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 $p_{\lambda }^{\mathrm{\prime }}<q<4$. 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 $A_1\cap R{H}_{\rho }$, where $\rho >\frac{4}{4-3p_{\lambda }}$.