Square function estimates for the Bochner-Riesz means

We consider the square function (known as Stein's square function) estimate associated with the Bochner-Riesz means. The previously known range of sharp estimate is improved. Our results are based on vector valued extensions of Bennett-Carbery-Tao's multilinear (adjoint) restriction estimate and adaption of induction argument due to Bourgain-Guth. Unlike the previous work by Bourgain-Guth on $L^p$ boundedness of Bochner-Riesz means in which oscillatory operators associated to the kernel had been 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 estimate for the square function.