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
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.
