Uniform sparse bounds for discrete quadratic phase Hilbert transforms

Abstract

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