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 ${\mathrm{BMO}}_L$ 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 ${\mathrm{BMO}}_L$ spaces in the theory of singular integration. For example we obtain ${\mathrm{BMO}}_L$ estimates and interpolation results for fractional powers, purely imaginary powers and spectral multipliers of self adjoint operators. We also demonstrate that the space ${\mathrm{BMO}}_L$ might coincide with or might be essentially different from the classical BMO space.