Sparse Bounds for Discrete Quadratic Phase Hilbert Transform

Abstract

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