On Convergence of Oscillatory Ergodic Hilbert Transforms

TL;DR

This paper establishes new sparse bounds and convergence results for oscillatory ergodic Hilbert transforms, including cases with polynomial phase modulation and random coefficients, advancing understanding in ergodic theory and harmonic analysis.

Contribution

It introduces sufficient conditions for sparse bounds of discrete singular integrals and proves convergence of modulated ergodic Hilbert transforms with polynomial phases and random variables.

Findings

Proved convergence of modulated ergodic Hilbert transforms with super-linear polynomial phases.

Established sparse bounds for random one-sided ergodic Hilbert transforms.

Provided new quantitative weighted inequalities for discrete singular integral operators.

Abstract

We introduce sufficient conditions on discrete singular integral operators for their maximal truncations to satisfy a sparse bound. The latter imply a range of quantitative weighted inequalities, which are new. As an application, we prove the following ergodic theorem: let $p(t)$ be a Hardy field function which grows "super-linearly" and stays "sufficiently far" from polynomials. We show that for each measure-preserving system, $(X,\Sigma,\mu,\tau)$, with $\tau$ a measure-preserving $\mathbb{Z}$-action, the modulated one-sided ergodic Hilbert transform \[ \sum_{n=1}^\infty \frac{e^{2\pi i p(n)}}{n} \tau^n f(x) \] converges $\mu$-a.e. for each $f \in L^r(X), \ 1 \leq r < \infty$. This affirmatively answers a question of J. Rosenblatt. In the second part of the paper, we establish almost sure sparse bounds for random one-sided ergodic Hilbert transforms, \[ \sum_{n=1}^\infty \frac{X_n}{n} \tau^n f(x), \] where $\{ X_n \}$ are uniformly bounded, independent, and mean-zero random variables.