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.

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In this paper, we give a new characterization of the Morrey-Campanato spaces by using the convolution ϕ t B * f (x) to replace the minimizing polynomial P B f of a function f in the Morrey-Campanato norm, where ϕ ∈ S(R n ) is an appropriate Schwartz function. (2000): 42B35, 47G10 Mathematics Subject Classification

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