On almost everywhere convergence of planar Bochner-Riesz means

TL;DR

This paper proves the almost everywhere convergence of planar Bochner-Riesz means for L^p functions within the optimal range, using sharp estimates for related maximal operators and refined L^2 bounds.

Contribution

It establishes the convergence in the optimal p-range by deriving new sharp estimates for maximal operators associated with Bochner-Riesz means.

Findings

Proves almost everywhere convergence for 5/3 ≤ p ≤ 2.

Establishes a sharp L^{5/3} estimate for a maximal operator.

Introduces a refined L^2 estimate of independent interest.

Abstract

We demonstrate the almost everywhere convergence of the planar Bochner-Riesz means for $L^p$ functions in the optimal range when $5/3\leq p\leq 2$. This is achieved by establishing a sharp $L^{5/3}$ estimate for a maximal operator closely associated with the Bochner-Riesz multiplier operator. The estimate depends on a new refined $L^2$ estimate, which may be of independent interest.