Endpoint estimates for commutators of singular integrals related to Schrödinger operators

  • Luong Dang Ky

    University of Quy Nhon, Quy Nhon, Binh Dinh, Vietnam

Abstract

Let L=Δ+VL= -\Delta+ V be a Schrödinger operator on Rd\mathbb R^d, d3d\geq 3, where VV is a nonnegative potential, V0V\ne 0, and belongs to the reverse H\"older class RHd/2RH_{d/2}. In this paper, we study the commutators [b,T][b,T] for TT in a class KL\mathcal K_L of sublinear operators containing the fundamental operators in harmonic analysis related to LL. More precisely, when TKLT\in \mathcal K_L, we prove that there exists a bounded subbilinear operator R=RT ⁣:HL1(Rd)×BMO(Rd)L1(Rd)\mathfrak R= \mathfrak R_T\colon H^1_L(\mathbb R^d)\times {\rm BMO}(\mathbb R^d)\to L^1(\mathbb R^d) such that

()T(S(f,b))R(f,b)[b,T](f)R(f,b)+T(S(f,b)),(\star)\qquad |T(\mathfrak S(f,b))|- \mathfrak R(f,b)\leq |[b,T](f)|\leq \mathfrak R(f,b) + |T(\mathfrak S(f,b))|,

where S\mathfrak S is a bounded bilinear operator from HL1(Rd)×BMO(Rd)H^1_L(\mathbb R^d)\times {\rm BMO}(\mathbb R^d) into L1(Rd)L^1(\mathbb R^d) which does not depend on TT. The subbilinear decomposition ()(\star) allows us to explain why commutators with the fundamental operators are of weak type (HL1,L1)(H^1_L,L^1), and when a commutator [b,T][b,T] is of strong type (HL1,L1)(H^1_L,L^1).

Also, we discuss the HL1H^1_L-estimates for commutators of the Riesz transforms associated with the Schrödinger operator LL.

Cite this article

Luong Dang Ky, Endpoint estimates for commutators of singular integrals related to Schrödinger operators. Rev. Mat. Iberoam. 31 (2015), no. 4, pp. 1333–1373

DOI 10.4171/RMI/871