Vol. 12, No. 5, 2019

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Sparse bounds for the discrete cubic Hilbert transform

Amalia Culiuc, Robert Kesler and Michael T. Lacey

Vol. 12 (2019), No. 5, 1259–1272

Consider the discrete cubic Hilbert transform defined on finitely supported functions f on by

H3f(n) = m0f(n m3) m .

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

|H3f,g| CΛr,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.

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Hilbert transform, cubic integers, sparse bound, exponential sum, circle method
Mathematical Subject Classification 2010
Primary: 11L03, 42A05
Received: 9 November 2017
Accepted: 5 July 2018
Published: 15 December 2018
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