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

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, \[ \mathcal{C}_df(x) := \sup_λ\left| \sum_{m \neq 0} f(x-m) \frac{e^{2πi λm^d}}{m} \right| \] is bounded on all ${\ell }^p,2-\frac{1}{d^2+1}<p<\infty$, for any $d\ge 2$. We also establish almost everywhere pointwise convergence of the modulated ergodic Hilbert transforms (as $\lambda \to 0$) \[ \sum_{m \neq 0} T^m f(x) \cdot \frac{e^{2πi λm^d}}{m} \] for any measure-preserving system $(X,\mu ,T)$, and any $f\in L^p(X),2-\frac{1}{d^2+1}<p<\infty$.

References in corpus (1)

• Maximal polynomial modulations of singular integrals

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