Sparse bounds for the discrete cubic Hilbert transform

We prove that there exists $r < 2$ and universal constant $C$ such that for all finitely supported $f, g$ on $ℤ$ there exists an $(r, r)$ -sparse form $Λ_{r, r}$ for which

| 〈 H_{3} f, g 〉 | \leq C Λ_{r, r} (f, g) .

This is the first result of this type concerning discrete harmonic analytic operators. It immediately implies some weighted inequalities, which are also new in this setting.

Keywords

Hilbert transform, cubic integers, sparse bound, exponential sum, circle method

Mathematical Subject Classification 2010

Primary: 11L03, 42A05

Milestones

Received: 9 November 2017

Accepted: 5 July 2018

Published: 15 December 2018

Authors

Amalia Culiuc
School of Mathematics Georgia Institute of Technology Atlanta, GA United States

Robert Kesler
School of Mathematics Georgia Institute of Technology Atlanta, GA United States

Michael T. Lacey
School of Mathematics Georgia Institute of Technology Atlanta, GA United States

Vol. 12, No. 5, 2019

Amalia Culiuc, Robert Kesler and Michael T. Lacey

Abstract

Keywords

Mathematical Subject Classification 2010

Milestones

Authors