Schrödinger type singular integrals: weighted estimates for p=1

Abstract

A critical radius function ρ assigns to each x ∈ Rd a positive number in a way that its variation at different points is somehow controlled by a power of the distance between them. This kind of function appears naturally in the harmonic analysis related to a Schr¨odinger operator −∆ + V with V a non-negative potential satisfying some specific reverse H¨older condition. For a family of singular integrals associated to such critical radius function, we prove boundedness results in the extreme case p = 1. On one side we obtain weighted weak (1, 1) results for a class of weights larger than Muckenhoupt class A1. On the other side, for the same weights, we prove continuity from appropriate weighted Hardy spaces into weighted L1 . To achieve the latter result we define weighted Hardy spaces by means of a ρ-localized maximal heat operator. We obtain a suitable atomic decomposition and a characterization via ρ-localized Riesz Transforms for these spaces. For the case of ρ derived from a Schr¨odinger operator, we obtain new estimates for of many of the operators appearing in [16].