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Analysis and Mathematical Physics

, Volume 9, Issue 1, pp 263–274 | Cite as

Uniform sparse bounds for discrete quadratic phase Hilbert transforms

  • Robert KeslerEmail author
  • Darío Mena Arias
Article

Abstract

For each \(\alpha \in \mathbb {T}\) consider the discrete quadratic phase Hilbert transform acting on finitely supported functions \(f : \mathbb {Z} \rightarrow \mathbb {C}\) according to
$$\begin{aligned} H^{\alpha }f(n):= \sum _{m \ne 0} \frac{e^{i\alpha m^2} f(n - m)}{m}. \end{aligned}$$
We prove that, uniformly in \(\alpha \in \mathbb {T}\), there is a sparse bound for the bilinear form \(\left\langle H^{\alpha } f , g \right\rangle \) for every pair of finitely supported functions \(f,g : \mathbb {Z}\rightarrow \mathbb {C}\). The sparse bound implies several mapping properties such as weighted inequalities in an intersection of Muckenhoupt and reverse Hölder classes.

Keywords

Discrete analysis Quadratic phase Sparse bounds 

Notes

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflicts of interest.

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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.School of MathematicsGeorgia Institute of TechnologyAtlantaUSA

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