On convergence of oscillatory ergodic Hilbert transforms


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, Σ, µ, τ), with τ a measure-preserving Z-action, the modulated one-sided ergodic Hilbert transform ∞ X e2πip(n) τnf (x) n n=1 converges µ-almost everywhere for each f ∈ Lr (X), 1 ≤ r < ∞. This affirmatively answers a question of J. Rosenblatt [22]. In the second part of the paper, we establish almost-sure sparse bounds for a random one-sided ergodic Hilbert ∞ nX=1Xnn τnf (x), where {Xn} are uniformly bounded, independent, and mean-zero random variables.

Publication Title

Indiana University Mathematics Journal