Endpoint estimates for commutators of singular integrals related to Schrödinger operators

Abstract

Let L = −∆ + V be a Schrödinger operator on R d , d ≥ 3, where V is a nonnegative potential, V = 0, and belongs to the reverse Hölder class RH d/2 . In this paper, we study the commutators [b, T ] for T in a class K L of sublinear operators containing the fundamental operators in harmonic analysis related to L. More precisely, when T ∈ K L , we prove that there exists a bounded subbilinearThe subbilinear decomposition (1) allows us to explain why commutators with the fundamental operators are of weak type (H 1 L , L 1 ), and when a commutator [b, T ] is of strong type (H 1 L , L 1 ). Also, we discuss the H 1 L -estimates for commutators of the Riesz transforms associated with the Schrödinger operator L. Contents2010 Mathematics Subject Classification. Primary: 42B35, 35J10 Secondary: 42B20.