Kyoto Journal of Mathematics

Vanishing mean oscillation spaces associated with operators satisfying Davies–Gaffney estimates

Yiyu Liang, Dachun Yang, and Wen Yuan

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Abstract

Let (X,d,μ) be a metric measure space, let L be a linear operator that has a bounded H-functional calculus and satisfies the Davies–Gaffney estimate, let Φ be a concave function on (0,) of critical lower type pΦ(0,1], and let ρ(t)t1/Φ1(t1) for all t(0,). In this paper, the authors introduce the generalized VMO space VMOρ,L(X) associated with L and establish its characterization via the tent space. As applications, the authors show that (VMOρ,L(X))=BΦ,L(X), where L denotes the adjoint operator of L in L2(X) and BΦ,L(X) the Banach completion of the Orlicz– Hardy space HΦ,L(X).

Article information

Source
Kyoto J. Math., Volume 52, Number 2 (2012), 205-247.

Dates
First available in Project Euclid: 24 April 2012

Permanent link to this document
https://projecteuclid.org/euclid.kjm/1335272979

Digital Object Identifier
doi:10.1215/21562261-1550958

Mathematical Reviews number (MathSciNet)
MR2914876

Zentralblatt MATH identifier
1245.42020

Subjects
Primary: 42B35: Function spaces arising in harmonic analysis
Secondary: 42B30: $H^p$-spaces 46E30: Spaces of measurable functions (Lp-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 30L99: None of the above, but in this section

Citation

Liang, Yiyu; Yang, Dachun; Yuan, Wen. Vanishing mean oscillation spaces associated with operators satisfying Davies–Gaffney estimates. Kyoto J. Math. 52 (2012), no. 2, 205--247. doi:10.1215/21562261-1550958. https://projecteuclid.org/euclid.kjm/1335272979


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References

  • [1] B. T. Anh, Functions of vanishing mean oscillation associated to non-negative self-adjoint operators satisfying Davies-Gaffney estimates, preprint, 2010.
  • [2] B. T. Anh and J. Li, Orlicz-Hardy spaces associated to operators satisfying bounded H functional calculus and Davies-Gaffney estimates, J. Math. Anal. Appl. 373 (2011), 485–501.
  • [3] T. Aoki, Locally bounded linear topological spaces, Proc. Imp. Acad. Tokyo 18 (1942), 588–594.
  • [4] P. Auscher, X. T. Duong, and A. McIntosh, Boundedness of Banach space valued singular integral operators and Hardy spaces, preprint, 2005.
  • [5] G. Bourdaud, Remarques sur certains sous-espaces de BMO(ℝn) et de bmo (ℝn), Ann. Inst. Fourier (Grenoble) 52 (2002), 1187–1218.
  • [6] R. R. Coifman, Y. Meyer, and E. M. Stein, Some new function spaces and their applications to harmonic analysis, J. Funct. Anal. 62 (1985), 304–335.
  • [7] R. R. Coifman and G. Weiss, Analyse harmonique non-commutative sur certains espaces homogènes, Lecture Notes in Math. 242, Springer, Berlin, 1971.
  • [8] R. R. Coifman and G. Weiss, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. 83 (1977), 569–645.
  • [9] D. Deng, X. T. Duong, L. Song, C. Tan, and L. Yan, Functions of vanishing mean oscillation associated with operators and applications, Michigan Math. J. 56 (2008), 529–550.
  • [10] X. T. Duong and J. Li, Hardy spaces associated to operators satisfying bounded H functional calculus and Davis-Gaffney estimates, preprint, 2009.
  • [11] X. T. Duong, J. Xiao, and L. Yan, Old and new Morrey spaces with heat kernel bounds, J. Fourier Anal. Appl. 12 (2007), 87–111.
  • [12] 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.
  • [13] 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.
  • [14] C. Fefferman and E. M. Stein, Hp spaces of several variables, Acta Math. 129 (1972), 137–193.
  • [15] 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.
  • [16] S. Hofmann and S. Mayboroda, Hardy and BMO spaces associated to divergence form elliptic operators, Math. Ann. 344 (2009), 37–116.
  • [17] S. Hofmann and S. Mayboroda, Correction to Hardy and BMO spaces associated to divergence form elliptic operators, arXiv:0907.0129v2 [math.AP]
  • [18] S. Janson, Generalizations of Lipschitz spaces and an application to Hardy spaces and bounded mean oscillation, Duke Math. J. 47 (1980), 959–982.
  • [19] R. Jiang and D. Yang, Generalized vanishing mean oscillation spaces associated with divergence form elliptic operators, Integral Equations Operator Theory 67 (2010), 123–149.
  • [20] R. Jiang and D. Yang, New Orlicz-Hardy spaces associated with divergence form elliptic operators, J. Funct. Anal. 258 (2010), 1167–1224.
  • [21] R. Jiang and D. Yang, Orlicz-Hardy spaces associated with operators satisfying Davies-Gaffney estimates, Commun. Contemp. Math. 13 (2011), 331–373.
  • [22] R. Jiang and D. Yang, Predual spaces of Banach completions of Orlicz-Hardy spaces associated with operators, J. Fourier Anal. Appl. 17 (2011), 1–35.
  • [23] R. Jiang, D. Yang, and Y. Zhou, Orlicz-Hardy spaces associated with operators, Sci. China Ser. A 52 (2009), 1042–1080.
  • [24] F. John and L. Nirenberg, On functions of bounded mean oscillation, Comm. Pure Appl. Math. 14 (1961), 415–426.
  • [25] A. McIntosh, “Operators which have an H functional calculus” in Miniconference on operator theory and partial differential equations (North Ryde, Australia, 1986), Proc. Centre Math. Anal. Austral. Nat. Univ. 14, Austral. Nat. Univ., Canberra, 1986, 210–231.
  • [26] S. Rolewicz, On a certain class of linear metric spaces, Bull. Acad. Polon. Sci. Ci. III 5 (1957), 471–473.
  • [27] E. Russ, “The atomic decomposition for tent spaces on spaces of homogeneous type” in CMA/AMSI Research Symposium Asymptotic Geometric Analysis, Harmonic Analysis, and Related Topics, Proc. Centre Math. Appl. Austral. Nat. Univ. 42, Austral. Nat. Univ., Canberra, 2007.
  • [28] D. Sarason, Functions of vanishing mean oscillation, Trans. Amer. Math. Soc. 207 (1975), 391–405.
  • [29] L. Song and M. Xu, VMO spaces associated with divergence form elliptic operators, Math. Z. 269 (2011), 927–943.
  • [30] B. E. Viviani, An atomic decomposition of the predual of BMO(ρ), Rev. Mat. Iberoamericana 3 (1987), 401–425.