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Let be a metric space with doubling measure and L a oneto-one operator of type ω having a bounded H ∞ -functional calculus in L 2 ( ) satisfying the reinforced ( L L ) off-diagonal estimates on balls, where L ∈ [1 2) and L ∈ (2 ∞]. Let:is an Orlicz function, (· ) ∈ A ∞ ( ) (the class of uniformly Muckenhoupt weights), its uniformly critical upper type index I( ) ∈ (0 1] and (· ) satisfies the uniformly reverse Hölder inequality of order ( L /I( )) , where ( L /I( )) denotes the conjugate exponent of L /I( ). In this paper, the authors introduce a Musielak-Orlicz-Hardy space H L ( ), 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 ( ) is also obtained. Furthermore, a sufficient condition for the equivalence between H L (R ) and the classical Musielak-Orlicz-Hardy space H (R ) is given. Moreover, for the Musielak-Orlicz-Hardy space H L (R ) associated with the second order elliptic operator in divergence form on R or the Schrödinger operator L := −∆ + V with 0 ≤ V ∈ L 1 loc (R ), 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 ∇L −1/2 is bounded from H L (R ) to the Musielak-Orlicz space L (R ) when ( ) ∈ (0 1], from H L (R ) to H (R ) when ( ) ∈ ( +1 1], and from H L (R ) to the weak MusielakOrlicz-Hardy space W H (R ) when ( ) = +1 is attainable and (· ) ∈ A 1 ( ), where ( ) denotes the uniformly critical lower type index of . KeywordsMusielak-Orlicz-Hardy space • molecule • atom • maximal function • Lusin area function • Schrödinger operator • elliptic operator • Riesz transform MSC: 42B35,

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Abstract. Let b be a BM O-function. It is well-known that the linear commutator [b, T ] of a Calderón-Zygmund operator T does not, in general, map continuouslyIn this paper, we find the largest subspace H 1 b (R n ) such that all commutators of Calderón-Zygmund operators are continuous fromWe also study the commutators [b, T ] for T in a class K of sublinear operators containing almost all important operators in harmonic analysis. When T is linear, we prove that there exists a bilinear operators R = R T mapping continuouslywhere S is a bounded bilinear operator fromIn the particular case of T a Calderón-Zygmund operator satisfying T 1 = T * 1 = 0 and b in BM O log (R n )-the generalized BMO type space that has been introduced by Nakai and Yabuta to characterize multipliers of BMO(R n ) -we prove that the commutatorWhen T is sublinear, we prove that there exists a bounded subbilinear operator R = R T :The bilinear decomposition (1) and the subbilinear decomposition (2) allow us to give a general overview of all known weak and strong L 1 -estimates.

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