
Keyword(s):  
Fourier Series ◽  
Hilbert Transform ◽  
Singular Integrals ◽  
Double Fourier Series ◽  
Variable Coefficients ◽  
Partial Sums ◽  
One Dimensional ◽  
Uniform Estimates ◽  
Almost Everywhere ◽  
The One

AbstractIt is an open problem to establish whether or not the partial sums operator SNN2f(x, y) of the Fourier series of f ∈ Lp, 1 < p < 2, converges to the function almost everywhere as N → ∞. The purpose of this paper is to identify the operator that, in this problem of a.e. convergence of Fourier series, plays the central role that the maximal Hilbert transform plays in the one-dimensional case. This operator appears to be a singular integral with variable coefficients which is a variant of the maximal double Hilbert transform.

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