  We consider Bochner-Riesz means on weighted $L^p$ spaces, at the critical
index $\lambda(p)=d(\frac 1p-\frac 12)-\frac 12$. For every $A_1$-weight we
obtain an extension of Vargas' weak type $(1,1)$ inequality in some range of
$p>1$. To prove this result we establish new endpoint results for sparse
domination. These are almost optimal in dimension $d=2$; partial results as
well as conditional results are proved in higher dimensions. For the means of
index $\lambda_*=\frac{d-1}{2d+2}$ we prove fully optimal sparse bounds.
