Abstract
In this work we obtain boundedness on L p, for 1<p<∞, of commutators T b f=bTf−T(bf) where T is any of the Riesz transforms or their conjugates associated to the Schrödinger operator −Δ+V with V satisfying an appropriate reverse Hölder inequality. The class where b belongs is larger than the usual BMO. We also obtain a substitute result for p=∞, under a slightly stronger condition on b.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Bongioanni, B., Harboure, E., Salinas, O.: Riesz transforms related to Schrödinger operators acting on BMO type spaces. J. Math. Anal. Appl. 357(1), 115–131 (2009)
Coifman, R.R., Rochberg, R., Weiss, G.: Factorization theorems for Hardy spaces in several variables. Ann. Math. (2) 103(3), 611–635 (1976)
Dziubański, J., Zienkiewicz, J.: Hardy spaces H 1 associated to Schrödinger operators with potential satisfying reverse Hölder inequality. Rev. Mat. Iberoam. 15(2), 279–296 (1999)
Dziubański, J., Garrigós, G., Martínez, T., Torrea, J., Zienkiewicz, J.: BMO spaces related to Schrödinger operators with potentials satisfying a reverse Hölder inequality. Math. Z. 249(2), 329–356 (2005)
Gehring, F.W.: The L p-integrability of the partial derivatives of a quasiconformal mapping. Acta Math. 130, 265–277 (1973)
Guo, Z., Li, P., Peng, L.: L p boundedness of commutators of Riesz transforms associated to Schrödinger operator. J. Math. Anal. Appl. 341(1), 421–432 (2008)
Harboure, E., Segovia, C., Torrea, J.L.: Boundedness of commutators of fractional and singular integrals for the extreme values of p. Ill. J. Math. 41(4), 676–700 (1997)
John, F., Nirenberg, L.: On functions of bounded mean oscillation. Commun. Pure Appl. Math. 14, 415–426 (1961)
Pérez, C.: Sharp estimates for commutators of singular integrals via iterations of the Hardy-Littlewood maximal function. J. Fourier Anal. Appl. 3(6), 743–756 (1997)
Pradolini, G., Salinas, O.: Commutators of singular integrals on spaces of homogeneous type. Czechoslov. Math. J. 57(1), 75–93 (2007)
Shen, Z.: L p estimates for Schrödinger operators with certain potentials. Ann. Inst. Fourier (Grenoble) 45(2), 513–546 (1995)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Fernando Soria.
This research is partially supported by Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET) and Universidad Nacional del Litoral (UNL), Argentina.
Rights and permissions
About this article
Cite this article
Bongioanni, B., Harboure, E. & Salinas, O. Commutators of Riesz Transforms Related to Schrödinger Operators. J Fourier Anal Appl 17, 115–134 (2011). https://doi.org/10.1007/s00041-010-9133-6
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00041-010-9133-6