# Sparse Bounds for Bochner-Riesz Multipliers

## TL;DR

This paper establishes optimal sparse bounds for Bochner-Riesz multipliers across various dimensions, improving understanding of their weighted estimates and extending partial results related to the Bochner-Riesz conjecture.

## Contribution

It provides the first sharp sparse bounds for Bochner-Riesz multipliers, including the critical case, using a novel single scale analysis approach.

## Key findings

- Sparse bounds hold for all $0< \, ext{delta} < (n-1)/2$
- Range of sparse bounds extends to the optimal as delta approaches the critical value
- Sharp weighted estimates for Bochner-Riesz multipliers in Muckenhoupt weights category

## Abstract

The Bochner-Riesz multipliers $ B_{\delta }$ on $ \mathbb R ^{n}$ are shown to satisfy a range of sparse bounds, for all $0< \delta < \frac {n-1}2 $. The range of sparse bounds increases to the optimal range, as $ \delta $ increases to the critical value, $ \delta =\frac {n-1}2$, even assuming only partial information on the Bochner-Riesz conjecture in dimensions $ n \geq 3$. In dimension $n=2$, we prove a sharp range of sparse bounds. The method of proof is based upon a `single scale' analysis, and yields the sharpest known weighted estimates for the Bochner-Riesz multipliers in the category of Muckenhoupt weights.