Discrete Analogoues in Harmonic Analysis: Maximally Monomially Modulated Singular Integrals Related to Carleson's Theorem

2018/03/26 by Krause, Ben
#Classical Analysis and ODEs (math.CA) #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.1803.09431

Abstract

Motivated by Bourgain's work on pointwise ergodic theorems, and the work of Stein and Stein-Wainger on maximally modulated singular integrals without linear terms, we prove that the maximally monomially modulated discrete Hilbert transform, C_df(x) := sup_λ| ∑_{m ≠ 0} f(x-m) \frace^{2πi λmd}m | is bounded on all ℓ^p, 2 - (1)/(d² + 1) < p < ∞, for any d ≥ 2. We also establish almost everywhere pointwise convergence of the modulated ergodic Hilbert transforms (as λ→ 0) ∑_{m ≠ 0} T^m f(x) ⋅ \frace^{2πi λmd}m for any measure-preserving system (X,μ,T), and any f ∈ L^p(X), 2 - (1)/(d² +1) < p < ∞.

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