Taiwanese Journal of Mathematics

WEIGHTED HARDY SPACES ASSOCIATED TO SELF-ADJOINT OPERATORS AND $BMO_{L,w}$

Suying Liu, Kai Zhao, and Shujuan Zhou

Full-text: Open access

Abstract

Let $L$ be a non-negative self-adjoint operator satisfying a pointwise Guassian estimate for its heat kernel. Let $w$ be some $A_s$ weight on $\mathbb{R}^n$. In this paper, we obtain a weighted $(p,q)-$atomic decomposition with $q\geq s$ for the weighted Hardy spaces $H^p_{L,w}(\mathbb{R}^n)$, $0\lt p\leq 1$. We also introduce the suitable weighted BMO spaces $BMO^p_{L,w}$. Then the duality between $H^1_{L,w}(\mathbb{R}^n)$ and $BMO_{L,w}$ is established.

Article information

Source
Taiwanese J. Math., Volume 18, Number 5 (2014), 1663-1678.

Dates
First available in Project Euclid: 10 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1499706532

Digital Object Identifier
doi:10.11650/tjm.18.2014.3759

Mathematical Reviews number (MathSciNet)
MR3265083

Zentralblatt MATH identifier
1357.42009

Subjects
Primary: 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.) 42B30: $H^p$-spaces
Secondary: 47F05: Partial differential operators [See also 35Pxx, 58Jxx] (should also be assigned at least one other classification number in section 47)

Keywords
weighted Hardy space self-adjoint operator weighted atom $BMO_{L,w}$

Citation

Liu, Suying; Zhao, Kai; Zhou, Shujuan. WEIGHTED HARDY SPACES ASSOCIATED TO SELF-ADJOINT OPERATORS AND $BMO_{L,w}$. Taiwanese J. Math. 18 (2014), no. 5, 1663--1678. doi:10.11650/tjm.18.2014.3759. https://projecteuclid.org/euclid.twjm/1499706532


Export citation

References

  • P. Auscher, X. T. Duong and A. McIntosh, Boundedness of Banach Space Valued Singular Integral Operators and Hardy Spaces, Unpublished preprint, 2005.
  • P. Auscher, A. McIntosh and E. Russ, Hardy spaces of differential forms on Riemannian manifolds, J. Geom. Anal., 18 (2008), 192-248.
  • P. Auscher and E. Russ, Hardy spaces and divergence operators on strongly Lipschitz domain of ${\mathbb R}^n$, J. Funct. Anal., 201 (2003), 148-184.
  • A. Bui and X. T. Duong, Weighted Hardy Spaces Associated to Operators and Boundedness of Singular Integrals, arXiv:1202.2063.
  • A. Bui and X. T. Duong, Weighted BMO Spaces Associated to Operators, arXiv: 1201.5828.
  • J. Cheeger, M. Cromov and M. Taylor, Finite propagation speed, kernel estimates for functions of the Laplacian and the geometry of complete Riemannian manifolds, J. Differential Geom., 17 (1982), 15-53.
  • R. R. Coifman, Y. Meyer and E. M. Stein, Some new functions and their applications to harmonic analysis, J. Funct. Anal., 62 (1985), 304-335.
  • T. Coulhon and A. Sikora, Gaussian heat kernel upper bounds via Phragmén-Lindelöf theorem, Proc. Lond. Math., 96 (2008), 507-544.
  • E. B. Davies, Heat Kernels and Spectral Theory, Cambridge Univ. Press, 1989.
  • J. Duoandikoetxea, Fourier Analysis, Grad. Stud. Math., 29, American Math. Soc., Providence, 2000.
  • X. T. Duong, E. M. Ouhabaz and A. Sikora, Plancherel-type estimates and sharp spectral multipliers, J. Funct. Anal., 196 (2002), 443-485.
  • X. T. Duong and L. X. Yan, New function spaces of BMO type, John-Nirenberg inequality, interpolation and applications, Comm. Pure Appl. Math., 58 (2005), 1375-1420.
  • X. T. Duong and L. X. Yan, Duality of Hardy and BMO spaces associated with operators with heat kernel bounds, J. Amer. Math. Soc., 18 (2005), 943-973.
  • C. Fefferman and E. M. Stein, $H^p$ spaces of several variables, Acta Math., 129 (1972), 137-195.
  • J. Garcia-Cuerva, Weighted $H^p$ spaces, Dissertationes Math., 162 (1979), 1-63.
  • J. Garcia-Cuerva and J. Rubio de Francia, Weighted Norm Inequalities and Related Topics, North-Holland, Amsterdam, 1985.
  • R. M. Gong and L. X. Yan, Weighted $L^p$ Estimates for the Area Integral Associated to Self-adjoint Operators, arxiv:1109.1662v1.
  • S. Hofmann, G. Lu, D. Mitrea, M. Mitrea and L. Yan, Hardy spaces associated to non-negative self-adjoint operators satisfying Davies-Gaffeney estimates, Memoirs Amer. Math. Soc., 214 (2011), no. 1007, 78 pp.
  • S. Hofmann and S. Mayboroda, Hardy and BMO spaces assocated to divergence form elliptic operators, J. Funct. Anal., 258 (2010), 1167-1224.
  • R. Jiang and D. C. Yang, New Orlicz-Hardy spaces associated to divergence form elliptic operators, Math. Ann., 344 (2009), 37-116.
  • S. Y. Liu and L. Song, An atomic decomposition of weighted Hardy spaces associated to self-adjoint operators, J. Funct. Anal., 265 (2013), 2709-2723.
  • A. McIntosh, Operators which have an $H_{\infty}$-calculus, Miniconference on Operator Theory and Partial Differential Equations (North Ryde, 1986), 210-231, Proceedings of the Centre for Mathematical Analysis, ANU, Canberra, 14 (1986).
  • E. M. Ouhabaz, Analysis of Heat Equations on Domains, London Math. Soc. Monographs, Vol. 31, Princeton Univ. Press, 2005.
  • L. Song and L. X. Yan, Riesz transforms associated to Schorödinger operators on weighted Hardy spaces, J. Funct. Anal., 259 (2010), 1466-1490.
  • E. M. Stein, Harmonic Analysis: Real Variable Methods, Orthogonality and Oscillatory Integrals, Princeton Univ. Press, Princeton, NJ, 1993.
  • E. M. Stein and G. Weiss, On the theory of harmonic functions of several variables. I. The theory of $H^{p}$-spaces, Acta Math., 103 (1960) 25-62.
  • J.-O. Strömberg and A. Torchinsky, Weighted Hardy Spaces, Lecture Notes in Mathematics, 1381 Springer-Verlag, Berlin/New York, 1989.
  • K. Yosida, Functional Analysis (Fifth edition), Spring-Verlag, Berlin, 1978.