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Uniform sparse bounds for discrete quadratic phase Hilbert transforms

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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.

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  1. School of Mathematics, Georgia Institute of Technology, 686 Cherry Street, Atlanta, GA, 30312, USA

    Robert Kesler & Darío Mena Arias

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  1. Robert Kesler
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  2. Darío Mena Arias
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Correspondence to Robert Kesler.

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Kesler, R., Arias, D.M. Uniform sparse bounds for discrete quadratic phase Hilbert transforms. Anal.Math.Phys. 9, 263–274 (2019). https://doi.org/10.1007/s13324-017-0195-3

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