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

We consider Bochner-Riesz means on weighted $L^p$ spaces, at the critical index $\lambda (p)=d(\frac{1}{p}-\frac{1}{2})-\frac{1}{2}$. 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 ${\lambda }_{\ast }=\frac{d-1}{2d+2}$ we prove fully optimal sparse bounds.