Inhomogeneous Besov spaces associated to operators with off-diagonal semigroup estimates

Let (X, d, μ) be a space of homogeneous type equipped with a distance d and a measure μ. 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_∞-calculus on L²(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)^-γ, γ > 0 on these Besov spaces. Finally, we carry out a comparison between our new Besov spaces and the classical Besov spaces.