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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Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.apde/1546657232

Digital Object Identifier
doi:10.2140/apde.2019.12.1259

Mathematical Reviews number (MathSciNet)
MR3892403

Zentralblatt MATH identifier
07006761

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

Keywords
Hilbert transform cubic integers sparse bound exponential sum circle method

Citation

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. https://projecteuclid.org/euclid.apde/1546657232


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