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Localized Morrey-Campanato spaces on metric measure spaces and applications to Schrüdinger operators

Published online by Cambridge University Press:  11 January 2016

Dachun Yang
School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, People’s, Republic of
Dongyong Yang
School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, People’s, Republic of
Yuan Zhou
School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, People’s, Republic of
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Let be a space of homogeneous type in the sense of Coifman and Weiss, and let be a collection of balls in . The authors introduce the localized atomic Hardy space the localized Morrey-Campanato space and the localized Morrey-Campanato-BLO (bounded lower oscillation) space with α ∊ ℝ and p ∊ (0, ∞) , and they establish their basic properties, including and several equivalent characterizations for In particular, the authors prove that when α > 0 and p ∊ [1, ∞), then and when p ∈(0,1], then the dual space of is Let ρ be an admissible function modeled on the known auxiliary function determined by the Schrödinger operator. Denote the spaces and , respectively, by and when is determined by ρ. The authors then obtain the boundedness from of the radial and the Poisson semigroup maximal functions and the Littlewood-Paley g-function, which are defined via kernels modeled on the semigroup generated by the Schrödinger operator. These results apply in a wide range of settings, for instance, the Schrödinger operator or the degenerate Schrödinger operator on ℝd, or the sub-Laplace Schrödinger operator on Heisenberg groups or connected and simply connected nilpotent Lie groups.

Research Article
Copyright © Editorial Board of Nagoya Mathematical Journal 2010


[1] Campanato, S., Proprietà di hölderianità di alcune classi di funzioni, Ann. Sc. Norm. Super. Pisa 17 (1963), 175188.Google Scholar
[2] Coifman, R. R. and Rochberg, R., Another characterization of BMO, Proc. Amer. Math. Soc. 79 (1980), 249254.CrossRefGoogle Scholar
[3] Coifman, R. R. and Weiss, G., Analyse Harmonique Non-commutative sur Certains Espaces Homogènes, Lecture Notes in Math. 242, Springer, Berlin, 1971.Google Scholar
[4] Coifman, R. R. and Weiss, G., Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. 83 (1977), 569645.CrossRefGoogle Scholar
[5] Duong, X. T., Xiao, J., and Yan, L., Old and new Morrey spaces with heat kernel bounds, J. Fourier Anal. Appl. 13 (2007), 87111.CrossRefGoogle Scholar
[6] Dziubanski, J., Note on H1 spaces related to degenerate Schrödinger operators, Illinois J. Math. 49 (2005), 12711297.CrossRefGoogle Scholar
[7] Dziubanski, J., Garrigös, G., Martônez, T., Torrea, J. L., and Zienkiewicz, J., BMO spaces related to Schrödinger operators with potentials satisfying a reverse Holder inequality, Math. Z. 249 (2005), 329356.CrossRefGoogle Scholar
[8] Dziubanski, J. and Zienkiewicz, J., Hardy space H1 associated to Schrödinger operator with potential satisfying reverse Holder inequality, Rev. Mat. Iberoam. 15 (1999), 279296.CrossRefGoogle Scholar
[9] Dziubanski, J. and Zienkiewicz, J., Hp spaces associated with Schrödinger operators with potentials from reverse Holder classes, Colloq. Math. 98 (2003), 538.CrossRefGoogle Scholar
[10] Fefferman, C., The uncertainty principle, Bull. Amer. Math. Soc. (N.S.) 9 (1983), 129206.CrossRefGoogle Scholar
[11] Goldberg, D., A local version of real Hardy spaces, Duke Math. J. 46 (1979), 2742.CrossRefGoogle Scholar
[12] Han, Y., Müller, D., and Yang, D., A theory of Besov and Triebel-Lizorkin spaces on metric measure spaces modeled on Carnot-Carathéodory spaces, Abstr. Appl. Anal. 2008, no. 89 3409.Google Scholar
[13] Hebisch, W. and Saloff-Coste, L., On the relation between elliptic and parabolic Harnack inequalities, Ann. Inst. Fourier (Grenoble) 51 (2001), 14371481.CrossRefGoogle Scholar
[14] Hu, G., Meng, Y., and Yang, D., Estimates for Marcinkiewicz integrals in BMO and Campanato spaces, Glasg. Math. J. 49 (2007), 167187.CrossRefGoogle Scholar
[15] Hu, G., Yang, D., and Yang, D., h1, bmo, blo and Littlewood-Paley g-functions with non-doubling measures, Rev. Mat. Iberoam. 25 (2009), 595667.CrossRefGoogle Scholar
[16] Huang, J. and Liu, H., Area integrals associated to Schrödinger operators, preprint.Google Scholar
[17] Lemarié-Rieusset, P. G., The Navier-Stokes equations in the critical Morrey-Campanato space, Rev. Mat. Iberoam. 23 (2007), 897930.CrossRefGoogle Scholar
[18] Li, H., Estimations Lp des opérateurs de Schrödinger sur les groupes nilpotents, J. Funct. Anal. 161 (1999), 152218.CrossRefGoogle Scholar
[19] Lin, C. and Liu, H., The BMO-type space BMOc associated with Schrödinger operatorson the Heisenberg group, preprint.Google Scholar
[20] Macías, R. A. and Segovia, C., Lipschitz functions on spaces of homogeneous type, Adv. Math. 33 (1979), 257270.CrossRefGoogle Scholar
[21] Nagel, A., Stein, E. M., and Wainger, S., Balls and metrics defined by vector fields I. Basic properties, Acta Math. 155 (1985), 103147.CrossRefGoogle Scholar
[22] Nakai, E., The Campanato, Morrey and Holder spaces on spaces of homogeneous type, Studia Math. 176 (2006), 119.Google Scholar
[23] Nakai, E., Orlicz-Morrey spaces and the Hardy-Littlewood maximal function, Studia Math. 188 (2008), 193221.CrossRefGoogle Scholar
[24] Peetre, J., On the theory of £ptλ spaces, J. Funct. Anal. 4 (1969), 7187.CrossRefGoogle Scholar
[25] Shen, Z., Lp estimates for Schrödinger operators with certain potentials, Ann. Inst. Fourier (Grenoble) 45 (1995), 513546.CrossRefGoogle Scholar
[26] Stein, E. M., Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, Princeton, 1993.Google Scholar
[27] Strömberg, J.-O. and Torchinsky, A., Weighted Hardy Spaces, Lecture Notes in Math. 1381, Springer, Berlin, 1989.Google Scholar
[28] Taibleson, M. H. and Weiss, G., “The molecular characterization of certain Hardy spaces,” in Representation Theorems for Hardy Spaces, Astérisque 77, Soc. Math. France, Paris, 1980, 67149.Google Scholar
[29] Triebel, H., Theory of Function Spaces, Vol. II, Birkhäuser, Basel, 1992.Google Scholar
[30] Varopoulos, N. T., Analysis on Lie groups, J. Funct. Anal. 76 (1988), 346410.Google Scholar
[31] Varopoulos, N. T., Saloff-Coste, L., and Coulhon, T., Analysis and Geometry on Groups, Cambridge University Press, Cambridge, 1992.Google Scholar
[32] Yang, D., Yang, D., and Zhou, Y., Localized BMO and BLO spaces on RD-spaces and applications to Schrödinger operators, Commun. Pure Appl. Anal. 9 (2010), 779812.CrossRefGoogle Scholar
[33] Yang, D. and Zhou, Y., Localized Hardy spaces H1related to admissible functions on RD-spaces and applications to Schröodinger operators, to appear in Trans. Amer. Math. Soc.Google Scholar
[34] Zhong, J., The Sobolev estimates for some Schrödinger type operators, Math. Sci. Res. Hot-Line 3 (1999), 148.Google Scholar