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Notes on boundedness of spectral multipliers on Hardy spaces associated to operators

Published online by Cambridge University Press:  11 January 2016

Bui The Anh
Bui The Anh*
Affiliation:
Department of Mathematics, University of Pedagogy, HoChiMinh City, [email protected]
*
Department of Mathematics, Macquarie University, NSW 2109, Australia[email protected]
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Abstract

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Let L be a nonnegative self-adjoint operator on L2 (X), where X is a space of homogeneous type. Assume that L generates an analytic semigroup e–tl whose kernel satisfies the standard Gaussian upper bounds. We prove that the spectral multiplier F(L) is bounded on for 0 < p < 1, the Hardy space associated to operator L, when F is a suitable function.

Keywords

34L0530H10
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 203 , September 2011 , pp. 109 - 122
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2011

References

[A1] Alexopoulos, G., Spectral multipliers on Lie groups of polynomial growth, Proc. Amer. Math. Soc. 120 (1994), 973–979.Google Scholar
[A2] Alexopoulos, G., Spectral multipliers for Markov chains, J. Math. Soc. Japan 56 (2004), 833–852.Google Scholar
[B] Blunck, S., A H¨ormander-type spectral multiplier theorem for operators without heat kernel, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 2 (2003), 449–459.Google Scholar
[C] Christ, M., Lp bounds for spectral multipliers on nilpotent groups, Trans. Amer. Math. Soc. 328 (1991), 73–81.Google Scholar
[CS] Cowling, M. and Sikora, A., A spectral multiplier theorem for a sublaplacian on SU(2), Math. Z. 238 (2001), 1–36.Google Scholar
[CW] Coifman, R. and Weiss, G., Analyse harmonique non-commutative sur certains espaces homogènes, Lecture Notes in Math. 242, Springer, Berlin, 1971.Google Scholar
[DeM] De, , Michele, L. and Mauceri, G., Hp multpliers on stratified groups, Ann. Mat. Pura Appl. 148 (1987), 353–366.Google Scholar
[D] Duong, X. T., From the L 1 norms of the complex heat kernels to a H¨ormander multiplier theorem for sub-Laplacians on nipotent Lie groups, Pacific J. Math. 173 (1996), 413–424.CrossRefGoogle Scholar
[DL] Duong, X. T. and Li, J., Hardy spaces associated to operators satisfying bounded H∞ functional calculus and Davies-Gaffney estimates, preprint, 2009.Google Scholar
[DOS] Duong, X. T., Ouhabaz, E. M., and Sikora, A., Plancherel-type estimates and sharp spectral multipliers, J. Funct. Anal. 196 (2002), 443–485.Google Scholar
[DP] Dziubański, J. and Preisner, M., Remarks on spectral multiplier theorems on Hardy spaces associated with semigroups of operators, Rev. Un. Mat. Argentina 50 (2009), 201–215.Google Scholar
[FS] Folland, G. B. and Stein, E. M., Hardy Spaces on Homogeneous Groups, Math. Notes 28, Princeton University Press, Princeton, 1982.Google Scholar
[He] Hebisch, W., A multiplier theorem for Schr¨odinger operators, Colloq. Math. 60/61 (1990), 659–664.CrossRefGoogle Scholar
[HLMMY] Hofmann, S., Lu, G., Mitrea, D., Mitrea, M., and Yan, L., Hardy spaces associated to nonnegative self-adjoint operators satisfying Davies-Gaffney estimates, preprint http://www.ams.org/journals/memo/0000–000-00/ (accessed 7 July 2011).Google Scholar
[HM] Hofmann, S. and Mayboroda, S., Hardy and BMO spaces associated to divergence form elliptic operators, Math. Ann. 344 (2009), 37–116.Google Scholar
[Ho] H¨ormander, L., The spectral function of an elliptic operator, Acta Math. 121 (1968), 193–218.CrossRefGoogle Scholar
[MSt] Müller, D. and Stein, E. M., On spectral multipliers for Heisenberg and related groups, J. Math. Pures Appl. 73 (1994), 413–440.Google Scholar