Schrödinger type singular integrals: weighted estimates for p=1

BONGIOANNI, BRUNO; CABRAL, ADRIÁN; HARBOURE, ELEONOR

MATHEMATISCHE NACHRICHTEN

WILEY-V C H VERLAG GMBH

Año: 2016 vol. 289 p. 1341 - 1369

0025-584X

A critical radius function ρ assigns to each xin R^d 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 Schrödinger operator -L+V with V a non-negative potential satisfying some specific reverse Hölder 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 A_1. On the other side, for the same weights, we prove continuity from appropriate weighted Hardy spaces into weighted L^1. 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 Schrödinger operator, we obtain new estimates for many of the operators appearing in [Z. Shen, L^p estimates for Schr¨odinger operators with certain potentials, Ann. Inst. Fourier (Grenoble) 45(2), 513?546(1995)].