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A local version of Hardy spaces associated with operators on metric spaces

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Abstract

Let (X, d, µ) be a metric measure space endowed with a distance d and a nonnegative Borel doubling measure µ. Let L be a second order self-adjoint positive operator on L 2(X). Assume that the semigroup e−tL generated by −L satisfies the Gaussian upper bounds on L 2(X). In this article we study a local version of Hardy space h 1 L (X) associated with L in terms of the area function characterization, and prove their atomic characters. Furthermore, we introduce a Moser type local boundedness condition for L, and then we apply this condition to show that the space h 1 L (X) can be characterized in terms of the Littlewood-Paley function. Finally, a broad class of applications of these results is described.

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References

  1. Arendt W, ter Elst A F M. Gaussian estimates for second order elliptic operators with boundary conditions. J Operator Theory, 1997, 38: 87–130

    MathSciNet  MATH  Google Scholar 

  2. Auscher P, Ben Ali B. Maximal inequalities and Riesz transform estimates on Lp spaces for Schrödinger operators with non-negative potentials. Ann Inst Fourier, 2007, 57: 1975–2013

    Article  MathSciNet  MATH  Google Scholar 

  3. Auscher P, McIntosh A, Russ E. Hardy spaces of differential forms on Riemannian manifolds. J Geom Anal, 2008, 18: 192–248

    Article  MathSciNet  MATH  Google Scholar 

  4. Auscher P, Russ E. Hardy spaces and divergence operators on strongly Lipschitz domain of ℝn. J Funct Anal, 2003, 201: 148–184

    Article  MathSciNet  MATH  Google Scholar 

  5. Christ M. AT(b) theorem with remarks on analytic capacity and the Cauchy integral. Colloq Math, 1990, LX/LXI: 601–628

    MathSciNet  Google Scholar 

  6. Coifman R, Weiss G. Analyse harmonique non-commutative sur certains espaces homogènes. In: Lecture Notes in Mathematics, vol. 242. Berlin-New York: Springer, 1971

    MATH  Google Scholar 

  7. Coulhon T, Sikora A. Gaussian heat kernel upper bounds via Phragmén-Lindelöf theorem. Proc Lond Math Soc, 2008, 96: 507–544

    Article  MathSciNet  MATH  Google Scholar 

  8. Duong X T, Hofmann S, Mitrea D, et al. Hardy spaces and regularity for the inhomogeneous Dirichlet and Neumann problems. In Rev Mat Iberoamericana, in press

  9. Duong X T, Yan L X. Duality of Hardy and BMO spaces associated with operators with heat kernel bounds. J Amer Math Soc, 2005, 18: 943–973

    Article  MathSciNet  MATH  Google Scholar 

  10. Dziubański J, Zienkiewicz J. Hardy space H 1 associated to Schrödinger operators with potential satisfying reverse Hölder inequality. Rev Mat Iberoamericana, 1999, 15: 279–296

    Article  MathSciNet  MATH  Google Scholar 

  11. Goldberg D. A local version of real Hardy spaces. Duke Math J, 1979, 46: 27–42

    Article  MathSciNet  MATH  Google Scholar 

  12. Gong R M, Yan L X. Littlewood-Paley and spectral multipliers on weighted L p spaces. Submitted, 2010

  13. Hofmann S, Lu G Z, Mitrea D, et al. Hardy spaces associated to non-negative self-adjoint operators satisfying Davies-Gaffney estimates. Mem Amer Math Soc, 2011, 214: 1007

    MathSciNet  Google Scholar 

  14. Hofmann S, Mayboroda S. Hardy and BMO spaces associated to divergence form elliptic operators. Math Ann, 2009, 344: 37–116

    Article  MathSciNet  MATH  Google Scholar 

  15. Jiang R J, Yang D C, Zhou Y. Localized Hardy spaces associated with operators. Appl Anal, 2009, 88: 1409–1427

    Article  MathSciNet  MATH  Google Scholar 

  16. Macías R A, Segovia C. A decomposition into atoms of distributions on spaces of homogeneous type. Adv Math, 1979, 33: 271–309

    Article  MATH  Google Scholar 

  17. Ouhabaz E M. Analysis of heat equations on domains. In: London Mathematical Society Monographs Series, vol. 31. Princeton, NJ: Princeton University Press, 2005

    Google Scholar 

  18. Russ E. The atomic decomposition for tent spaces on spaces of homogeneous type. In: Asymptotic Geometric Analysis, Harmonic Analysis, and Related Topics, 125–135. Proceedings of the Centre for Mathematical Analysis, Australian National University, 42. Canberra: Australian National University, 2007

    Google Scholar 

  19. Saloff-Coste L. Aspects of Sobolev-type inequalities. In: London Mathematical Society Lecture Note Series, vol. 289. Cambridge: Cambridge University Press, 2002

    Google Scholar 

  20. Sikora A. Riesz transform, Gaussian bounds and the method of wave equation. Math Z, 2004, 247: 643–662

    Article  MathSciNet  MATH  Google Scholar 

  21. Taylor M. Hardy spaces and bmo on manifolds with bounded geometry. J Geom Anal, 2009, 19: 137–190

    Article  MathSciNet  MATH  Google Scholar 

  22. Varopoulos N, Saloff-Coste L, Coulhon T. Analysis and Geometry on Groups. Cambridge: Cambridge University Press, 1993

    Book  Google Scholar 

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Authors and Affiliations

  1. School of Mathematics and Information Science, Guangzhou University, Guangzhou, 510006, China

    RuMing Gong

  2. Key Laboratory of Mathematics and Interdisciplinary Sciences of Guangdong Higher Education Institutes, Guangzhou University, Guangzhou, 510006, China

    RuMing Gong

  3. Department of Mathematics, Sun Yat-sen University, Guangzhou, 510275, China

    Ji Li & LiXin Yan

Authors
  1. RuMing Gong
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  2. Ji Li
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  3. LiXin Yan
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Correspondence to RuMing Gong.

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Gong, R., Li, J. & Yan, L. A local version of Hardy spaces associated with operators on metric spaces. Sci. China Math. 56, 315–330 (2013). https://doi.org/10.1007/s11425-012-4428-5

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