New bounds for Stein’s square function in ℝ³

Abstract

We prove an L p ( R 3 ) L^p(\mathbb {R}^3) bounds on Stein’s square function for p ≥ 3.25 p\geq 3.25 . As an application, it improves the maximal Bochner–Riesz conjecture to the same range of p p . The main method we used is polynomial partitioning.