Bochner-Riesz Means at the Critical Index: Weighted and Sparse Bounds

Abstract

We consider Bochner-Riesz means on weighted Lp spaces, at the critical index λ(p)=d(1/p−1/2)−1/2. For every A₁-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 λ∗=(d−1)/(2d+2) we prove fully optimal sparse bounds.