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 Schrodinger operator ̈ − + V with V a non-negative potential satisfying some specific reverse Holder condition. For a family of singular integrals associated with 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 Schrodinger operator, we obtain new estimates for many of the operators appearing in [27].