Square function estimates for the Bochner–Riesz means

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.