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(1/p-1/2)-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(& lowast;)=d-1/2d+2 we prove fully optimal sparse bounds.