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ENDPOINT ESTIMATES FOR COMMUTATORS OF RIESZ TRANSFORMS ASSOCIATED WITH SCHRÖDINGER OPERATORS

  • PENGTAO LI (a1) and LIZHONG PENG (a2)
Abstract
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.

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Copyright
Corresponding author
For correspondence; e-mail: li_ptao@163.com
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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.

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References
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[1] J. Dziubanski , G. Garrigós , T. Martinez , J. L. Torrea and J. Zienkiewicz , ‘BMO spaces related to Schrödinger operators with potentials satisfying a reverse Hölder inequality’, Math. Z. 249 (2005), 329356.

[3] Z. Guo , P. Li and L. Peng , ‘Lp boundedness of commutators of Riesz transforms associated to Schrödinger operator’, J. Math. Anal. Appl. 341 (2008), 421432.

[5] H. Li , ‘Estimations Lp des opérateurs de Schrödinger sur les groupes nilpotents’, J. Funct. Anal. 161 (1999), 152218.

[6] C. Pérez , ‘Endpoint estimates for commutators of singular integral operators’, J. Funct. Anal. 128 (1995), 163185.

[7] Z. Shen , ‘Lp estimate for Schrödinger operators with certain potentials’, Ann. Inst. Fourier 45 (1995), 513546.

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Bulletin of the Australian Mathematical Society
  • ISSN: 0004-9727
  • EISSN: 1755-1633
  • URL: /core/journals/bulletin-of-the-australian-mathematical-society
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