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 BMOL space associated with L was recently introduced in [15] and [16]. We discuss applications of the new BMOL spaces in the theory of singular integration. For example we obtain BMOL estimates and interpolation results for fractional powers, purely imaginary powers and spectral multipliers of self adjoint operators. We also demonstrate that the space BMOL might coincide with or might be essentially different from the classical BMO space.