Musielak-Orlicz Hardy spaces associated to operators satisfying Davies-Gaffney estimates and bounded holomorphic functional calculus

Let X be a metric space with doubling measure and L be an operator which satisfies Davies-Gaffney heat kernel estimates and has a bounded H1 functional calculus on L2(X). In this paper, we develop a theory of Musielak-Orlicz Hardy spaces associated to L, including a molecular decomposition, square function characterization and duality of Musielak-Orlicz Hardy spaces HL_∞(X). Finally, we show that L has a bounded holomorphic functional calculus on HL_∞(X) and the Riesz transform is bounded from HL_∞(X) to L1(w).