Revista Matemática Iberoamericana

Comparison of the classical BMO with the BMO spaces associated with operators and applications

Donggao Deng , Xuan Thinh Duong , Adam Sikora , and Lixin Yan

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Let $L$ be a generator of a semigroup satisfying the Gaussian upper bounds. A new ${\rm BMO}_L$ space associated with $L$ was recently introduced in [Duong, X. T. and Yan, L.: {New function spaces of BMO type, the John-Nirenberg inequality, interpolation and applications}. \textit{Comm. Pure Appl. Math.} {\bf 58} (2005), 1375-1420] and [Duong, X. T. and Yan, L.: {Duality of Hardy and BMO spaces associated with operators with heat kernels bounds}. \textit{J. Amer. Math. Soc.} {\bf 18} (2005), 943-973]. We discuss applications of the new ${\rm BMO}_L$ spaces in the theory of singular integration. For example we obtain ${\rm BMO}_L$ estimates and interpolation results for fractional powers, purely imaginary powers and spectral multipliers of self adjoint operators. We also demonstrate that the space ${\rm BMO}_L$ might coincide with or might be essentially different from the classical BMO space.

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Rev. Mat. Iberoamericana, Volume 24, Number 1 (2008), 267-296.

First available in Project Euclid: 16 July 2008

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Primary: 42B35: Function spaces arising in harmonic analysis 42B25: Maximal functions, Littlewood-Paley theory 47B38: Operators on function spaces (general)

BMO space Hardy space Dirichlet and Neumann Laplacians semigroup Gaussian bounds fractional powers purely imaginary powers spectral multiplier


Deng , Donggao; Duong , Xuan Thinh; Sikora , Adam; Yan , Lixin. Comparison of the classical BMO with the BMO spaces associated with operators and applications. Rev. Mat. Iberoamericana 24 (2008), no. 1, 267--296.

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