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ENDPOINT ESTIMATES FOR COMMUTATORS OF RIESZ TRANSFORMS ASSOCIATED WITH SCHRÖDINGER OPERATORS

Part of: General theory of linear operators Communication, information Special classes of linear operators

Published online by Cambridge University Press:  16 August 2010

PENGTAO LI  and
LIZHONG PENG
PENGTAO LI*
Affiliation:
Department of Mathematics, Faculty of Science and Technology, University of Macau, Av. Padre Tomás Pereira, Taipa, Macau, PR China (email: [email protected])
LIZHONG PENG
Affiliation:
LMAM School of Mathematical Sciences, Peking University, Beijing 100871, PR China (email: [email protected])
*
✉For correspondence; e-mail: [email protected]
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Abstract

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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 b∈BMO (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.

Keywords

commutatorH1LBMOSchrödinger operatorRiesz transform

MSC classification

Secondary: 47B32: Operators in reproducing-kernel Hilbert spaces (including de Branges, de Branges-Rovnyak, and other structured spaces) 47A75: Eigenvalue problems 94A40: Channel models (including quantum)
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 82 , Issue 3 , December 2010 , pp. 367 - 389
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2010

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.

References

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