No abstract

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,

Let ϕ : R n × [0, ∞) → [0, ∞) be a growth function such that ϕ(x, ·) is nondecreasing, ϕ(x, 0) = 0, ϕ(x, t) > 0 when t > 0, lim t→∞ ϕ(x, t) = ∞, and ϕ(·, t) is a Muckenhoupt A ∞ (R n ) weight uniformly in t. In this paper, the authors establish the Lusin area function and the molecular characterizations of the Musielak-Orlicz Hardy space H ϕ (R n ) introduced by Luong Dang Ky via the grand maximal function. As an application, the authors obtain the ϕ-Carleson measure characterization of the Musielak-Orlicz BMO-type space BMO ϕ (R n ), which was proved to be the dual space of H ϕ (R n )[1] and Orlicz in [24], which is widely used in various branches of analysis (see, for example, [25,26] and their references). Moreover, as a development of the theory of Orlicz spaces, Orlicz-Hardy spaces and their dual spaces were studied by Strömberg [30] and Janson [14] on R n and, quite recently, Orlicz-Hardy spaces associated with divergence form elliptic operators by Jiang and Yang [15].Furthermore, the classical BMO space (the space of functions with bounded mean oscillation), originally introduced by John and Nirenberg [16], plays an important role in the study of partial differential equations and harmonic analysis. In particular, Fefferman and Stein [9] proved that BMO(R n ) is the dual space of H 1 (R n ) and also obtained the Carleson measure characterization of BMO(R n ). Moreover, the generalized BMO-type space BMO ρ (R n ) was studied in [30,14,13] and it was proved therein to be the dual space of the Orlicz-Hardy space H Φ (R n ), where the function Φ : [0, ∞) → [0, ∞) satisfies the following assumptions:). Here and in what follows, Φ −1 denotes the inverse function of Φ. Observe that Φ may not be convex and hence may not be an Orlicz function in the classical sense. Meanwhile, the Carleson measure characterization of BMO ρ (R n ) was obtained in [13]. Recently, a new Musielak-Orlicz Hardy space H ϕ (R n ) was introduced by Ky [17], via the grand maximal function, which includes both the Orlicz-Hardy space in [30,14] and the weighted Hardy space H p ω (R n ) with p ∈ (0, 1] and ω ∈ A ∞ (R n ) in [11,31]. Here and in what follows, ϕ : R n × [0, ∞) → [0, ∞) is a growth function such that ϕ(x, ·), for any fixed x ∈ R n , satisfies (1.1) with uniformly upper type 1 and lower type p for some p ∈ (0, 1] (see Section 2 for the definitions of uniformly upper or lower types), and ϕ(·, t) is a Muckenhoupt A ∞ (R n ) weight uniformly in t, and A q (R n ) with q ∈ [1, ∞] denotes the class of Muckenhoupt weights (see, for example, [12] for their definitions and properties). In [17], Ky first established the atomic characterization of H ϕ (R n ) and further introduced the Musielak-Orlicz BMO-type space BMO ϕ (R n ), which was proved to be the dual space of H ϕ (R n ). Furthermore, some interesting applications of these spaces were also presented in [2,4,5,17,18,19]. Moreover, the local Musielak-Orlicz Hardy space, h ϕ (R n ), and its dual space, bmo ϕ (R n ), were studied in [33] and some applications of h ϕ (R n ) and bmo ϕ (R n...

Let X be a metric space with doubling measure and L a nonnegative selfadjoint operator in L 2 (X ) satisfying the Davies-Gaffney estimates. Let ϕ : X × [0, ∞) → [0, ∞) be a function such that ϕ(x, ·) is an Orlicz function, ϕ(·, t) ∈ A ∞ (X ) (the class of uniformly Muckenhoupt weights), its uniformly critical upper type index I(ϕ) ∈ (0, 1] and it satisfies the uniformly reverse Hölder inequality of order 2/[2 − I(ϕ)]. In this paper, the authors introduce a Musielak-Orlicz-Hardy space H ϕ, L (X ), by the Lusin area function associated with the heat semigroup generated by L, and a Musielak-Orlicz BMO-type space BMO ϕ, L (X ), which is further proved to be the dual space of H ϕ, L (X ) and hence whose ϕ-Carleson measure characterization is deduced. Characterizations of H ϕ, L (X ), including the atom, the molecule and the Lusin area function associated with the Poisson semigroup of L, are presented. Using the atomic characterization, the authors characterize H ϕ, L (X ) in terms of the Littlewood-Paley g * λ -function g * λ, L and establish a Hörmandertype spectral multiplier theorem for L on H ϕ, L (X ). Moreover, for the Musielak-Orlicz-Hardy space H ϕ, L (R n ) associated with the Schrödinger operator L := −∆ + V , where 0 ≤ V ∈ L 1 loc (R n ), the authors obtain its several equivalent characterizations in terms of the non-tangential maximal function, the radial maximal function, the atom and the molecule; finally, the authors show that the Riesz transform ∇L −1/2 is bounded from H ϕ, L (R n ) to the Musielak-Orlicz space L ϕ (R n ) when i(ϕ) ∈ (0, 1], and from H ϕ, L (R n ) to the Musielak-Orlicz-Hardy space H ϕ (R n ) when i(ϕ) ∈ ( n n+1 , 1], where i(ϕ) denotes the uniformly critical lower type index of ϕ.

Let Φ be a concave function on (0, ∞) of strictly lower type pΦ ∈ (0, 1] and ω ∈ A loc ∞ (R n ) (the class of local weights introduced by V. S. Rychkov). We introduce the weighted local Orlicz-Hardy. We also introduce the BMO-type space bmoρ, ω (R n ) and establish the duality between h Φ ω (R n ) and bmoρ, ω (R n ). Characterizations of h Φ ω (R n ), including the atomic characterization, the local vertical and the local nontangential maximal function characterizations, are presented. Using the atomic characterization, we prove the existence of finite atomic decompositions achieving the norm in some dense subspaces of h Φ ω (R n ), from which, we further deduce that for a given admissible triplet (ρ, q, s)ω and a β-quasi-Banach space B β with β ∈ (0, 1], if T is a B β -sublinear operator, and maps all (ρ, q, s)ω-atoms and (ρ, q)ω-single-atoms with q < ∞ (or all continuous (ρ, q, s)ω-atoms with q = ∞) into uniformly bounded elements of B β , then T uniquely extends to a bounded B β -sublinear operator from h Φ ω (R n ) to B β . As applications, we show that the local Riesz transforms are bounded on h Φ ω (R n ), the local fractional integrals are bounded from 2010 Mathematics Subject Classification: Primary 46E30; Secondary 42B35, 42B30, 42B25, 42B20, 35S05, 47G30, 47B06.