Musielak-Orlicz-Hardy Spaces Associated with Operators Satisfying Reinforced Off-Diagonal Estimates

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,

Isomorphisms and several characterizations of Musielak-Orlicz-Hardy spaces associated with some Schrödinger operators

“…This kind of Musielak-Orlicz-Hardy spaces associated with operators generalizes the (Orlicz-)Hardy space and the (weighted) Hardy space associated with operators, which has attracted great interests in recent years. Such function spaces associated with operators play important roles in the study for the boundedness of singular integrals associated with some differential operators, which may not fall within the scope of the classical Calderón-Zygmund theory (see, for example, [3], [5], [11], [12], [13], [17], [16], [18], [29], [30]). …”

“…By using the atomic characterization of H ϕ,L (R n ) obtained in [3], Theorem 5.4 (see also Lemma 3.2 below), the molecular characterization of H ϕ (R n ) established in [14], Theorem 4.13 (see also Lemma 3.4 below), the definitions of L-harmonic functions, the radial maximal function characterization of H ϕ,L (R n ) associated with the Poisson semigroup {e −t √ L } t>0 obtained in [3], Theorem 8.3 (see also Lemma 3.6 below) and Lemma 1.3, we complete the proof of Theorem 1.4.…”

“…By applying the atomic and molecular characterizations of H ϕ,L (R n ) established in Theorem 1.6, the atomic characterization of H ϕ,L (R n ) obtained in [3], Theorem 5.4 (see also Lemma 3.2 below), the molecular characterizations of H ϕ (R n ) obtained in [14], Theorem 4.13 (see also Lemma 3.4 below) and [6], Lemmas 2.11 and 2.13 (see also Lemma 4.1 below), we prove Theorem 1.7. (i) When ϕ(x, t) := t for all x ∈ R n and t ∈ [0, ∞), then i(ϕ) = 1, q(ϕ) = 1 and r(ϕ) = ∞.…”

“…Observe that i(ϕ) need not be attainable, namely, ϕ need not be of uniformly lower type i(ϕ) (see, for example, [3], [4], [30]). …”

“…Moreover, we need some properties of ϕ in Definition 2.4, which are useful in the proof of Theorems 1.4, 1.6 and 1.7. Then we have the following properties for ϕ from [3], Lemma 2.5, based on the corresponding results of [20], [9], [10]. …”