Vol. 2, No. 1, 2020

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Sparse bounds for the discrete spherical maximal functions

Robert Kesler, Michael T. Lacey and Darío Mena

Vol. 2 (2020), No. 1, 75–92
Abstract

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.

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Keywords
sparse bounds, spherical averages, discrete, spherical maximal function
Mathematical Subject Classification 2010
Primary: 11K70, 42B25
Milestones
Received: 8 April 2019
Accepted: 5 July 2019
Published: 9 November 2019
Authors
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
Darío Mena
Escuela de Matemática
Universidad de Costa Rica
San José
Costa Rica