Resumen de Endpoint estimates for commutators of singular integrals related to Schrödinger operators

Luong Dang Ky

• Let L=−Δ+V be a Schrödinger operator on Rd, d≥3, where V is a nonnegative potential, V≠0, and belongs to the reverse H\"older class RHd/2. In this paper, we study the commutators [b,T] for T in a class KL of sublinear operators containing the fundamental operators in harmonic analysis related to L. More precisely, when T∈KL, we prove that there exists a bounded subbilinear operator R=RT:H1L(Rd)×BMO(Rd)→L1(Rd) such that (⋆)|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 H1L(Rd)×BMO(Rd) into L1(Rd) which does not depend on T. The subbilinear decomposition (⋆) allows us to explain why commutators with the fundamental operators are of weak type (H1L,L1), and when a commutator [b,T] is of strong type (H1L,L1).

  Also, we discuss the H1L-estimates for commutators of the Riesz transforms associated with the Schrödinger operator L.