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Potential Analysis

, Volume 41, Issue 3, pp 849–867 | Cite as

Preduals of Quadratic Campanato Spaces Associated to Operators with Heat Kernel Bounds

  • Liang Song
  • Jie Xiao
  • Xuefang Yan
Article

Abstract

Let L be a nonnegative, self-adjoint operator on \(L^{2}(\mathbb {R}^{n})\) with the Gaussian upper bound on its heat kernel. As a generalization of the square Campanato space \(\mathcal {L}^{2,\lambda }_{-\Delta }(\mathbb R^{n})\), in Duong et al. (J. Fourier Anal. Appl. 13:87–111, 2007) the quadratic Campanato space \(\mathcal {L}_{L}^{2,\lambda }(\mathbb {R}^{n})\) is defined by a variant of the maximal function associated with the semigroup {e t L } t≥0. On the basis of Dafni and Xiao (J. Funct. Anal. 208:377–422, 2004) and Yang and Yuan (J. Funct. Anal. 255:2760–2809, 2008) this paper addresses the preduality of \(\mathcal {L}_{L}^{2,\lambda }(\mathbb {R}^{n})\) through an induced atom (or molecular) decomposition. Even in the case L = −Δ the discovered predual result is new and natural.

Keywords

Quadratic Campanato space Self-adjoint operator Heat semigroup Hausdorff capacity Choquet integral Atom Molecule 

Mathematics Subject Classifications (2010)

42B35 47B38 

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Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Department of MathematicsSun Yat-sen UniversityGuangzhouPeople’s Republic of China
  2. 2.Department of Mathematics and StatisticsMemorial UniversitySt. John’sCanada
  3. 3.College of Mathematics and Information ScienceHeibei Normal UniversityShijiazhuangPeople’s Republic of China

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