Sparse Bounds for Discrete Quadratic Phase Hilbert Transform

TL;DR

This paper establishes uniform sparse bounds for the discrete quadratic phase Hilbert Transform, leading to new weighted inequalities and mapping properties for functions in $ ext{ell}^2$ spaces.

Contribution

It introduces the first uniform sparse bounds for the discrete quadratic phase Hilbert Transform across all parameters, enhancing understanding of its boundedness and weighted inequalities.

Findings

Proves uniform sparse bounds for $H^{ ext{alpha}}$ across all $ ext{alpha} \

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Abstract

Consider the discrete quadratic phase Hilbert Transform acting on $\ell^{2}$ finitely supported functions $$ H^{\alpha} f(n) : = \sum_{m \neq 0} \frac{e^{2 \pi i\alpha m^2} f(n - m)}{m}. $$ We prove that, uniformly in $\alpha \in \mathbb{T}$, there is a sparse bound for the bilinear form $\langle H^{\alpha} f , g \rangle$. The sparse bound implies several mapping properties such as weighted inequalities in an intersection of Muckenhoupt and reverse H\"older classes.