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Hardy spaces of spaces of homogeneous type


Authors: Xuan Thinh Duong and Lixin Yan
Journal: Proc. Amer. Math. Soc. 131 (2003), 3181-3189
MSC (2000): Primary 42B20, 42B30, 47G10
DOI: https://doi.org/10.1090/S0002-9939-03-06868-0
Published electronically: February 14, 2003
MathSciNet review: 1992859
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Abstract: Let $X$ be a space of homogeneous type, and $L$ be the generator of a semigroup with Gaussian kernel bounds on $L^2(X)$. We define the Hardy spaces $H^p_s(X)$ of $X$ for a range of $p$, by means of area integral function associated with the Poisson semigroup of $L$, which is proved to coincide with the usual atomic Hardy spaces $H^p_{at}(X)$ on spaces of homogeneous type.


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  • [AR] P. Auscher and E. Russ, Hardy spaces and divergence operators on strongly Lipschitz domain of ${\mathbb R}^n$, preprint.
  • [CD] T. Coulhon and X.T. Duong, Maximal regularity and kernel bounds: observations on a theorem by Hieber and Prüss, Adv. Differential Equations, 5 (2000), 343-368. MR 2001d:34087
  • [CF] S.Y.A. Chang and R. Fefferman, The Calderón-Zygmund decomposition on product domain, Amer. J. Math., 104 (1982), 445-468. MR 84a:42028
  • [Ch] M. Christ, A Tb theorem with remarks on analytic capacity and the Cauchy integral, Colloq. Math., 61 (1990), 601-628. MR 92k:42020
  • [CKS] D-C. Chang, S.G. Krantz, and E.M. Stein, $H^p$ theory on a smooth domain in ${\mathbb R}^n$ and elliptic boundary value problems, J. Funct. Anal., 114 (1993), 286-347. MR 94j:46032
  • [CW] R. Coifman and G. Weiss, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc., 83 (1977), 569-645. MR 56:6264
  • [D] E.B. Davies, Heat Kernels and Spectral Theorey, Cambridge Univ. Press, 1989. MR 90e:35123
  • [DR] X.T. Duong and D.W. Robinson, Semigroup kernels, Poisson bounds, and holomorphic functional calculus, J. Funct. Anal., 142 (1996), 89-128. MR 97j:47056
  • [FeS] C. Fefferman and E.M. Stein, $H^p$ spaces of several variables, Acta Math., 129 (1972), 137-195. MR 56:6263
  • [FoS] G.B. Folland and E.M. Stein, Hardy spaces on homogeneous group, Math. Notes, 28, Princeton University Press, (1982). MR 84h:43027
  • [HS] Y.S. Han, E. Sawyer, Littlewood-Paley theory on spaces of homogeneous type and the classical function spaces, Mem. Amer. Math. Soc. 110 (1994), no. 530. MR 96a:42016
  • [Mc] A. McIntosh, Operators which have an $H_{\infty}$-calculus, Miniconference on Operator Theory and Partial Differential Equations (Proc. Centre Math. Analysis, 14, A.N.U., Canberra, 1986), 210-231. MR 88k:47019
  • [MS] R.A Maciac and C. Segovia, Lipschitz functions on spaces of homogeneous type, Adv. in Math., 33 (1979), 257-270.
  • [Sc] L. Saloff-Coste, Analyse sur les groupes de Lie à croissance polynômiale, Arkiv för Mat., 28 (1990), 315-331. MR 92d:22014
  • [Si] A. Sikora, Sharp pointwise estimates on heat kernels, Quart. J. Math. Oxford, 47 (1996), 371-382. MR 97m:58189

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Additional Information

Xuan Thinh Duong
Affiliation: Department of Mathematics, Macquarie University, New South Wales 2109, Australia
Email: duong@ics.mq.edu.au

Lixin Yan
Affiliation: Department of Mathematics, Macquarie University, New South Wales 2109, Australia – and – Department of Mathematics, Zhongshan University, Guangzhou, 10275, People’s Republic of China
Email: lixin@ics.mq.edu.au

DOI: https://doi.org/10.1090/S0002-9939-03-06868-0
Keywords: Spaces of homogeneous type, Hardy spaces, semigroup, Calder\'on-type reproducing formula, atomic decomposition
Received by editor(s): January 24, 2002
Received by editor(s) in revised form: May 16, 2002
Published electronically: February 14, 2003
Additional Notes: Both authors were partially supported by a grant from Australia Research Council, and the second author was also partially supported by the NSF of China
Communicated by: Andreas Seeger
Article copyright: © Copyright 2003 American Mathematical Society

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