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


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.


Discrete analysis Quadratic phase Sparse bounds 


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Conflict of interest

The authors declare that they have no conflicts of interest.


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© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.School of MathematicsGeorgia Institute of TechnologyAtlantaUSA

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