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

Abstract

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.