## Arkiv för Matematik

• Ark. Mat.
• Volume 48, Number 2 (2010), 301-310.

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

#### Abstract

Let L=−Δ+V be a Schrödinger operator on ℝd, d≥3. We assume that V is a nonnegative, compactly supported potential that belongs to Lp(ℝd), for some p> d/2. Let Kt be the semigroup generated by −L. We say that an L1(ℝd)-function f belongs to the Hardy space $H^{1}_{L}$ associated with L if sup t>0|Ktf| belongs to L1(ℝd). We prove that $f\in H^{1}_{L}$ if and only if RjfL1(ℝd) for j=1,…, d, where Rj=(/xj)L−1/2 are the Riesz transforms associated with L.

#### Note

Supported by the Polish Ministry of Science and High Education—grant N N201 397137, the European Commission Marie Curie Host Fellowship for the Transfer of Knowledge “Harmonic Analysis, Nonlinear Analysis and Probability" MTKD-CT-2004-013389.

#### Article information

Source
Ark. Mat., Volume 48, Number 2 (2010), 301-310.

Dates
Revised: 20 January 2010
First available in Project Euclid: 31 January 2017

https://projecteuclid.org/euclid.afm/1485907118

Digital Object Identifier
doi:10.1007/s11512-010-0121-5

Mathematical Reviews number (MathSciNet)
MR2672611

Zentralblatt MATH identifier
1202.42046

Rights

#### Citation

Dziubański, Jacek; Preisner, Marcin. Riesz transform characterization of Hardy spaces associated with Schrödinger operators with compactly supported potentials. Ark. Mat. 48 (2010), no. 2, 301--310. doi:10.1007/s11512-010-0121-5. https://projecteuclid.org/euclid.afm/1485907118

#### References

• 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.
• 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.
• Duoandikoetxea, J., Fourier Analysis, Graduate Studies in Mathematics 29, American Mathematical Society, Providence, RI, 2001.
• Duong, X. T., Ouhabaz, E. M. and Yan, L., Endpoint estimates for Riesz transforms of magnetic Schrödinger operators, Ark. Mat. 44 (2006), 261–275.
• Dziubański, J. and Zienkiewicz, J., Hardy space H1 associated to Schrödinger operator with potential satisfying reverse Hölder inequality, Rev. Mat. Iberoamericana 15 (1999), 279–296.
• Dziubański, J. and Zienkiewicz, J., Hardy space H1 for Schrödinger operators with compactly supported potentials, Ann. Mat. Pura Appl. 184 (2005), 315–326.
• Fefferman, C. and Stein, E. M., Hp spaces of several variables, Acta Math. 129 (1972), 137–193.
• Sikora, A., Riesz transform, Gaussian bounds and the method of wave equation, Math. Z. 247 (2004), 643–662.
• Stein, E. M., Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, Princeton, NJ, 1993.