Let (X, d, mu) be a space of homogeneous type equipped with a distance d and a measure mu. Assume that L is a closed linear operator which generates an analytic semigroup e(-tL), t > 0. Also assume that L has a bounded H-infinity-calculus on L-2 (X) and satisfies the L-P - L-q semigroup estimates (which is weaker than the pointwise Gaussian or Poisson heat kernel bounds). The aim of this paper is to establish a theory of inhomogeneous Besov spaces associated to such an operator L. We prove the molecular decompositions for the new Besov spaces and obtain the boundedness of the fractional powers (I + L)(-gamma), gamma > 0 on these Besov spaces. Finally, we carry out a comparison between our new Besov spaces and the classical Besov spaces.
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