Revista Matemática Iberoamericana

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Volume 24, Issue 1, 2008, pp. 267–296
DOI: 10.4171/RMI/536

Published online: 2008-04-30

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

Donggao Deng[1], Xuan Thinh Duong[2], Adam Sikora[3] and Lixin Yan[4]

(1) Zhongshan University, Guangzhou, Guangdong, China
(2) Macquarie University, Sydney, Australia
(3) Macquarie University, Sydney, Australia
(4) Zhongshan University, Guangzhou, China

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.

Keywords: 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. Iberoam. 24 (2008), 267-296. doi: 10.4171/RMI/536