Integral Equations and Operator Theory

, Volume 67, Issue 1, pp 123–149 | Cite as

Generalized Vanishing Mean Oscillation Spaces Associated with Divergence Form Elliptic Operators

  • Renjin Jiang
  • Dachun Yang


Let L be a divergence form elliptic operator with complex bounded measurable coefficients, ω a positive concave function on (0, ∞) of strictly critical lower type p ω ∈(0, 1] and ρ(t) = t −1/ω −1(t −1) for \({t\in (0,\infty).}\) In this paper, the authors introduce the generalized VMO spaces \({{\mathop{\rm VMO}_ {\rho, L}({\mathbb R}^n)}}\) associated with L, and characterize them via tent spaces. As applications, the authors show that \({({\rm VMO}_{\rho,L} ({\mathbb R}^n))^\ast=B_{\omega,L^\ast}({\mathbb R}^n)}\), where L * denotes the adjoint operator of L in \({L^2({\mathbb R}^n)}\) and \({B_{\omega,L^\ast}({\mathbb R}^n)}\) the Banach completion of the Orlicz–Hardy space \({H_{\omega,L^\ast}({\mathbb R}^n)}\). Notice that ω(t) = t p for all \({t\in (0,\infty)}\) and \({p\in (0,1]}\) is a typical example of positive concave functions satisfying the assumptions. In particular, when p = 1, then ρ(t) ≡ 1 and \({({\mathop{\rm VMO}_{1, L}({\mathbb R}^n)})^\ast=H_{L^\ast}^1({\mathbb R}^n)}\), where \({H_{L^\ast}^1({\mathbb R}^n)}\) was the Hardy space introduced by Hofmann and Mayboroda.

Mathematics Subject Classification (2000)

Primary 42B35 Secondary 42B30 46E30 


Divergence form elliptic operator Gaffney estimate Orlicz function Orlicz–Hardy space BMO VMO CMO molecule dual 


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Copyright information

© Birkhäuser / Springer Basel AG 2010

Authors and Affiliations

  1. 1.School of Mathematical SciencesBeijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of EducationBeijingPeople’s Republic of China
  2. 2.Department of Mathematics and StatisticsUniversity of JyväskyläJyväskyläFinland

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