Weighted decoupling estimates and the Bochner-Riesz means

TL;DR

The paper addresses the problem of almost everywhere convergence of Bochner-Riesz means $S_t^\ abla f$ for $f\in L^p(\mathbb{R}^d)$ with $1<p<2$, focusing on dimensions $d=2,3$. It develops weighted decoupling theory, establishing a multilinear weighted decoupling inequality and a weighted refined decoupling estimate, then proves a broad-narrow bound (Theorem thm:main0) via wave packets and transversality analysis. These weighted decoupling tools are then used in the Bochner-Riesz context to derive improved weak-type estimates for the associated maximal operator, following the Gan-Wu framework, and to obtain improved convergence results in low dimensions, including explicit bounds in dimension $d=2$ and an enhanced bound in dimension $d=3$. The work combines a broad-narrow strategy, refined decoupling (GIOW; du2023weighted), and parabolic rescaling to bridge decoupling theory with maximal-function control for Bochner-Riesz means, yielding sharper thresholds for almost-everywhere convergence and advancing the understanding of weighted decoupling effects on harmonic-analytic convergence problems.

Abstract

We prove new weighted decoupling estimates. As an application, we give an improved sufficient condition for almost everywhere convergence of the Bochner-Riesz means of arbitrary $L^p$ functions for $1<p<2$ in dimensions 2 and 3.

Theorems & Definitions (24)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.3
• Theorem 2.1: Multilinear restriction
• Theorem 2.2: Refined decoupling
• Theorem 2.3
• proof
• Theorem 2.4: du2023weighted
• Theorem 2.5: Weighted refined decoupling
• proof