Advertisement

Arkiv för Matematik

, Volume 48, Issue 2, pp 301–310 | Cite as

Riesz transform characterization of Hardy spaces associated with Schrödinger operators with compactly supported potentials

  • Jacek Dziubański
  • Marcin PreisnerEmail author
Article

Abstract

Let L=−Δ+V be a Schrödinger operator on ℝ d , d≥3. We assume that V is a nonnegative, compactly supported potential that belongs to L p (ℝ d ), for some p>d /2. Let K t be the semigroup generated by −L. We say that an L 1(ℝ d )-function f belongs to the Hardy space \(H^{1}_{L}\) associated with L if sup t>0|K t f| belongs to L 1(ℝ d ). We prove that \(f\in H^{1}_{L}\) if and only if R j fL 1(ℝ d ) for j=1,…,d, where R j =(/ x j )L −1/2 are the Riesz transforms associated with L.

Keywords

Hardy Space Atomic Decomposition Perturbation Formula Classical Hardy Space Supp Versus 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Auscher, P. and Ben Ali, B., Maximal inequalities and Riesz transform estimates on Lp spaces for Schrödinger operators with nonnegative potentials, Ann. Inst. Fourier (Grenoble)57 (2007), 1975–2013. zbMATHMathSciNetGoogle Scholar
  2. 2.
    Burkholder, D. L., Gundy, R. F. and Silverstein, M. L., A maximal function characterization of the class Hp, Trans. Amer. Math. Soc.157 (1971), 137–153. zbMATHMathSciNetGoogle Scholar
  3. 3.
    Duoandikoetxea, J., Fourier Analysis, Graduate Studies in Mathematics 29, American Mathematical Society, Providence, RI, 2001. zbMATHGoogle Scholar
  4. 4.
    Duong, X. T., Ouhabaz, E. M. and Yan, L., Endpoint estimates for Riesz transforms of magnetic Schrödinger operators, Ark. Mat.44 (2006), 261–275. zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Dziubański, J. and Zienkiewicz, J., Hardy space H1 associated to Schrödinger operator with potential satisfying reverse Hölder inequality, Rev. Mat. Iberoamericana15 (1999), 279–296. zbMATHMathSciNetGoogle Scholar
  6. 6.
    Dziubański, J. and Zienkiewicz, J., Hardy space H1 for Schrödinger operators with compactly supported potentials, Ann. Mat. Pura Appl.184 (2005), 315–326. zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Fefferman, C. and Stein, E. M., Hp spaces of several variables, Acta Math.129 (1972), 137–193. zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Sikora, A., Riesz transform, Gaussian bounds and the method of wave equation, Math. Z.247 (2004), 643–662. zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Stein, E. M., Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, Princeton, NJ, 1993. zbMATHGoogle Scholar

Copyright information

© Institut Mittag-Leffler 2010

Authors and Affiliations

  1. 1.Instytut MatematycznyUniwersytet WrocławskiWrocławPoland
  2. 2.Instytut MatematycznyUniwersytet WrocławskiWrocławPoland

Personalised recommendations