# Uniform sparse bounds for discrete quadratic phase Hilbert transforms

Uniform sparse bounds for discrete quadratic phase Hilbert transforms For each $$\alpha \in \mathbb {T}$$ α ∈ T consider the discrete quadratic phase Hilbert transform acting on finitely supported functions $$f : \mathbb {Z} \rightarrow \mathbb {C}$$ f : Z → 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} H α f ( n ) : = ∑ m ≠ 0 e i α m 2 f ( n - m ) m . We prove that, uniformly in $$\alpha \in \mathbb {T}$$ α ∈ T , there is a sparse bound for the bilinear form $$\left\langle H^{\alpha } f , g \right\rangle$$ H α f , g for every pair of finitely supported functions $$f,g : \mathbb {Z}\rightarrow \mathbb {C}$$ f , g : Z → C . The sparse bound implies several mapping properties such as weighted inequalities in an intersection of Muckenhoupt and reverse Hölder classes. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Analysis and Mathematical Physics Springer Journals

# Uniform sparse bounds for discrete quadratic phase Hilbert transforms

, Volume 9 (1) – Sep 25, 2017
12 pages

Publisher
Springer Journals
Subject
Mathematics; Analysis; Mathematical Methods in Physics
ISSN
1664-2368
eISSN
1664-235X
DOI
10.1007/s13324-017-0195-3
Publisher site
See Article on Publisher Site

### Abstract

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

### Journal

Analysis and Mathematical PhysicsSpringer Journals

Published: Sep 25, 2017