Musielak-orlicz-hardy spaces associated with operators satisfying reinforced off-diagonal estimates

Let X be a metric space with doubling measure and L a oneto- one operator of type ! having a bounded H1-functional calculus in L2(X) satisfying the reinforced (pL; qL) off-diagonal estimates on balls, where pL ∈ [1; 2) and qL ∈ (2;∞]. Let Φ : X × [0;1) ! [0;1) be a function such that '(x; ·) is an Orlicz function, '(·; t) ∈ A1(X) (the class of uniformly Muckenhoupt weights), its uniformly critical upper type index I(Φ) ∈ (0; 1] and '(·; t) satisfies the uniformly reverse Hölder inequality of order (qL/I(Φ))0, where (qL/I(Φ))0 denotes the conjugate exponent of qL/I(Φ). In this paper, the authors introduce a Musielak-Orlicz-Hardy space H'; L(X), via the Lusin-Area function associated with L, and establish its molecular characterization. In particular, when L is nonnegative self-Adjoint and satisfies the Davies-Gaffney estimates, the atomic characterization of H'; L(X) is also obtained. Furthermore, a sufficient condition for the equivalence between H'; L(Rn) and the classical Musielak-Orlicz-Hardy space H'(Rn) is given. Moreover, for the Musielak-Orlicz-Hardy space H'; L(Rn) associated with the second order elliptic operator in divergence form on Rn or the Schrödinger operator L := -Δ + V with 0 ΔV ∈ L1 loc(Rn), the authors further obtain its several equivalent characterizations in terms of various non-Tangential and radial maximal functions; finally, the authors show that the Riesz transform rL-1/2 is bounded from H'; L(Rn) to the Musielak-Orlicz space L'(Rn) when i(Φ) ∈ (0; 1], from HΦ; L(Rn) to H'(Rn) when i(Φ) ∈ ( n/n+1 ; 1], and from HΦ; L(Rn) to the weak Musielak- Orlicz-Hardy space WH'(Rn) when i(') = n n+1 is attainable and '(·; t) ∈ A1(X), where i(Φ) denotes the uniformly critical lower type index of Φ.