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 H_∞ 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 H_L,ω(X). Finally, we show that L has a bounded holomorphic functional calculus on H_L,ω(X) and the Riesz transform is bounded from H_L,ω(X) to L¹(ω).

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