## Kyoto Journal of Mathematics

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

#### Abstract

Let $(\mathcal{X},d,\mu)$ be a metric measure space, let $L$ be a linear operator that has a bounded $H_{\infty}$-functional calculus and satisfies the Davies–Gaffney estimate, let $\Phi$ be a concave function on $(0,\infty)$ of critical lower type $p_{\Phi}^{-}\in(0,1]$, and let $\rho(t)\equiv t^{-1}/\Phi^{-1}(t^{-1})$ for all $t\in(0,\infty)$. In this paper, the authors introduce the generalized VMO space $\operatorname{VMO}_{\rho,L}({\mathcal{X}})$ associated with $L$ and establish its characterization via the tent space. As applications, the authors show that $(\operatorname{VMO}_{\rho,L}({\mathcal{X}}))^{*}=B_{\Phi,L^{*}}({\mathcal{X}})$, where $L^{*}$ denotes the adjoint operator of $L$ in $L^{2}({\mathcal{X}})$ and $B_{\Phi,L^{*}}({\mathcal{X}})$ the Banach completion of the Orlicz– Hardy space $H_{\Phi,L^{*}}({\mathcal{X}})$.

#### Article information

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

Dates
First available in Project Euclid: 24 April 2012

https://projecteuclid.org/euclid.kjm/1335272979

Digital Object Identifier
doi:10.1215/21562261-1550958

Mathematical Reviews number (MathSciNet)
MR2914876

Zentralblatt MATH identifier
1245.42020

#### 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

#### 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.