Skip to main content
Log in

Hardy spaces associated to the Schrödinger operator on strongly Lipschitz domains of \({\mathbb{R}^d}\)

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

Abstract

Let L = −Δ + V be a Schrödinger operator and Ω be a strongly Lipschitz domain of \({\mathbb R^{d}}\) , where Δ is the Laplacian on \({\mathbb R^{d}}\) and the potential V is a nonnegative polynomial on \({\mathbb R^{d}}\) . In this paper, we investigate the Hardy spaces on Ω associated to the Schrödinger operator L.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Auscher P., Russ E.: Hardy spaces and divergence operators on strongly Lipschitz domain of \({\mathbb{R}^n}\) . J. Funct. Anal. 201, 148–184 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  2. Auscher P., Tchamitchian P.: Square roots of elliptic second order divergence operators on strongly Lipschitz domains: L p theory. Math. Ann. 320(3), 577–623 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  3. Auscher P., McIntosh A., Russ E.: Hardy spaces of differential forms on Riemmannian manifolds. J. Geom. Anal. 18, 192–248 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chang D.C.: The dual of Hardy spaces on a bounded domain in \({\mathbb{R}^n}\) . Forum. Math. 6, 65–81 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  5. Chang D.C., Krantz S.G., Stein E.M.: H p theory on a smooth domain in \({\mathbb{R}^n}\) and elliptic boundary value problems. J. Funct. Anal. 114, 286–347 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  6. Chang D.C., Dafni G., Stein E.M.: Hardy spaces, BMO and boundary value problems for the Laplacian on a smooth domain in \({\mathbb{R}^n}\) . Trans. Am. Math. Soc. 351, 1605–1661 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  7. Coifman R.R., Meyer Y., Stein E.M.: Some new function spaces and their applications to harmonic analysis. J. Funct. Anal. 62, 304–335 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  8. Duong X.T., Yan L.X.: Duality of Hardy and BMO spaces associated with operators with heat kernel bounds. J. Am. Math. Soc. 18, 943–973 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  9. Dziubański J.: Atomic decomposition of H p spaces associated with some Schrödinger operators. Indiana Univ. Math. J. 47, 75–98 (1998)

    MathSciNet  MATH  Google Scholar 

  10. Dziubański J., Garrigós G., Martínez T., Torrea J.L., Zienkiewicz J.: BMO spaces related to Schrödinger operator with potential satisfying reverse Hölder inequality. Math. Z. 249, 329–356 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  11. Dziubański J., Zienkiewicz J.: Hardy spaces associated with some Schrödinger operators. Studia. Math. 126, 149–160 (1997)

    MathSciNet  MATH  Google Scholar 

  12. Dziubański J., Zienkiewicz J.: H p spaces associated with Schrödinger operators with potentials from reverse Hölder classes. Colloq. Math. 98, 5–38 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  13. Fefferman C.: The uncertainty principle. Bull. Am. Math. Soc. (N.S.) 9, 129–206 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  14. Fefferman C., Stein E.M.: H p spaces of several variables. Acta. Math. 129, 137–193 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  15. Garcia-Cuerva J., Rubio de Francia J.L.: Weighted Norm Inequalities and Related Topics, North-Holland Mathematics Studies, 116. North-Holland Publishing Co., Amsterdam (1985)

    Google Scholar 

  16. Huang, J.Z., Liu, H.P.: Littlewood-Paley functions associated with the Schrödinger operators. Mathematische Annalen (submitted)

  17. Huang, J.Z., Liu, H.P.: Area integral associated with the Schrödinger operators. Indiana Univ. Math. J. (submitted)

  18. Li H.: Estimations L p des opérateurs de Schrödinger sur les groupes nilpotents. J. Funct. Anal. 161, 152–218 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  19. Peng, L.Z.: The dual spaces of \({\lambda_\alpha(\mathbb{R}^n)}\) . Thesis in Peking University (1981)

  20. Russ, E.: The atomic decomposition for tent spaces on spaces of homogeneous type, CMA/AMSI Research Symposium “Asymptotic Geometric Analysis, Harmonic Analysis and Related Topics”, Proceedings of the Centre for Math. and Appl., vol. 42, pp. 125–135. Australian National University, Canberra (2007)

  21. Shen Z.: L p estimates for Schrödinger operators with certain potentials. Ann. Inst. Fourier (Grenoble) 45, 513–546 (1995)

    MathSciNet  MATH  Google Scholar 

  22. Reed M., Simon B.: Methods of Modern Mathematical Physics I: Functional Analysis. Academic Press, New York (1972)

    MATH  Google Scholar 

  23. Stein E.M.: Topics in Harmonic Analysis, Annals of Math Study. Princeton University Press, Princeton (1970)

    Google Scholar 

  24. Wang W.S.: The characterization of \({\lambda_\alpha(\mathbb{R}^n)}\) and the predual spaces of tent space. Acta. Sci. Natu. Univ. Pek. 24, 535–550 (1988)

    MATH  Google Scholar 

  25. Zhong, J.: Harmonic analysis for some Schrödinger type operators. Ph.D. Thesis, Princeton University, Princeton (1993)

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Jizheng Huang.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Huang, J. Hardy spaces associated to the Schrödinger operator on strongly Lipschitz domains of \({\mathbb{R}^d}\) . Math. Z. 266, 141–168 (2010). https://doi.org/10.1007/s00209-009-0558-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-009-0558-z

Keywords

Mathematis Subject Classification (2000)

Navigation