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AN ATOMIC DECOMPOSITION FOR HARDY SPACES ASSOCIATED TO SCHRÖDINGER OPERATORS

Part of: Partial differential operators Harmonic analysis in several variables

Published online by Cambridge University Press:  28 September 2011

LIANG SONG ,
CHAOQIANG TAN  and
LIXIN YAN
LIANG SONG
Affiliation:
Department of Mathematics, Sun Yat-sen (Zhongshan) University, Guangzhou 510275, PR China (email: songl@mail.sysu.edu.cn)
CHAOQIANG TAN*
Affiliation:
Department of Mathematics, Shantou University, Shantou Guangdong 515063, PR China (email: cqtan@stu.edu.cn)
LIXIN YAN
Affiliation:
Department of Mathematics, Sun Yat-sen (Zhongshan) University, Guangzhou 510275, PR China (email: mcsylx@mail.sysu.edu.cn)
*
✉For correspondence; e-mail: cqtan@stu.edu.cn
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Abstract

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Let L=−Δ+V be a Schrödinger operator on ℝn where V is a nonnegative function in the space L1loc(ℝn) of locally integrable functions on ℝn. In this paper we provide an atomic decomposition for the Hardy space H1L(ℝn) associated to L in terms of the maximal function characterization. We then adapt our argument to give an atomic decomposition for the Hardy space H1L(ℝn×ℝn) on product domains.

Keywords

Hardy spaceSchrödinger operatoratomnontangential maximal functionproduct spaces

MSC classification

Secondary: 42B30: $H^p$-spaces 47F05: Partial differential operators (should also be assigned at least one other classification number in section 47)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 91 , Issue 1 , August 2011 , pp. 125 - 144
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2011

Footnotes

L. Song is supported by NNSF of China (No. 11001276 and 10926136) and NSF of Guangdong (No. 9451027501002491). C. Q. Tan is supported by NNSF of China (No. 11026215), NSF of Guangdong (No. 10451503101006384) and Specialized Research Fund for the Doctoral Program of Higher Education (No. 20104402120002). L. X. Yan is supported by NNSF of China (No. 10771221 and 10925106).

References

[1]Auscher, P., McIntosh, A. and Russ, E., ‘Hardy spaces of differential forms on Riemannian manifolds’, J. Geom. Anal. 18 (2008), 192–248.Google Scholar
[2]Auscher, P. and Russ, E., ‘Hardy spaces and divergence operators on strongly Lipschitz domain of ℝn’, J. Funct. Anal. 201 (2003), 148–184.Google Scholar
[3]Bernicot, F. and Zhao, J., ‘New abstract Hardy spaces’, J. Funct. Anal. 255 (2008), 1761–1796.CrossRefGoogle Scholar
[4]Chang, S.-Y. A. and Fefferman, R., ‘A continuous version of duality of H 1 with BMO on the bidisc’, Ann. of Math. (2) 112 (1980), 179–201.Google Scholar
[5]Chang, S.-Y. A. and Fefferman, R., ‘The Calderón–Zygmund decomposition on product domains’, Amer. J. Math. 104 (1982), 445–468.CrossRefGoogle Scholar
[6]Cheeger, J., Gromov, M. and Taylor, M., ‘Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds’, J. Differential Geom. 17 (1982), 15–53.Google Scholar
[7]Coifman, R., ‘A real-variable characterization of H p’, Studia Math. 51 (1974), 269–274.CrossRefGoogle Scholar
[8]Córdoba, A. and Fefferman, R., ‘A geometric proof of the strong maximal theorem’, Ann. of Math. (2) 102 (1975), 95–100.Google Scholar
[9]Coulhon, T. and Sikora, A., ‘Gaussian heat kernel upper bounds via Phragmén–Lindelöf theorem’, Proc. Lond. Math. Soc. 96 (2008), 507–544.CrossRefGoogle Scholar
[10]Davies, E. B., Heat Kernels and Spectral Theory (Cambridge University Press, Cambridge, 1989).CrossRefGoogle Scholar
[11]Deng, D. G., Song, L., Tan, C. Q. and Yan, L. X., ‘Duality of Hardy and BMO spaces associated with operators with heat kernel bounds on product domains’, J. Geom. Anal. 17 (2007), 455–483.CrossRefGoogle Scholar
[12]Duong, X. T. and Yan, L. X., ‘Duality of Hardy and BMO spaces associated with operators with heat kernel bounds’, J. Amer. Math. Soc. 18 (2005), 943–973.Google Scholar
[13]Dziubański, J. and Zienkiewicz, J., ‘Hardy spaces H 1 associated with Schrödinger operator with potential satisfying reverse Hölder inequality’, Rev. Mat. Iberoam. 15 (1999), 279–296.Google Scholar
[14]Fefferman, R., ‘Calderón–Zygmund theory for product domains: H p spaces’, Proc. Natl. Acad. Sci. USA 83 (1986), 840–843.Google Scholar
[15]Fefferman, C. and Stein, E. M., ‘H p spaces of several variables’, Acta Math. 129 (1972), 137–193.Google Scholar
[16]Goldberg, D., ‘A local version of real Hardy spaces’, Duke Math. J. 46 (1979), 27–42.Google Scholar
[17]Hofmann, S., Lu, G. Z., Mitrea, D., Mitrea, M. and Yan, L. X., ‘Hardy spaces associated to nonnegative self-adjoint operators satisfying Davies–Gaffney estimates’, Mem. Amer. Math. Soc., to appear.Google Scholar
[18]Hofmann, S. and Mayboroda, S., ‘Hardy and BMO spaces associated to divergence form elliptic operators’, Math. Ann. 344 (2009), 37–116.Google Scholar
[19]Jessen, B., Marcinkiewicz, J. and Zygmund, A., ‘Note on the differentiability of multiple integrals’, Fund. Math. 25 (1935), 217–234.Google Scholar
[20]Journé, J. L., ‘A covering lemma for product spaces’, Proc. Amer. Math. Soc. 96 (1986), 593–598.Google Scholar
[21]Latter, R. H., ‘A characterization of H p(ℝn) in terms of atoms’, Studia Math. 62 (1978), 93–101.CrossRefGoogle Scholar
[22]McIntosh, A., ‘Operators which have an H ∞ functional calculus’, in: Miniconference on Operator Theory and Partial Differential Equations, 1986, Proceedings of the Centre for Mathematical Analysis, 14 (Australian National University, Canberra, 1986), pp. 210–231.Google Scholar
[23]Merryfield, K., ‘On the area integral, Carleson measures and H p in the polydisc’, Indiana Univ. Math. J. 34 (1985), 663–685.Google Scholar
[24]Ouhabaz, E. M., Analysis of Heat Equations on Domains, London Mathematical Society Monographs Series, 31 (Princeton University Press, Princeton, NJ, 2005).Google Scholar
[25]Simon, B., ‘Maximal and minimal Schrödinger forms’, J. Operator Theory 1 (1979), 37–47.Google Scholar
[26]Stein, E. M. and Weiss, G., ‘On the theory of harmonic functions of several variables, I: the theory of H p spaces’, Acta Math. 103 (1960), 25–62.CrossRefGoogle Scholar
[27]Wilson, J. M., ‘On the atomic decomposition for Hardy spaces’, Pacific J. Math. 116 (1985), 201–207.Google Scholar