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We consider Bochner-Riesz means on weighted $L^p$ spaces, at the criticalindex $\lambda(p)=d(\frac 1p-\frac 12)-\frac 12$. For every $A_1$-weight weobtain 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 sparsedomination. These are almost optimal in dimension $d=2$; partial results aswell as conditional results are proved in higher dimensions. For the means ofindex $\lambda_*=\frac{d-1}{2d+2}$ we prove fully optimal sparse bounds.

Bochner-Riesz means at the critical index: Weighted and sparse bounds