Lusin Area Function and Molecular Characterizations of Musielak–orlicz Hardy Spaces and Their Applications

Abstract

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...

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

“…It is well known that [14], Lemma 2.4, or Lemma 2.5 below). Thus, we can introduce the critical indices for ϕ ∈ A ∞ (R n ) as follows:…”

“…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) = ∞.…”

Riesz Transform Characterizations of Musielak-Orlicz Hardy Spaces

“…Here, A q (R n ) with q ∈ [1, ∞] denotes the class of Muckenhoupt weights (see, for example, [11,12,13] for their definitions and properties). Moreover, the space H ϕ (R n ) also has already found many applications in analysis (see, for example, [3,4,17,22,23] and their references).…”

“…For example, for all (x, t) ∈ R n × [0, ∞), ϕ(x, t) := ω(x)Φ(t) satisfies Assumption (ϕ) if ω ∈ A ∞ (R n ) and Φ is an Orlicz function of lower type p for some p ∈ (0, 1] and upper type 1. A typical example of such an Orlicz function Φ is Φ(t) := t p , with p ∈ (0, 1], for all t ∈ [0, ∞); see, for example, [17,22,23] for more examples. Another typical example of functions satisfying Assumption (ϕ) is ϕ(x, t) := t α [ln(e+|x|)] β +[ln(e+t)] γ for all x ∈ R n and t ∈ [0, ∞) with any α ∈ (0, 1] and β, γ ∈ [0, ∞) (see [23] for further details).…”