Bochner–Riesz means at the critical index: weighted and sparse bounds

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