Sparse Endpoint Estimates for Bochner-Riesz Multipliers on the Plane

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 }}$.