Taiwanese Journal of Mathematics


The Anh Bui, Jun Cao, Luong Dang Ky, Dachun Yang, and Sibei Yang

Full-text: Open access


Let $L$ be a nonnegative self-adjoint operator on $L^2(\mathbb{R}^n)$ satisfying the reinforced $(p_L, p_L')$ off-diagonal estimates, where $p_L\in[1,2)$ and $p_L'$ denotes its conjugate exponent. Assume that $p\in(0,1]$ and the weight $w$ satisfies the reverse Hölder inequality of order $(p'_L/p)'$. In particular, if the heat kernels of the semigroups $\{e^{-tL}\}_{t\gt 0}$ satisfy the Gaussian upper bounds, then $p_L=1$ and hence $w\in A_\infty({\mathbb R}^n)$. In this paper, the authors introduce the weighted Hardy spaces $H^p_{L,\,w}(\mathbb{R}^n)$ associated with the operator $L$, via the Lusin area function associated with the heat semigroup generated by $L$. Characterizations of $H^p_{L,\,w}(\mathbb{R}^n)$, in terms of the atom and the molecule, are obtained. As applications, the boundedness of singular integrals such as spectral multipliers, square functions and Riesz transforms on weighted Hardy spaces $H^p_{L,\,w}(\mathbb{R}^n)$ are investigated. Even for the Schrödinger operator $-\Delta+V$ with $0\le V\in L_{\rm{loc}}^1 (\mathbb{R}^n)$, the obtained results in this paper essentially improve the known results by extending the narrow range of the weights into the whole $A_\infty(\mathbb{R}^n)$ weights.

Article information

Taiwanese J. Math., Volume 17, Number 4 (2013), 1127-1166.

First available in Project Euclid: 10 July 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 42B30: $H^p$-spaces
Secondary: 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.) 42B35: Function spaces arising in harmonic analysis 42B25: Maximal functions, Littlewood-Paley theory 42B15: Multipliers 47B06: Riesz operators; eigenvalue distributions; approximation numbers, s- numbers, Kolmogorov numbers, entropy numbers, etc. of operators

weighted Hardy space atom molecule reinforced off-diagonal estimate singular integral spectral multiplier Riesz transform


Bui, The Anh; Cao, Jun; Ky, Luong Dang; Yang, Dachun; Yang, Sibei. WEIGHTED HARDY SPACES ASSOCIATED WITH OPERATORS SATISFYING REINFORCED OFF-DIAGONAL ESTIMATES. Taiwanese J. Math. 17 (2013), no. 4, 1127--1166. doi:10.11650/tjm.17.2013.2719. https://projecteuclid.org/euclid.twjm/1499706109

Export citation


  • P. Auscher, On necessary and sufficient conditions for $L^p$-estimates of Riesz transforms associated to elliptic operators on $\RR^n$ and related estimates, Mem. Amer. Math. Soc., 186 (2007), no. 871, xviii+75 pp.
  • P. Auscher and B. Ben Ali, Maximal inequalities and Riesz transform estimates on $L^p$ spaces for Schrödinger operators with nonnegative potentials, Ann. Inst. Fourier $($Grenoble$)$, 57 (2007), 1975-2013.
  • P. Auscher and J. Martell, Weighted norm inequalities, off-diagonal estimates and elliptic operators. II. Off-diagonal estimates on spaces of homogeneous type, J. Evol. Equ., 7 (2007), 265-316.
  • P. Auscher, X. T. Duong and A. McIntosh, Boundedness of Banach space valued singular integral operators and Hardy spaces, Unpublished manuscript.
  • P. Auscher, A. McIntosh and E. Russ, Hardy spaces of differential forms on Riemannian manifolds, J. Geom. Anal., 18 (2008), 192-248.
  • S. Blunck and P. Kunstmann, Generalized Gaussian estimates and the Legendre transform, J. Operator Theory, 53 (2005), 351-365.
  • T. A. Bui, Weighted norm inequalities for spectral multipliers without Gaussian estimates, preprint.
  • T. A. Bui and X. T. Duong, Weighted Hardy spaces associated to operators and the boundedness of singular integrals, preprint.
  • T. A. Bui and J. Li, Orlicz-Hardy spaces associated to operators satisfying bounded $H_\vc$ functional calculus and Davies-Gaffney estimates, J. Math. Anal. Appl., 373 (2011), 485-501.
  • J. Cao, D.-C. Chang, D. Yang and S. Yang, Boundedness of generalized Riesz transforms on Orlicz-Hardy spaces associated to operators, Submitted.
  • R. R. Coifman, A real variable characterization of $H^p$, Studia Math., 51 (1974), 269-274.
  • S. Chanillo and R. L. Wheeden, Some weighted norm inequalities for the area integral, Indiana Univ. Math. J., 36 (1987), 277-294.
  • 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.
  • R. R. Coifman and G. Weiss, Analyse Harmonique Non-commutative sur Certains Espaces Homog\ptmrs ènes, Lecture Notes in Math. 242, Springer, Berlin, 1971.
  • R. R. Coifman and G. Weiss, Extensions of Hardy spaces and their use on analysis, Bull. Amer. Math. Soc., 83 (1977), 569-645.
  • D. Cruz-Uribe and C. J. Neugebauer, The structure of the reverse Hölder classes, Trans. Amer. Math. Soc., 347 (1995), 2941-2960.
  • T. Coulhon and X. T. Duong, Riesz transforms for $1\leq p\leq 2$, Trans. Amer. Math. Soc., 351 (1999), 1151-1169.
  • T. Coulhon and A. Sikora, Gaussian heat kernel upper bounds via the Phragmén-Lindelöf theorem, Proc. Lond. Math. Soc., 96 (2008), 507-544.
  • E. B. Davies, Uniformly elliptic operators with measurable coefficients, J. Funct. Anal., 132 (1995), 141-169.
  • J. Duoandikoetxea, Fourier Analysis, Grad. Stud. Math., Vol. 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. Yan, New function spaces of BMO type, the John-Nirenberg inequality, interpolation, and applications, Comm. Pure Appl. Math., 58 (2005), 1375-1420.
  • X. T. Duong and L. Yan, Duality of Hardy and BMO spaces associated with operators with heat kernel bounds, J. Amer. Math. Soc., 18 (2005), 943-973.
  • X. T. Duong and L. Yan, Spectral multipliers for Hardy spaces associated to nonnegative self-adjoint operators satisfying Davies-Gaffney estimates, J. Math. Soc. Japan, 63 (2011), 295-319.
  • J. Dziubański and M. Preisner, Remarks on spectral multiplier theorems on Hardy spaces associated with semigroups of operators, Revista de la unión Matemática Argentina, 50 (2009), 201-215.
  • C. Fefferman and E. M. Stein, $H^p$ spaces of several variables, Acta Math., 129 (1972), 137-193.
  • M. Gaffney, The conservation property of the heat equation on Riemannian manifolds, Comm. Pure Appl. Math., 12 (1959), 1-11.
  • J. Garc\ptmrs ía-Cuerva, Weighted $H^p$ spaces, Dissertationes Math. $($Rozprawy Mat.$)$, 162 (1979), 1-63.
  • E. Harboure, O. Salinas and B. Viviani, A look at BMO$_\varphi(w)$ through Carleson measures, J. Fourier Anal. Appl., 13 (2007), 267-284.
  • S. Hofmann, G. Lu, D. Mitrea, M. Mitrea and L. Yan, Hardy spaces associated to non-negative self-adjoint operators satisfying Davies-Gaffney estimates, Mem. Amer. Math. Soc., 214 (2011), no. 1007, vi+78 pp.
  • S. Hofmann and J. M. Martell, $L^p$ bounds for Riesz transforms and square roots associated to second order elliptic operators, Publ. Mat., 47 (2003), 497-515.
  • S. Hofmann and S. Mayboroda, Hardy and BMO spaces associated to divergence form elliptic operators, Math. Ann., 344 (2009), 37-116.
  • S. Hofmann, S. Mayboroda and A. McIntosh, Second order elliptic operators with complex bounded measurable coefficients in $L^p$, Sobolev and Hardy spaces, Ann. Sci. École Norm. Sup. (4), 44 (2011), 723-800.
  • R. Jiang and D. Yang, Orlicz-Hardy spaces associated with operators satisfying Davies-Gaffney estimates, Commun. Contemp. Math., 13 (2011), 331-373.
  • R. Jiang and D. Yang, New Orlicz-Hardy spaces associated with divergence form elliptic operators, J. Funct. Anal., 258 (2010), 1167-1224.
  • R. Johnson and C. J. Neugebauer, Homeomorphisms preserving $A_p$, Rev. Mat. Ibero., 3 (1987), 249-273.
  • R. Latter, A characterization of $H^p(\mathbb R^n)$ in terms of atoms, Studia Math., 62 (1978), 93-101.
  • G. Mauceri and S. Meda, Vector-valued multipliers on stratified groups, Rev. Mat. Ibero., 6 (1990), 141-154.
  • A. McIntosh, Operators which have an $H_\infty$-calculus, Miniconference on operator theory and partial differential equations, Proc. Centre Math. Analysis, ANU, Canberra, 14 (1986), 210-231.
  • B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc., 165 (1972), 207-226.
  • E. M. Ouhabaz, Analysis of Heat Equations on Domains, Princeton University Press, Princeton, N. J., 2005.
  • L. Song and L. Yan, Riesz transforms associated to Schrödinger operators on weighted Hardy spaces, J. Funct. Anal., 259 (2010), 1466-1490.
  • 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.
  • H. Wang, Riesz transforms associated with Schrödinger operators acting on weighted Hardy spaces, arXiv: 1102.5467.
  • D. Yang and S. Yang, Musielak-Orlicz Hardy spaces associated with operators and their applications, J. Geom. Anal., (2012), DOI 10.1007/s12220-012-9344-y or arXiv: 1201.5512.