Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-26T09:58:31.465Z Has data issue: false hasContentIssue false
  • English
  • Français

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]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

[1]Dziubanski, J., Garrigós, G., Martinez, T., Torrea, J. L. and Zienkiewicz, J., ‘BMO spaces related to Schrödinger operators with potentials satisfying a reverse Hölder inequality’, Math. Z. 249 (2005), 329–356.CrossRefGoogle Scholar
[2]Dziubański, J. and Zienkiewicz, J., ‘Hardy space H 1 associated to Schrödinger operator with potential satisfying reverse Hölder inequality’, Rev. Mat. Iberoamericana 15 (1999), 279–296.CrossRefGoogle Scholar
[3]Guo, Z., Li, P. and Peng, L., ‘L p boundedness of commutators of Riesz transforms associated to Schrödinger operator’, J. Math. Anal. Appl. 341 (2008), 421–432.CrossRefGoogle Scholar
[4]Harboure, E., Segovia, C. and Torrea, J., ‘Boundedness of commutators of fractional and singular integrals for the extreme values of p’, Illinois J. Math. 41 (1997), 676–700.CrossRefGoogle Scholar
[5]Li, H., ‘Estimations L p des opérateurs de Schrödinger sur les groupes nilpotents’, J. Funct. Anal. 161 (1999), 152–218.CrossRefGoogle Scholar
[6]Pérez, C., ‘Endpoint estimates for commutators of singular integral operators’, J. Funct. Anal. 128 (1995), 163–185.CrossRefGoogle Scholar
[7]Shen, Z., ‘L p estimate for Schrödinger operators with certain potentials’, Ann. Inst. Fourier 45 (1995), 513–546.CrossRefGoogle Scholar
[8]Stein, E. M., Harmonic Analysis: Real Variable Methods, Orthogonality and Oscillatory Integrals (Princeton University, Princeton, NJ, 1993).Google Scholar
[9]Zhong, J., ‘Harmonic analysis for some Schrödinger type operators’, PhD Thesis, Princeton University, 1993.Google Scholar