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

Let L be a generator of a semigroup satisfying the Gaussian upper bounds. A new BMO_L space associated with L was recently introduced in [15] and [16]. We discuss applications of the new BMO_L spaces in the theory of singular integration. For example we obtain 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 BMO_L might coincide with or might be essentially different from the classical BMO space.