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.
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Auscher P., Russ E.: Hardy spaces and divergence operators on strongly Lipschitz domain of \({\mathbb{R}^n}\) . J. Funct. Anal. 201, 148–184 (2003)
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)
Auscher P., McIntosh A., Russ E.: Hardy spaces of differential forms on Riemmannian manifolds. J. Geom. Anal. 18, 192–248 (2008)
Chang D.C.: The dual of Hardy spaces on a bounded domain in \({\mathbb{R}^n}\) . Forum. Math. 6, 65–81 (1994)
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)
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)
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)
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)
Dziubański J.: Atomic decomposition of H p spaces associated with some Schrödinger operators. Indiana Univ. Math. J. 47, 75–98 (1998)
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)
Dziubański J., Zienkiewicz J.: Hardy spaces associated with some Schrödinger operators. Studia. Math. 126, 149–160 (1997)
Dziubański J., Zienkiewicz J.: H p spaces associated with Schrödinger operators with potentials from reverse Hölder classes. Colloq. Math. 98, 5–38 (2003)
Fefferman C.: The uncertainty principle. Bull. Am. Math. Soc. (N.S.) 9, 129–206 (1983)
Fefferman C., Stein E.M.: H p spaces of several variables. Acta. Math. 129, 137–193 (1972)
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)
Huang, J.Z., Liu, H.P.: Littlewood-Paley functions associated with the Schrödinger operators. Mathematische Annalen (submitted)
Huang, J.Z., Liu, H.P.: Area integral associated with the Schrödinger operators. Indiana Univ. Math. J. (submitted)
Li H.: Estimations L p des opérateurs de Schrödinger sur les groupes nilpotents. J. Funct. Anal. 161, 152–218 (1999)
Peng, L.Z.: The dual spaces of \({\lambda_\alpha(\mathbb{R}^n)}\) . Thesis in Peking University (1981)
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)
Shen Z.: L p estimates for Schrödinger operators with certain potentials. Ann. Inst. Fourier (Grenoble) 45, 513–546 (1995)
Reed M., Simon B.: Methods of Modern Mathematical Physics I: Functional Analysis. Academic Press, New York (1972)
Stein E.M.: Topics in Harmonic Analysis, Annals of Math Study. Princeton University Press, Princeton (1970)
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)
Zhong, J.: Harmonic analysis for some Schrödinger type operators. Ph.D. Thesis, Princeton University, Princeton (1993)
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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
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DOI: https://doi.org/10.1007/s00209-009-0558-z