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 }}$.
Categories: math.CA