Sparse Bounds for Discrete Quadratic Phase Hilbert Transform

Abstract

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

9 pages