Hardy Spaces, Commutators of Singular Integral Operators Related to Schrödinger Operators and Applications

Let L = −∆+ V be a Schrödinger operator on R, d ≥ 3, where V is a nonnegative function, V 6= 0, and belongs to the reverse Hölder class RHd/2. The purpose of this paper is three-fold. First, we prove a version of the classical theorem of Jones and Journé on weak∗-convergence in H L(R ). Secondly, we give a bilinear decomposition for the product space H L(R )×BMOL(R). Finally, we study the commutators [b, T ] for T belongs to a class KL of sublinear operators containing almost all fundamental operators in harmonic analysis related to L. More precisely, when T ∈ KL, we prove that there exists a bounded subbilinear operator R = RT : H 1 L(R d)×BMO(Rd) → L(R) such that (1) |T (S(f, b))| −R(f, b) ≤ |[b, T ](f)| ≤ R(f, b) + |T (S(f, b))|, where S is a bounded bilinear operator from H L(R ) × BMO(R) into L(R) which does not depend on T . In the particular case of the Riesz transforms Rj = ∂xjL , j = 1, ..., d, and b ∈ BMO(R), we prove that the commutators [b, Rj ] are bounded on H 1 L(R ) iff b ∈ BMO L (R)– a new space of BMO type, which coincides with the space LMO(R) when L = −∆+ 1. Furthermore, ‖b‖BMOlog L ≈ ‖b‖BMO + d ∑ j=1 ‖[b, Rj]‖H1 L →H L . The subbilinear decomposition (1) explains why almost all commutators of the fundamental operators are of weak type (H L, L ), and when a commutator [b, T ] is of strong type (H L, L ).