Pure and Applied Analysis

Sparse bounds for the discrete spherical maximal functions

Robert Kesler, Michael T. Lacey, and Darío Mena

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We prove sparse bounds for the spherical maximal operator of Magyar, Stein and Wainger. The bounds are conjecturally sharp, and contain an endpoint estimate. The new method of proof is inspired by ones by Bourgain and Ionescu, is very efficient, and has not been used in the proof of sparse bounds before. The Hardy–Littlewood circle method is used to decompose the multiplier into major and minor arc components. The efficiency arises as one only needs a single estimate on each element of the decomposition.

Article information

Pure Appl. Anal., Volume 2, Number 1 (2020), 75-92.

Received: 8 April 2019
Accepted: 5 July 2019
First available in Project Euclid: 13 December 2019

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11K70: Harmonic analysis and almost periodicity 42B25: Maximal functions, Littlewood-Paley theory

sparse bounds spherical averages discrete spherical maximal function


Kesler, Robert; Lacey, Michael T.; Mena, Darío. Sparse bounds for the discrete spherical maximal functions. Pure Appl. Anal. 2 (2020), no. 1, 75--92. doi:10.2140/paa.2020.2.75. https://projecteuclid.org/euclid.paa/1576206323

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