Sparse Endpoint Estimates for Bochner-Riesz Multipliers on the Plane

Abstract

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