Littlewood–Paley Characterization for Musielak–Orlicz–Hardy Spaces Associated with Self-Adjoint Operators

Let (X, d, μ) be a metric measure space endowed with a metric d and a non-negative Borel doubling measure μ. Let L be a nonnegative self-adjoint operator on L2(X). Assume that the (heat) kernel associated to the semigroup e tL satisfies a Gaussian upper bound. In this paper, we prove that the Musielak–Orlicz–Hardy space Hφ,L(X) associated with L in terms of the Lusin-area function and the Musielak–Orlicz–Hardy space HL,G,φ(X) associated with L in terms of the Littlewood–Paley function coincide and their norms are equivalent. To do this, we first establish the discrete characterization of these two spaces. It improves the known results in the literature.


Introduction
e metric measure space (X, d, μ) is a set X equipped with a metric d and a non-negative Borel doubling measure μ on X. Let f ∈ L 2 (X) and L be a densely defined operator on L 2 (X) which satisfies the following two conditions: (i) (H1) L is a non-negative self-adjoint operator on L 2 (X).
(ii) (H2) e kernel of e − tL , denoted by p t (x, y), is a measurable function on X × X satisfying the Gaussian estimates, i.e., there exist C 1 , C 2 > 0 such that holds for all t > 0 and ). e Littlewood-Paley function G L (f) and Lusin-area function S L (f) associated with the heat semigroup generated by L are given by In this paper, we focus on the characterization of the Musielak-Orlicz-Hardy spaces H φ,L and H L,G,φ , where the operator L satisfies (H1) and (H2) and φ is a growth function (cf. Definition 6 below). Definition 1. Suppose that the operator L satisfies (H1) and (H2) and φ is a growth function. A function f ∈ H 2 (X) is said to be in H φ,L (X) if S L (f) ∈ L φ (X) (cf. Definition 7 below). Moreover, we define e Musielak-Orlicz-Hardy space H φ,L (X) is defined to be the complement space of H φ,L (X).
Definition 2. Suppose that the operator L satisfies (H1) and (H2) and φ is a growth function. A function f ∈ H 2 (X) is said to be in H L,G,φ (X) if G L (f) ∈ L φ (X). Moreover, we define e Musielak-Orlicz-Hardy space H L,G,φ (X) is defined to be the complement space of H L,G,φ (X).
Recently, the study of Hardy spaces associated with operators has been attracting great interest. It was initiated by Auscher et al. who studied the Hardy space H 1 L (R n ) with operators L in [1], where the heat kernel of L satisfies the pointwise Poisson upper bounded condition. Later on, Duong and Yan [2,3] presented the adapted BMO theory on condition that the heat kernel of L satisfies the pointwise Gaussian estimate. In [4], Yan established the theory of Hardy space H p L (R n ) for 0 < p < 1 associated with the operator L satisfying Davies-Gaffney estimates.
It is a natural question to ask the behavior of weighted Hardy space H p L,ω (R n ) associated with an operator L and an appropriate weight ω. A pioneering investigation work of the weighted Hardy space H 1 L,ω (R n ) associated with the Schrödinger operator L was the paper by Song and Yan [5]. In 2016, Duong et al. [6] considered the weighted Hardy spaces H p L,S,ω (R n ) and H p L,G,ω (R n ) on homogeneous space X for 0 < p ≤ 1 and obtained the equivalence of these two kinds by adding Moser-type conditions, where the operator L has the kernel satisfying Gaussian upper bound. Shortly after that, the equivalence of these two kinds spaces was characterized by Hu [7] without assuming the Moser-type boundedness condition.
In 2014, Ky [8] introduced the Musielak-Orlicz-Hardy space H φ (R n ) by using growth function φ. Naturally, the Musielak-Orlicz-Hardy space H φ,L which is defined by means of the Lusin-area function associated with an operator L was introduced and studied in [9], where L satisfies Davies-Gaffney estimates. Unfortunately, the characterization of H φ,L required an extra assumption that φ satisfies the uniformly reverse Hölder condition (cf. [9]). Motivated by the above, we are concerned with the Musielak-Orlicz spaces H φ,L (X) and H L,G,φ (X) which we define by means of the Lusin-area function and the Littlewood-Paley function on homogeneous space X. Our aim in the present paper is to prove that the two kinds of Musielak-Orlicz spaces are equivalent. Our main result is stated as follows.
Theorem 1. Suppose that the operator L satisfies (H1) and (H2) and φ is a growth function of uniformly lower type p 1 . en, the spaces H φ,L (X) and H L,G,φ (X) coincide and their norms are equivalent. eorem 1 obtains the behavior of Littlewood-Paley g-function G L on H φ,L and partly improves the result in [9]. To make it clear, we first establish the discrete characterization of the Musielak-Orlicz spaces H φ,L (X) and H L,G,φ (X) and state these results as follows.
Theorem 2. Suppose that the operator L satisfies (H1) and (H2) and φ is a growth function of uniformly lower type p 1 .
en, for all M ∈ N with M > (nq(φ)/2p 1 ), f has an AT L,M -expansion such that Theorem 3. Suppose that the operator L satisfies (H1) and (H2) and φ is a growth function of uniformly lower type p 1 .
eorems 2 and 3 extend the results in [6,7], respectively. Also, we extend the results in [9] by removing the assumption of uniformly reverse Hölder condition. As a consequence of eorems 2 and 3, we immediately get eorem 1. e paper is organized as follows. Section 2 contains some basic definitions and lemmas concerning metric measure spaces, growth functions, Musielak-Orlicz space, and AT L,M -family. e aim of Section 3 is to prove eorem 2 and establish the characterization of Musielak-Orlicz-Hardy space H φ,L . We develop a method to unify the different control terms of inner integral. e aim of Section 4 is to prove eorem 3 and set up the characterization of Musielak-Orlicz-Hardy space H L,G,φ . We borrow the ideas from [6,10]. Consequently, we get that the characterization of Musielak-Orlicz-Hardy space by means of H φ,L and H L,G,φ is equivalent.
Most of the notations we use are standard. C denotes a positive constant that may change from line to line and we use the subscript for the sake of eliminating confusion. We write A � B if there exist constants C 1 and C 2 which are independent of A and B such that C 1 B ≤ A ≤ C 2 B. For a measurable set A, |A| denotes the Lebesgue measure of A and χ A is the characteristic function.

Basic Concepts and Lemmas
is a set X equipped with a metric d and a non-negative Borel doubling measure μ on X. Fix x ∈ X and let r ∈ (0, ∞), and we denote the open ball centered at x with radius r by and set V(x, r) � μ (B(x, r)).

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Definition 3. A space of homogeneous type (X, d, μ) is a set X with a metric d and a non-negative measure μ on X, so that there exists a constant C D ∈ [1, ∞) such that for all x ∈ X and r > 0, Definition 3 was introduced by Coifman and Weiss [11]. e property of μ in (8) is the doubling condition and it implies the strong n homogeneity property, i.e., for some constant C > 0 and homogeneity n, holds uniformly for all λ ∈ [1, ∞), x ∈ X, and r > 0. Let C D be as in (8) and set m � log 2 C D , and Grigor'yan et al. have shown that (see [12]) holds for all x, y ∈ X and 0 < r ≤ R < ∞. It is easy to verify, by doubling condition (8), that for any N > n, there exists a constant C N such that for all x ∈ X and t > 0, e dyadic cube decomposition on spaces of homogeneous type comes from Christ [13] as follows: en, there exists a collection of open subsets Q k α ⊂ X: k ∈ Z, α ∈ I k and constants δ ∈ (0, 1) and e sets Q k α are analogues of the Euclidean dyadic cubes; it may help to think of Q k α as being essentially a cube of ball of diameter roughly δ k with center z k α . We then set ℓ(Q k α ) � C 1 δ k . It is worthy pointing out that the precise value of C 1 is non-essential (cf. Christ [13]). Here and in what follows, we assume C 1 � δ − 1 .

Growth Functions.
We first recall the Orlicz function. A non-decreasing function Φ: [9]). e function Φ is said to be of upper type p (resp., lower Given a function φ: then φ is said to be of uniformly upper type p (resp., uniformly lower type p). Moreover, φ is said to be of positive uniformly upper type (resp., uniformly lower type) if it is of uniformly upper type (resp., uniformly lower type) p for some p ∈ (0, ∞).

Journal of Function Spaces
Here the first supremum is taken over all t ∈ (0, ∞) and the second one is taken over all balls B ⊂ X.
be the critical indices of φ. Moreover, we denote for any measurable subset E of X and t ∈ [0, ∞). Let M be the Hardy-Littlewood maximal function on X, namely, for all x ∈ X, where the supremum is taken over all balls B containing x. e following lemma on the properties of A ∞ (X) is Lemma 2.8 in [9].

Lemma 3. Let φ be a growth function and set
en, φ is a growth function, which is equivalent to φ, and φ(x, ·) is continuous and strictly increasing.

Musielak-Orlicz Spaces.
In this section, we study the Musielak-Orlicz spaces associated with the growth function φ.
e Musielak-Orlicz space L φ (X) denotes the set of all measurable function f on X with X φ(x, |f(x)|)dμ(x) < ∞ and the Luxembourg norm We have the following Fefferman-Stein vector-valued inequality of Musielak-Orlicz type (cf. [15]).

Lemma 4.
Let p ∈ (1, ∞], φ be a Musielak-Orlicz function with uniformly lower type p 1 and upper type p 2 , q ∈ (1, ∞), Corollary 1. Let p, φ be as in Lemma 4. en, for all r ∈ (0, (p 1 /q(φ))) and f j j∈Z ∈ L φ (ℓ p , X), there exists a constant C > 0 such that Proof. Fix r ∈ (0, (p 1 /q(φ))) and let φ( We claim that φ is of uniformly lower type p 1 /r and upper type p 2 /r. In fact, there exist constants C 1 , C 2 > 0 such that It finishes the proof of Corollary 1. In this section, we assume that the space X satisfies the strong homogeneity property (9) with homogeneous dimension n. In view of Lemma 1, the space X possesses a dyadic decomposition analogous to the Euclidean dyadic cubes, i.e., there exists a collection of open subsets Q k α ⊂ X: k ∈ Z, α ∈ I k such that for every k ∈ Z, where I k is some index set and Q k α has the properties as in Lemma 1. Such open subsets Q k α ⊂ X: k ∈ Z, α ∈ I k are said to be a family of dyadic cubes of X (cf. [6]).
Here, D(T) denotes the domain of an unbounded operator T and T k , which is the k-fold composition of T with itself, in the sense of unbounded operators.
For a function f in L 2 (X), if there exists sequence s � s Q Q: dyadic , 0 ≤ s Q < ∞, and an AT L,M -family a Q Q: dyadic in L 2 (X) such that we say that f has an AT L,M -expansion. en, we denote the function related to the sequence s � s Q Q:dyadic by W f (x) and With the notation above, we have the following characterization of L 2 (X).
Moreover, let Q k α and δ be as in Lemma 1. en,

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(32) Proof. e proof of Proposition 1 can be found in [6, eorem 3.2].

The Proof of Theorem 2
In this section, we establish a characterization of the Musielak-Orlicz-Hardy space H φ,L , where the operator L satisfies (H1) and (H2) and φ is a growth function.
For every v ∈ (0, ∞) and x ∈ X, let Γ v (x) � (y, t) ∈ X × (0, ∞): d(x, y) < vt be the cone of aperture v and vertex x ∈ X. For any closed subset F of X, we denote the union of all cones with vertices in F by When v � 1, Γ(x) and R(F) stand for Γ 1 (x) and R 1 (x), respectively. Given an open subset O of X, we establish Lemma 5 of R(O ∁ ) on the geometric properties. We also remark that Aguilera and Segovia [16] obtained the same result in the case of Euclidean space. (X, d, μ) is a space of homogeneous type and there exists a constant C D > 1 such that (8)

Proof. It suffices to show that the lemma holds when
It is easy to see that (z, t) ∈ R(F) since d(z, z) � 0 < t in the case z ∈ F and then R v (F * ) ⊂ R(F) holds. e proof of (i) is reduced to the verification in the case z ∈ O.
Suppose z ∈ O and let δ � dist(z, F). en, 0 < δ < ∞ and B(z, δ) ⊂ O since F is closed and non-empty. For every (z, t) ∈ R v (F * ), we have y ∈ F * such that d(z, y) < vt. us, writing r � δ + d(z, y), we get B(z, δ) ⊂ B(y, r) and Hence, By using (10) twice, we have and then It follows that δ < t since v > 1. Recalling the definition of δ, we get x ∈ F such that d(x, z) < t, which implies (z, t) ∈ R(F). It completes the proof of (i).
Next, we prove (ii). Given (z, t) ∈ R v (F * ), we get y ∈ F * such that d(z, y) < vt. us, B(z, t) ⊂ B(y, (1 + v)t) and We obtain since V(z, t) � μ(B(z, t) ∩ O) + μ(B(z, t) ∩ F), and complete the proof of (ii). It finishes the proof of Lemma 5. □ For all v ∈ (0, ∞), f ∈ L 2 (X) and x ∈ X, the variant Lusin-area function associated with L is given by Lemmas 6 and 7 extend the results in [14,16] for the operator S L,v . L satisfies (H1) and  (H2). Let φ ∈ A p (X) for 1 ≤ p < ∞ and O, O * , F, F * be as in Lemma 5. en, there exists a finite constant C, which is independent of O, such that for all λ ∈ (0, ∞) and f ∈ L 2 (X),

Lemma 6. Suppose that the operator
where S L is the short hand of S L,1 .

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Proof. Given x ∈ F * and (y, t) ∈ Γ v (x), we observe that d(x, y) < vt and hence by (10), It follows that (46) en, applying Lemma 2 to the sets B(y, t) and B(y, vt), B(y, t) ∩ F and B(y, t), respectively, we get (49) (50) Finally, in view of R v (F * ) ⊂ R(F) (see Lemma 5), it follows immediately that (50) is bounded by where we use the fact that for (y, t) ∈ Γ v (x) in the last line. It finishes the proof of Lemma 6.

Lemma 7. Suppose that the operator L satisfies (H1) and (H2). Let φ be a growth function and φ ∈
holds for all v ∈ (0, ∞) and all measurable functions f.

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(57) erefore, we employ (57) together with the assumption v ∈ (1, ∞), Lemma 3, and the uniformly upper type 1 of φ to get It finishes the proof of Lemma 7.
□ Lemma 8 says that the sequence s Q k α α∈I k can be majorized by the Hardy-Littlewood maximal operator M on (X, d, μ) (cf. [17], pp.147, where we take r � 1). Lemma 8. Suppose 0 < q ≤ 1 and N > (n/q). Fix k ∈ Z and let s Q k α α∈I k be as in Proposition 1. en, for any subsequence I k ′ ⊂ I k and for every x ∈ X, where y k α is the center of Q k α and C depends only on n and N − (n/q).
We now turn to prove (60). Given (x, k) ∈ X × Z, by Lemma 1, there exists a unique α ∈ I k such that x ∈ Q k α . We denote such Q k α by Q k x and write where constants δ ∈ (0, 1) satisfy Lemma 1 and the last line is obtained by using Proposition 1. Moreover, for any fixed (x, k) ∈ X × Z, Lemma 1 also tells us that there are z k x ∈ Q k x and constants C 1 ∈ (0, 1), for all t ∈ (δ k+1 , δ k ). Consequently, where we use (10) and the fact that 1). Hence, (63) and (65) yield us, by using Lemma 7, we deduce that It remains to establish the reverse inequality of (67). Let δ be as in Lemma 1. In view of Proposition 1, we write and get

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(69) We firstly estimate the inner integral of I 1 . For any k > j and α ∈ I k , noting a Q k Since M > (nq(φ)/2p 1 ) with n given as in (9), we can choose some q satisfying Corollary 1 such that 2M > (n/q).
us, there is some N > 0 such that 2M > N > (n/q). en, applying Definition 8, the upper bound of the kernel (t 2 L) M+1 e − t 2 L (cf. [18], Proposition 3.1), and (11), we get where z k α is the center of Q k α . Since d(x, y) < t, we further have Hence, Lemma 8 yields the inner integral of I 1 which is bounded by 10 Journal of Function Spaces Secondly, we estimate the inner integral of I 2 . For any k ≤ j and α ∈ I k , we write en, using Definition 8, Gaussian estimate (1), and inequality (11), we obtain Since d(x, y) < t ≤ ℓ(Q k α ), we further have Hence, Lemma 8 yields the inner integral of I 2 which is bounded by Fix j ∈ Z, and we let β > 0 and we have where In view of inequalities (73)-(81), taking β � 2M − N, τ � 1, and β � 2, τ � − 1 respectively, we get erefore, (82) and Corollary 1 yield which gives the reverse inequality of (67). It finishes the proof of eorem 2.

The Proof of Theorem 3
In this section, we establish a characterization of the Musielak-Orlicz-Hardy space H L,G,φ , where the operator L satisfies (H1) and (H2) and φ is a growth function. Our proof will borrow some ideals from Duong et al. [6]. We first recall some basic definitions and facts about Fefferman-Stein type maximal function, referring to [7] for a complete account.
We also need Lemma 10, and its proof is standard, which we omit here.

Lemma 10.
Let n and m be as in (9) and (10), and N > n + m.
en, there exists a constant C > 0 such that holds for all measurable functions f on (X, d, μ), t > 0, and each y ∈ X.