Sparse Endpoint Estimates for Bochner-Riesz Multipliers on the Plane

TL;DR

This paper establishes sparse bounds for Bochner-Riesz multipliers on the plane at critical indices, leading to new endpoint weighted weak type estimates and advancing the understanding of their boundedness properties.

Contribution

It provides the first sparse bounds for Bochner-Riesz multipliers at the critical index, extending previous weak boundedness results with quantitative endpoint estimates.

Findings

Sparse bounds for $ B_{\lambda} $ at critical indices

New endpoint weighted weak type estimates for specific weights

Quantification of endpoint weak $L^{p_\lambda}$ boundedness

Abstract

For $ 0< \lambda < \frac{1}2$, let $ B_{\lambda }$ be the Bochner-Riesz multiplier of index $ \lambda $ on the plane. Associated to this multiplier is the critical index $1 < p_\lambda = \frac{4} {3+2 \lambda } < \frac{4}3$. We prove a sparse bound for $ B_{\lambda }$ with indices $ (p_\lambda , q)$, where $ p_\lambda ' < q < 4$. This is a further quantification of the endpoint weak $L^{p_\lambda}$ boundedness of $ B_{\lambda }$, due to Seeger. Indeed, the sparse bound immediately implies new endpoint weighted weak type estimates for weights in $ A_1 \cap RH_{\rho }$, where $ \rho > \frac4 {4 - 3 p_{\lambda }}$.