English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

Motivated by Bourgain's work on pointwise ergodic theorems, and the work ofStein and Stein-Wainger on maximally modulated singular integrals withoutlinear terms, we prove that the maximally monomially modulated discrete Hilberttransform, \[ \mathcal{C}_df(x) := \sup_\lambda \left| \sum_{m \neq 0} f(x-m)\frac{e^{2\pi i \lambda m^d}}{m} \right| \] is bounded on all $\ell^p, \ 2 -\frac{1}{d^2 + 1} < p < \infty$, for any $d \geq 2$. We also establish almosteverywhere pointwise convergence of the modulated ergodic Hilbert transforms(as $\lambda \to 0$) \[ \sum_{m \neq 0} T^m f(x) \cdot \frac{e^{2\pi i \lambdam^d}}{m} \] for any measure-preserving system $(X,\mu,T)$, and any $f \inL^p(X), \ 2 - \frac{1}{d^2 +1} < p < \infty$.

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