Weighted norm inequalities of Bochner–Riesz means

Abstract

Let w be a Muckenhoupt weight and H w p ( R n ) be the weighted Hardy spaces. We use the atomic decomposition of H w p ( R n ) and their molecular characters to show that the Bochner–Riesz means T R δ are bounded on H w p ( R n ) for 0 < p ⩽ 1 and δ > max { n / p − ( n + 1 ) / 2 , [ n / p ] r w ( r w − 1 ) −1 − ( n + 1 ) / 2 } , where r w is the critical index of w for the reverse Hölder condition. We also prove the H w p − L w p boundedness of the maximal Bochner–Riesz means T ∗ δ for 0 < p ⩽ 1 and δ > n / p − ( n + 1 ) / 2 .