Square function estimates for the Bochner–Riesz means

Lee, Sanghyuk

doi:10.2140/apde.2018.11.1535

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Square function estimates for the Bochner–Riesz means

Sanghyuk Lee

Vol. 11 (2018), No. 6, 1535–1586

DOI: 10.2140/apde.2018.11.1535

Abstract

We consider the square-function (known as Stein’s square function) estimate associated with the Bochner–Riesz means. The previously known range of the sharp estimate is improved. Our results are based on vector-valued extensions of Bennett, Carbery and Tao’s multilinear (adjoint) restriction estimate and an adaptation of an induction argument due to Bourgain and Guth. Unlike the previous work by Bourgain and Guth on $L^{p}$ boundedness of the Bochner–Riesz means in which oscillatory operators associated to the kernel were studied, we take more direct approach by working on the Fourier transform side. This enables us to obtain the correct order of smoothing, which is essential for obtaining the sharp estimates for the square functions.

Keywords

square function, Bochner–Riesz means

Mathematical Subject Classification 2010

Primary: 42B15

Secondary: 35B65

Milestones

Received: 4 October 2017

Accepted: 12 January 2018

Published: 3 May 2018

Authors

Sanghyuk Lee
School of Mathematical Sciences Seoul National University Seoul South Korea

Vol. 11, No. 6, 2018