Bulletin of the Australian Mathematical Society

Research Article

ENDPOINT ESTIMATES FOR COMMUTATORS OF RIESZ TRANSFORMS ASSOCIATED WITH SCHRÖDINGER OPERATORS

PENGTAO LIa1 c1 and LIZHONG PENGa2

a1 Department of Mathematics, Faculty of Science and Technology, University of Macau, Av. Padre Tomás Pereira, Taipa, Macau, PR China (email: li_ptao@163.com)

a2 LMAM School of Mathematical Sciences, Peking University, Beijing 100871, PR China (email: lzpeng@pku.edu.cn)

Abstract

In this paper, we discuss the H1L-boundedness of commutators of Riesz transforms associated with the Schrödinger operator L=−△+V, where H1L(Rn) is the Hardy space associated with L. We assume that V (x) is a nonzero, nonnegative potential which belongs to Bq for some q>n/2. Let T1=V (x)(−△+V )−1, T2=V1/2(−△+V )−1/2 and T3 =(−△+V )−1/2. We prove that, for bBMO (Rn) , the commutator [b,T3 ] is not bounded from H1L(Rn) to L1 (Rn) as T3 itself. As an alternative, we obtain that [b,Ti ] , ( i=1,2,3 ) are of (H1L,L1weak) -boundedness.

(Received November 09 2008)

2000 Mathematics subject classification

  • primary 47B32;
  • 47A75; secondary 42C40;
  • 94A40

Keywords and phrases

  • commutator;
  • H1L;
  • BMO;
  • Schrödinger operator;
  • Riesz transform

Correspondence:

c1 For correspondence; e-mail: li_ptao@163.com

Footnotes

The first author was supported by the Macau Government Science and Technology Development Fund FDCT/014/2008/A1; the second author was supported by NNSF of China No. 10471002 and RFDP of China No. 20060001010.