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 \(\lambda _*= \frac{d-1}{2d+2}\) we prove fully optimal sparse bounds.