Analysis & PDE

  • Anal. PDE
  • Volume 12, Number 5 (2019), 1259-1272.

Sparse bounds for the discrete cubic Hilbert transform

Amalia Culiuc, Robert Kesler, and Michael T. Lacey

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

H 3 f ( n ) = m 0 f ( n m 3 ) 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

| H 3 f , g | 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.

Article information

Anal. PDE, Volume 12, Number 5 (2019), 1259-1272.

Received: 9 November 2017
Accepted: 5 July 2018
First available in Project Euclid: 5 January 2019

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11L03: Trigonometric and exponential sums, general 42A05: Trigonometric polynomials, inequalities, extremal problems

Hilbert transform cubic integers sparse bound exponential sum circle method


Culiuc, Amalia; Kesler, Robert; Lacey, Michael T. Sparse bounds for the discrete cubic Hilbert transform. Anal. PDE 12 (2019), no. 5, 1259--1272. doi:10.2140/apde.2019.12.1259.

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