Littlewood-Paley Characterization for Musielak-Orlicz-Hardy Spaces Associated with Operators
aa r X i v : . [ m a t h . C A ] M a r Littlewood-Paley Characterization for Musielak-Orlicz-HardySpaces Associated with Operators
Jiawei Shen · Zhitian Chen · ShunchaoLong Abstract
Let X be a space of homogeneous type. Assume that L is an non-negative second-orderself-adjoint operator on L ( X ) with (heart) kernel associated to the semigroup e − tL that satisfies theGaussian upper bound. In this paper, the authors introduce a new characterization of the Musielak-Orlicz-Hardy Space H ϕ, L ( X ) associated with L in terms of the Lusin area function where ϕ isa growth function. Further, the authors prove that the Musielak-Orlicz-Hardy Space H L , G ,ϕ ( X )associated with L in terms of the Littlewood-Paley function is coincide with H ϕ, L ( X ) and theirnorms are equivalent. Keywords
Musielak-Orlicz Hardy spaces · Heat semigroup · Gaussian estimate · Nonnegativeself-adjoint operator · Area and Littlewood-Paley functions
Mathematics Subject Classification · · · · ·
1. Introduction
Recently, the study of the Hardy spaces associated with operators has been in the spotlight.This topic was initiated by Auscher et al. [2], who studied the Hardy space H L ( R n ) associated withthe operator L whose heat kernel satisfies the pointwise Poisson upper bounded condition. Lateron, the adapted BMO theory has been presented by Duong and Yan [4, 5], under the assumptionthat the heat kernel associated to L satisfies the pointwise Gaussian estimate. The theory of theHardy space H pL ( R n ) for 0 < p < L satisfying the Davies-Ga ff neyestimates was established by Yan [13]. It is then quite natural to consider the weighted Hardyspaces H pL ,ω associated with an operator L and a weight function ω . Song and Yan [12] firstintroduced the weighted Hardy space H L ,ω ( R n ) associated with the Schrdinger operator L for ω ∈ A ∞ ( R n ). Recently, Duong et al. [6] considered two kinds of weighted Hardy spaces on thehomogeneous spaces X associated with an operator whose kernel satisfying the Gaussian upperbound. For 0 < p ≤ ω ∈ A ∞ , they first studied the weighted Hardy space H pL , S ,ω ( X )which defined in terms of the Lusin area function, and secondly turned to consider the weightedHardy space H pL , G ,ω ( X ) which defined in terms of the Littlewood-Paley function. Finally, theyobtained the equivalence between the two kinds of weighted Hardy spaces by adding the Moser Shunchao Long: [email protected] School of Mathematics and Computational Science, Xiangtan University, Xiangtan, 411105, China
Preprint submitted to Elsevier March 21, 2019 ype condition for L . Subsequently, the equivalence of these two kinds of weighted Hardy spaceswas demonstrated by Hu [10] without using the additional Moser type estimate.On the other hannd, Ky [11] presented a new Musielak-Orlicz-Hardy space, H ϕ ( R n ), definedvia a growth function ϕ (see Sect.2 below for the definition of growth function). As an naturalgeneralization, the Musielak-Orlicz-Hardy space H ϕ, L , defined via the Lusin area function asso-ciated with an operator L that satisfies the Davies-Ga ff ney estimate, which contains the weightedHardy space H pL , S , w ( X ) in [6], had been introduced and systematically studied by Yang et al. in[14] later on. Characterizations of H ϕ, L , including the atom, the molecule, etc. was obtained in[14]. However, to characterize H ϕ, L , Yang et al. needed to impose an extra assumption that thegrowth function ϕ satisfies the uniformly reverse H¨older condition.Throughout this article ( X , d , µ ) is a metric measure space endowed with a distance d and anon-negative Borel doubling measure µ . And we assume that L is a densely defined operator on L ( X ) and satisfies the following two conditions in di ff erent sections of this paper.( H1 ) L is a second-order non-negative self-adjoint operator on L ( X );( H2 ) The kernel of e − tL , denote by p t ( x , y ), is a measurable function on X × X and satisfies theGaussian estimates, namely, there exist positive constants C and C such that, for all t > x , y ∈ X , | p t ( x , y ) | ≤ C V (cid:16) x , √ t (cid:17) exp − d ( x , y ) C t ! , (1)where V (cid:16) x , √ t (cid:17) = µ (cid:16) B (cid:16) x , √ t (cid:17)(cid:17) .Given an operator L that satisfying ( H1 ) and ( H2 ) and a function f ∈ L ( X ), we considerthe following Littlewood-Paley function G L ( f ) and Lusin area function S L ( f ) associated with theheat semigroup generated by LG L ( f ) ( x ) : = Z ∞ | t Le − t L f ( x ) | dtt ! / (2)and S L ( f ) ( x ) : = Z ∞ Z d ( x , y ) < t | t Le − t L f ( y ) | d µ ( y ) µ ( B ( x , t )) dtt ! / . (3)In this paper, Musielark-Orlicz Hardy spaces H ϕ, L and H L , G ,ϕ will be concerned. Their defini-tions are as follows. Definition 1.1.
Let L satisfies ( H1 ) and ( H2 ), and ϕ be a growth function. A function f ∈ H ( X ) is said to be in ˜ H ϕ, L ( X ) if S L ( f ) ∈ L ϕ ( X ) . Moreover, define k f k H ϕ, L ( X ) : = k S L ( f ) k L ϕ ( X ) : = inf ( λ ∈ (0 , ∞ ) ; Z X ϕ x , S L ( f ) ( x ) λ ! d µ ( x ) ≤ ) . The Musielak-Orlicz-Hardy space H ϕ, L ( X ) is defined to be the completion of ˜ H ϕ, L ( X ) with thequasi-norm k·k H ϕ, L ( X ) . 2 efinition 1.2. Let L satisfies ( H1 ) and ( H2 ), and ϕ be a growth function. A function f ∈ H ( X ) is said to be in ˜ H L , G ,ϕ ( X ) if G L ( f ) ∈ L ϕ ( X ) . Moreover, define k f k H L , G ,ϕ ( X ) : = k G L ( f ) k L ϕ ( X ) : = inf ( λ ∈ (0 , ∞ ) ; Z X ϕ x , G L ( f ) ( x ) λ ! d µ ( x ) ≤ ) . The Musielak-Orlicz-Hardy space H L , G ,ϕ ( X ) is defined to be the completion of ˜ H L , G ,ϕ ( X ) with thequasi-norm k·k H L , G ,ϕ ( X ) .What deserves to be mentioned the most is that the Musielak-Orlicz Hardy space H ϕ, L intro-duced in [14] is associated with L satisfying the Davies-Ga ff ney estimates, while the operator L indefinition 1.1 and definition 1.2 satisfies the stronger Gaussian estimates.Motivated by the work of [6, 14, 10], the first contribution of this paper is to establish a dis-crete characterization for the two kinds of Musielak-Orlicz-Hardy spaces H ϕ, L ( X ) and H L , G ,ϕ ( X )defined above. This generalizes the results presented in [6, 10] since H ϕ, L ( X ) contains H pL , S , w ( X )and H L , G ,ϕ ( X ) contains H pL , G , w ( X ). Also, by removing the uniformly reverse H¨older condition ongrowth function ϕ , our work improves a part of results of Yang and Yang [14]. The second goal ofthis article is to prove that H ϕ, L ( X ) and H L , G ,ϕ ( X ) are equivalent, which improves the result aboutthe behavior of Littlewood-Paley g -function G L on H ϕ, L proved in [14, Theorem 6.3].Our main approach is inspired by the results in [8, 6].The layout of this article is as follows.We first recall some basic facts and known results in Sect. 2. In Sect.3, we first establish discretecharacterizations for H ϕ, L ( X ) and H L , G ,ϕ ( X ) and then obtain the consistency between H ϕ, L ( X ) and H L , G ,ϕ ( X ) in the sense of norm as a corollary.Throughout this paper, we mean by writting a (cid:27) b that variables a and b are equivalent, namely,there exist positive constants C and C independent of a and b such that C b ≤ a ≤ C b .
2. Preliminaries
Let ( X , d , µ ) be a metric measure space, namely, d is a metric and µ a nonnegative Borel regularmeasure on X . Throughout out this paper, for any fixed x ∈ X and r ∈ (0 , ∞ ), we denote the openball centered at x with radius r by B ( x , r ) : = { y ∈ X ; d ( x , y ) < r } , and we set V ( x , r ) : = µ ( B ( x , r )). Moreover, we assume that X is of homogeneous type, that is,there exists a constant C D ∈ [1 , ∞ ) such that, for all x ∈ X and r ∈ (0 , ∞ ), V ( x , r ) ≤ C D V ( x , r ) < ∞ . (4)Condition (4) is also called the doubling condition which implies that the following stronghomogeneity property that, for some positive constants C and n , V ( x , λ r ) ≤ C λ n V ( x , r ) (5)uniformly for all λ ∈ [1 , ∞ ), x ∈ X , and r ∈ (0 , ∞ ). And as is shown by Grigor’yan et al. [9], let C D be as in (4) and m = log C D , then for all x , y ∈ X and 0 < r ≤ R < ∞ we have V ( x , R ) ≤ C D " R + d ( x , y ) r m V ( y , r ) . (6)3sing the doubling condition (4), it is trivial to show that for any N > n , there exists a constant C N such that for all x ∈ X and t > Z X (cid:16) + t − d ( x , y ) (cid:17) − N d µ ( y ) ≤ C N V ( x , t ) . (7)We further have the following dyadic cubes decomposition on spaces of homogeneous typeconstructed by Christ [3]. Lemma 2.1.
Let ( X , d , µ ) be a space of homogeneous type. Then, there exist a collection n Q k α ⊂ X : k ∈ Z , α ∈ I k o of open subsets, where I k is some index set, and constants δ ∈ (0 , , andC , C > , such that (i) µ (cid:16) X \∪ α Q k α (cid:17) = , for each fixed k and Q k α ∩ Q k β = ∅ if α , β ; (ii) for any α, β, k , l with k ≤ l, either Q l β ⊂ Q k α or Q l β ∩ Q k α = ∅ ; (iii) for each ( k , α ) and each l < k, there exists a unique β ∈ I l such that Q k α ⊂ Q l β ; (iv) diam (cid:16) Q k α (cid:17) ≤ C δ k ; (v) each Q k α contains some ball B (cid:16) z k α , C δ k (cid:17) , where z k α ∈ X. We can think of Q k α as being a dyadic cube with diameter roughly δ k centered at y Q k α , and wethen set ℓ ( Q k α ) = C δ k . The precise value C is nonessential, and as was proved by Christ [3], inwhat follows, we without loss of generality assume C = δ − . Recall from [14] that a nonnegative nondecreasing function Φ defined on [0 , + ∞ ) is said to bean Orlicz function if Φ (0) = Φ ( t ) > t ∈ (0 , ∞ ) and lim t →∞ Φ ( t ) = ∞ . The function Φ is said to be of upper type p (resp., lower type p ) for some p ∈ [0 , ∞ ), if there exists a positiveconstant C such that, for all t ∈ [1 , ∞ ) (resp., t ∈ [0 , s ∈ [0 , ∞ ), Φ ( st ) ≤ Ct p Φ ( s ). And itis trivial that an Orlicz function is of upper type 1, if it is of upper type p ∈ (0 , ϕ : X × [0 , + ∞ ) → [0 , + ∞ ) be a function, such that, for any x ∈ X , ϕ ( x , · ) is an Orliczfunction. We say that ϕ is of uniformly upper type p (resp., uniformly lower type p ) for some p ∈ [0 , ∞ ), if there exists a positive constant C such that, for all x ∈ X , t ∈ [1 , ∞ ) (resp., t ∈ [0 , s ∈ [0 , ∞ ), ϕ ( x , st ) ≤ Ct p ϕ ( x , s ) . (8)As in Ky [11], a function ϕ : X × [0 , + ∞ ) → [0 , + ∞ ) is said to be uniformly locally integrable ,if for all t ∈ [0 , ∞ ), x ϕ ( x , t ) is measurable and for all bounded subsets K ⊂ X , Z K sup t ∈ (0 , ∞ ) ϕ ( x , t ) "Z K ϕ ( y , t ) d µ ( y ) − d µ ( x ) < ∞ . Following [14, 15], we next recall the definition of the
Uniform Muchenhoupt Class and itsproperties. 4 efinition 2.1.
Let ϕ : X × [0 , + ∞ ) → [0 , + ∞ ) be uniformly locally integrable. The function ϕ ( · , t ) is said to satisfy the uniformly Muckenhoupt condition for some q ∈ [1 , ∞ ), denoted by ϕ ∈ A q ( X ), if, when q ∈ (1 , ∞ ), A q ( ϕ ) : = sup t ∈ (0 , ∞ ) sup B ⊂ X ( µ ( B ) Z B ϕ ( x , t ) d µ ( x ) ) × ( µ ( B ) Z B (cid:2) ϕ ( y , t ) (cid:3) − q ′ / q d µ ( y ) ) q / q ′ < ∞ , where 1 / q + / q ′ =
1, or A ( ϕ ) : = sup t ∈ (0 , ∞ ) sup B ⊂ X µ ( B ) Z B ϕ ( x , t ) d µ ( x ) ( esssup y ∈ B (cid:2) ϕ ( y , t ) (cid:3) − ) < ∞ . We further define A ∞ ( X ) : = S q ∈ [1 , ∞ ) A q ( X ) and let q ( ϕ ) : = inf n q ∈ [1 , ∞ ) ; ϕ ∈ A q ( X ) o to be the critical indices of ϕ .The following properties of A ∞ ( X ) and their proofs are similar to those in Yang et al. [15], andwe omit the details. In what follows, we use the notation ϕ ( E , t ) : = Z E ϕ ( x , t ) d µ ( x )for any measurable subset E of X and t ∈ [0 , ∞ ). And M denotes the Hardy-Liitlewood maximalfunction on X , i.e., for all x ∈ X , M ( f ) ( x ) : = sup B ∋ x µ ( B ) Z B | f ( y ) | d µ ( y ) , where the supremum is taken over all balls B ∋ x . Lemma 2.2. A ( X ) ⊂ A p ( X ) ⊂ A q ( X ) , for ≤ p ≤ q < ∞ . If ϕ ∈ A p ( X ) with p ∈ (1 , ∞ ) , then there exists some q ∈ (1 , p ) such that ϕ ∈ A q ( X ) . If ϕ ∈ A p ( X ) with p ∈ (1 , ∞ ) , then there exists a positive constant C such that, for allmeasurable functions f on X and t ∈ [0 , ∞ ) , Z X (cid:2) M ( f ) ( x ) (cid:3) p ϕ ( x , t ) d µ ( x ) ≤ C Z X | f ( x ) | p ϕ ( x , t ) d µ ( x ) . If ϕ ∈ A p ( X ) with p ∈ [1 , ∞ ) , then there exists a positive constant C such that, for all ballsB ⊂ X and measurable subset E ⊂ B and t ∈ [0 , ∞ ) , ϕ ( B , t ) ϕ ( E , t ) ≤ C h µ ( B ) µ ( E ) i p . We now introduce the growth functions and their properties which can be found in [11, 14].
Definition 2.2.
A function ϕ : X × [0 , + ∞ ) → [0 , + ∞ ) is called a growth function if the followingremain true:(i) ϕ is a Musielak-Orlicz function , namely,(i) the function ϕ ( x , · ) : [0 , + ∞ ) → [0 , + ∞ ) is an Orlicz function for all x ∈ X ;5i) b the funtion ϕ ( · , t ) is an measurable function for all t ∈ [0 , ∞ ).(ii) ϕ ∈ A ∞ ( X ) . (iii) The function ϕ is of positive uniformly upper type p for some p ∈ (0 ,
1] and of uniformlylower type p for some p ∈ (0 , Lemma 2.3.
Let ϕ be a growth function. Set ˜ ϕ ( x , t ) : = R t ϕ ( x , s ) dss for all ( x , t ) ∈ X × [0 , ∞ ) .Then ˜ ϕ is a growth function, which is equivalent to ϕ ; moreover, ˜ ϕ ( x , · ) is continuous and strictlyincreasing.2.3. Musielak-Orlicz Space In this subsection we recall the Musielak-Orlicz Space and obtain a vector-valued inequality.In what follows, we always assume ϕ is a growth function .The Musielak-Orlicz space L ϕ ( X ) contains all measurable functions f which satisfy R X ϕ ( x , | f ( x ) | ) d µ ( x ) < ∞ with Luxembourg norm k f k L ϕ ( X ) : = in f ( λ ∈ (0 , ∞ ) ; Z X ϕ x , | f ( x ) | λ ! d µ ( x ) ≤ ) . The following Lemma of Musielak-Orlicz Fe ff erman-Stein vector-valued inequality is ob-tained by Obtained by Yiyu et al. [16]. In what follows, the space L ϕ ( ℓ p , X ) is defined to bethe set of all n f j o j ∈ Z satisfying hP j (cid:12)(cid:12)(cid:12) f j (cid:12)(cid:12)(cid:12) p i / p ∈ L ϕ ( X ) and we let (cid:13)(cid:13)(cid:13)(cid:13)n f j o j ∈ Z (cid:13)(cid:13)(cid:13)(cid:13) L ϕ ( ℓ p , X ) : = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:20)X j (cid:12)(cid:12)(cid:12) f j (cid:12)(cid:12)(cid:12) p (cid:21) / p (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ϕ ( X ) . Lemma 2.4.
Let p ∈ (1 , ∞ ] , ϕ be a Musielak-Orlicz function with uniformly lower type p andupper type p , and ϕ ∈ A q ( X ) for q ∈ (1 , ∞ ) . If q ( ϕ ) < p ≤ p < ∞ , then there exists a positiveconstant C such that, for all n f j o j ∈ Z ∈ L ϕ ( ℓ p , X ) , Z X ϕ x , (cid:20)X j M (cid:16) f j (cid:17) ( x ) p (cid:21) / p ! d µ ( x ) ≤ C Z X ϕ x , (cid:20)X j (cid:12)(cid:12)(cid:12) f j ( x ) (cid:12)(cid:12)(cid:12) p (cid:21) / p ! d µ ( x ) . Corollary 2.1.
Let p and ϕ be as in Lemma 2.4, then for all r ∈ (0 , p / q ( ϕ )) , we have Z X ϕ x , (cid:20)X j M (cid:16) f j (cid:17) ( x ) p (cid:21) / rp ! d µ ( x ) ≤ C Z X ϕ x , (cid:20)X j (cid:12)(cid:12)(cid:12) f j ( x ) (cid:12)(cid:12)(cid:12) p (cid:21) / rp ! d µ ( x ) . Proof
For any fixed r ∈ (0 , p / q ( ϕ )), let ˜ ϕ ( x , t ) = ϕ (cid:16) x , t / r (cid:17) . We claim that ˜ ϕ is of uniformlylower type p r and upper type p r . By assumption, there exists a constant C , such that˜ ϕ ( x , st ) = ϕ (cid:16) x , s / r t / r (cid:17) ≤ C t p / r ϕ (cid:16) x , s / r (cid:17) = C t p / r ˜ ϕ ( x , s )for all t ∈ [0 , x ∈ X and s ∈ [0 , ∞ ). In the mean time, there exists another constant C , such that˜ ϕ ( x , st ) = ϕ (cid:16) x , s / r t / r (cid:17) ≤ C t p / r ϕ (cid:16) x , s / r (cid:17) = C t p / r ˜ ϕ ( x , s )for all t ∈ [1 , ∞ ), x ∈ X and s ∈ [0 , ∞ ). 6ince q ( ˜ ϕ ) = q ( ϕ ), for arbitrary r ∈ (0 , p / q ( ϕ )), it is trivial that q ( ˜ ϕ ) = q ( ϕ ) < p r ≤ p r < ∞ . By employing Lemma 2.4, we obtain Z X ϕ x , (cid:20)X j M (cid:16) f j (cid:17) ( x ) p (cid:21) / rp ! d µ ( x ) = Z X ˜ ϕ x , (cid:20)X j M (cid:16) f j (cid:17) ( x ) p (cid:21) / p ! d µ ( x ) ≤ C Z X ˜ ϕ x , (cid:20)X j (cid:12)(cid:12)(cid:12) f j ( x ) (cid:12)(cid:12)(cid:12) p (cid:21) / p ! d µ ( x ) = C Z X ϕ x , (cid:20)X j (cid:12)(cid:12)(cid:12) f j ( x ) (cid:12)(cid:12)(cid:12) p (cid:21) / rp ! d µ ( x ) . (cid:3) AT L , M -Family Associated with Operator L Recalling that X is a space that satisfies the strong homogeneity property (5) with homogeneousdimension n . In the view of Lemma 2.1, there exists a collection n Q k α ⊂ X ; k ∈ Z , α ∈ I k } of opensubsets, where I k is the index set, such that for every k ∈ Z , X = [ α ∈ I k Q k α with properties of Q k α as in Lemma 2.1. In what follows, such open subsets n Q k α ⊂ X ; k ∈ Z , α ∈ I k } is said to be a family of dyadic cubes of X . And we now turn to introduce the AT L , M -familyassociated with an operator L whose definition can also be found in [6]. Definition 2.3.
Suppose that an operator L satisfies ( H1 ) and ( H2 ) and M ∈ N . A collection offunctions (cid:8) a Q (cid:9) Q : Dyadic is said to be an AT L , M -family associated with L, if for each dyadic cube Q,there exists a function b Q ∈ D (cid:16) L M (cid:17) such that (i) a Q = L M (cid:0) b Q (cid:1) ;(ii) supp (cid:16) L k (cid:0) b Q (cid:1)(cid:17) ⊂ Q, k = , , · · · , M; (iii) (cid:16) ℓ ( Q ) L (cid:17) k (cid:0) b Q (cid:1) ≤ ℓ ( Q ) M µ ( Q ) − / , k = , , · · · , M. Here, D ( L ) denotes the domain of operator L , and by L k the k -fold composition of L with itself.With this definition, we can decompose an L function into AT L , M -family. Given a function f ∈ L ( X ), we say that f has an AT L , M - expansion , if there exists sequence s = (cid:8) s Q (cid:9) Q : Dyadic ,0 ≤ s Q < ∞ and an AT L , M -family (cid:8) a Q (cid:9) Q : Dyadic in L ( X ) such that f = X Q : Dyadic s Q a Q . (9)We then define by W f ( x ) the function related to the sequence s = (cid:8) s Q (cid:9) Q : Dyadic , 0 ≤ s Q < ∞ as W f ( x ) : = (cid:18)X Q : Dyadic µ ( Q ) − | s Q | X Q ( x ) (cid:19) / . (10)7 roposition 2.1. Given an operator L that satisfies ( H1 )-( H2 ) and f ∈ L ( X ) . Then for allM ∈ N , f has an AT L , M -expansion. Moreover, let Q k α and δ be as in Lemma 2.1, we haves Q k α = Z δ k δ k + Z Q k α | t Le − t L f ( y ) | d µ ( y ) dtt / . The proof of Proposition 2.1 can be found in [6, Proof of Theorem 3.2]. We omit the details.
3. Musielark-Orlicz Hardy Space H ϕ, L and its Equivalent Characterization In this section, we begin to study the Musielar-Orlicz-Hardy spce, and in what follows, wealways assume the operator L satisfies ( H1 ) and ( H2 ), and ϕ is a growth function which is definedin Definition 2.2. With some basic notations set forth in Sect. 2, we first establish the followingcharacterization for the Hardy space H ϕ, L . Theorem 3.1.
Suppose L is an operator that satisfies ( H1 ) and ( H2 ). Let ϕ be a growth func-tion with uniformly lower type p and f ∈ H ϕ, L ( X ) ∩ L ( X ) , then for all natural number M > nq ( ϕ ) / (2 p ) , f has an AT L , M -expansion such that k f k H ϕ, L ( X ) (cid:27) (cid:13)(cid:13)(cid:13) W f (cid:13)(cid:13)(cid:13) L ϕ ( X ) . Before we prove Theorem 3.1, we need to introduce some notions and establish some resultsas follows.For any v ∈ (0 , ∞ ) and x ∈ X , let Γ ν ( x ) : = { ( y , t ) ∈ X × (0 , ∞ ) ; d ( x , y ) < ν t } be the cone ofaperture ν with vertex x ∈ X . For any closed subset F of X , denote by R ν ( F ) the union of allcones with vertices in F , i.e., R ν ( F ) = S x ∈ F Γ ν ( x ). In what follows, we denote Γ ( x ) and R ( F )simply by Γ ( x ) and R ( F ), respectively. For any open subset O of X , we establish the followinggeometric property of R (cid:16) O ∁ (cid:17) which generalizes a similar result obtained by Aguilera and Segovia[1, Lemma 1] in the case of Euclidean space. Lemma 3.1.
Suppose that ( X , d , µ ) is a space of homogeneous type with constant C D > suchthat (4) holds. Let O be an open subset of X , F = O ∁ and X O its characteristic function. If for ν > , we define O ∗ as O ∗ : = n x ∈ X ; M ( X O ) ( x ) > (4 ν ) − C D o and let F ∗ = ( O ∗ ) ∁ , then we have (i) R ν ( F ∗ ) is contained in R ( F ) . (ii) If ( z , t ) ∈ R ν ( F ∗ ) , then there exists some constant C ν such thatV ( z , t ) < C ν µ ( B ( z , t ) ∩ F ) . Proof
The lemma is trivial if R ν ( F ∗ ) = ∅ . We then with no loss of generality assume that R ν ( F ∗ ) , ∅ , which implies that O , X . We then first prove (i). If ( z , t ) ∈ R ν ( F ∗ ), then either z ∈ F or z ∈ O . In the first case it is apparent that ( z , t ) ∈ R ( F ), since d ( z , z ) = < t .If we are in the second case, i.e., z ∈ O , let δ be the distance from z to the closed and non-empty set F . This number δ is positive and finite, and B ( z , δ ) is contained in O . The assumption8hat ( z , t ) ∈ R ν ( F ∗ ) implies that there is y ∈ F ∗ with d ( z , y ) < ν t . Thus, writing r = δ + d ( z , y ), weget B ( z , δ ) ⊂ B ( y , r ) and also B ( z , δ ) ⊂ B ( z , δ ) ∩ O ⊂ B ( y , r ) ∩ O , which together with the definition of O ∗ , implies that V ( z , δ ) ≤ µ ( B ( y , r ) ∩ O ) ≤ (4 ν ) − C D V ( y , r )since y ∈ F ∗ .By using (6) twice, we also have V ( y , r ) ≤ C D (cid:16) r δ − (cid:17) log C D V ( y , δ ) ≤ C D (cid:16) r δ − (cid:17) log C D (cid:16) + δ − d ( y , z ) (cid:17) log C D V ( z , δ ) = (cid:16) r δ − (cid:17) C D V ( z , δ ) . From these inequalities, we get that δ ≤ r ν . Recalling that r = δ + d ( z , y ) and d ( z , y ) < ν t , we obtain δ ≤ δ + d ( z , y )2 ν < δ + ν t ν and since ν >
1, it follows that δ < t . Then by the very definition of δ , there exists an x ∈ F ,satisfying d ( x , z ) < t , which means that ( z , t ) ∈ R ( F ). This proves (i).We then turn our attention to (ii). If ( z , t ) ∈ R ν ( F ∗ ), there is y ∈ F ∗ such that d ( z , y ) < ν t . Then B ( z , t ) ⊂ B ( y , (1 + ν ) t ) and since y ∈ F ∗ , we get µ ( B ( z , t ) ∩ O ) ≤ µ ( B ( y , (1 + ν ) t ) ∩ O ) ≤ (4 ν ) − C D V ( y , (1 + ν ) t ) , and therefore µ ( B ( z , t ) ∩ O ) ≤ (4 ν ) − C D C D (1 + ν ) log C D V ( y , t ) ≤ C D (4 ν ) − C D (1 + ν ) log C D (cid:16) + t − d ( y , z ) (cid:17) log C D V ( z , t ) < + ν ν ! C D V ( z , t ) . Now from V ( z , t ) = µ ( B ( z , t ) ∩ O ) + µ ( B ( z , t ) ∩ F ), we obtain − + ν ν ! C D V ( z , t ) < µ ( B ( z , t ) ∩ F ) , which implies (ii). (cid:3) Next we introduce the following variant of Lusin-area function associated with L . For all ν ∈ (0 , ∞ ), f ∈ L ( X ) and x ∈ X , let S L ,ν ( f ) ( x ) : = Z ∞ Z d ( x , y ) <ν t | t Le − t L ( f ) ( y ) | d µ ( y ) V ( x , t ) dtt ! / .
9e also have the following two Lemmas for the variant of Lusin-area function that associated with L which generalize the results of Aguilera and Segovia [1, Lemma 2] and Yang et al. [15, Lemma3.3.5]. Lemma 3.2.
Assume that L satisfies ( H1 ) and ( H2 ). Let ϕ ∈ A p ( X ) , ≤ p < ∞ , and O be anopen subset of X. If O ∗ is the set associated to O as in Lemma 3.1 with some ν > , then thereexists a finite constant C, which is independent of O, such that for all λ ∈ (0 , ∞ ) and f ∈ L ( X ) , Z F ∗ | S L ,ν ( f ) ( x ) | ϕ ( x , λ ) d µ ( x ) ≤ C Z F | S L ( f ) ( x ) | ϕ ( x , λ ) d µ ( x ) , where F ∗ = ( O ∗ ) ∁ and F = O ∁ . Proof
For any x ∈ F ∗ , ( y , t ) ∈ Γ ν ( x ), we observe that d ( x , y ) < ν t , and hence by (6), V ( x , t ) − ≤ C D (cid:16) + t − d ( x , y ) (cid:17) log C D V ( y , t ) − < C D (1 + ν ) log C D V ( y , t ) − . It follows that Z F ∗ | S L ,ν ( f ) ( x ) | ϕ ( x , λ ) d µ ( x ) ≤ C D (1 + ν ) log C D Z F ∗ Z Γ ν ( x ) | t Le − t L ( f ) ( y ) | d µ ( y ) dtV ( y , t ) t ! ϕ ( x , λ ) d µ ( x ) = C ν, D Z R ν ( F ∗ ) | t Le − t L ( f ) ( y ) | V ( y , t ) − ϕ ( B ( y , ν t ) ∩ F ∗ , λ ) d µ ( y ) dtt . (11)We then employ Lemma 2.2 (iv) to the set E = B ( y , t ) and B = B ( y , ν t ), to get ϕ ( B ( y , ν t ) , λ ) ≤ C (2 ν ) log C D ϕ ( B ( y , t ) , λ ) . (12)Applying Lemma 2.2 (iv) once again to E = B ( y , t ) ∩ F and B = B ( y , t ), we get ϕ ( B ( y , t ) , λ ) ≤ C V ( y , t ) µ ( B ( y , t ) ∩ F ) ! p ϕ ( B ( y , t ) ∩ F , λ ) . (13)Therefore, from (12) and (13), plus part (ii) of Lemma 3.1, we have ϕ ( B ( y , ν t ) , λ ) ≤ C ϕ ( B ( y , t ) ∩ F , λ ) . From this estimate it follows that the last integral in (11) is bounded by C Z R ν ( F ∗ ) | t Le − t L ( f ) ( y ) | V ( y , t ) − ϕ ( B ( y , t ) ∩ F , λ ) d µ ( y ) dtt . Finally, in the view of Lemma 3.1 (i), we observe that R ν ( F ∗ ) ⊂ R ( F ), it follows immediately that10he last integral above is bounded by C Z R ( F ) | t Le − t L ( f ) ( y ) | V ( y , t ) − ϕ ( B ( y , t ) ∩ F , λ ) d µ ( y ) dtt = C Z F Z Γ ν ( x ) | t Le − t L ( f ) ( y ) | d µ ( y ) dtV ( y , t ) t ! ϕ ( x , λ ) d µ ( x ) ≤ C Z F | S L ( f ) ( x ) | ϕ ( x , λ ) d µ ( x ) , where the last inequality follows from the fact that V ( y , t ) − ≤ C D (cid:16) + t − d ( x , y ) (cid:17) log C D V ( x , t ) − < C D V ( x , t ) − for ( y , t ) ∈ Γ ν ( x ). This proves the lemma. (cid:3) Lemma 3.3.
Assume that L satisfies ( H1 ) and ( H2 ). Let q ∈ (1 , ∞ ) , ϕ be as in Definition 2.2 and ϕ ∈ A q ( X ) . Then for all ν ∈ (0 , ∞ ) there exists a positive constant C ν such that, for all measurablefunctions f , Z X ϕ (cid:0) x , S L ,ν ( f ) ( x ) (cid:1) d µ ( x ) ≤ C ν Z X ϕ ( x , S L ( f ) ( x )) d µ ( x ) . Proof If ν ∈ (0 , ν ∈ (1 , ∞ ). For all λ ∈ (0 , ∞ ), let O λ : = { x ∈ X ; S L ( f ) ( x ) > λ } and O ∗ λ : = n x ∈ X ; M (cid:0) X O λ (cid:1) ( x ) > (4 ν ) − log C D o . where M is the Hardy-Littlewood maximal function. Since ϕ ∈ A q ( X ), it follows from Lemma2.2 (iii), ϕ (cid:0) O ∗ λ , λ (cid:1) = ϕ (cid:16)n x ∈ X ; M (cid:0) X O λ (cid:1) ( x ) > (4 ν ) − log C D o , λ (cid:17) ≤ Z X (4 ν ) q log C D (cid:0) M (cid:0) X O λ (cid:1) ( x ) (cid:1) q ϕ ( x , λ ) d µ ( x ) ≤ C ϕ ( O λ , λ ) . (14)Let F λ : = O ∁ λ , F ∗ λ : = ( O ∗ λ ) ∁ and apply Lemma 3.2 to get Z F ∗ λ | S L ,ν ( f ) ( x ) | ϕ ( x , λ ) d µ ( x ) ≤ C Z F λ | S L ( f ) ( x ) | ϕ ( x , λ ) d µ ( x ) . (15)11hus, from (14) and (15), it follows that ϕ (cid:0)(cid:8) x ∈ X ; S L ,ν ( f ) ( x ) > λ (cid:9) , λ (cid:1) ≤ ϕ (cid:0) O ∗ λ , λ (cid:1) + ϕ (cid:0)(cid:8) x ∈ F ∗ λ ; S L ,ν ( f ) ( x ) > λ (cid:9) , λ (cid:1) ≤ C ϕ ( O λ , λ ) + λ − Z F ∗ λ | S L ,ν ( f ) ( x ) | ϕ ( x , λ ) d µ ( x ) ≤ C " ϕ ( O λ , λ ) + λ − Z F λ | S L ( f ) ( x ) | ϕ ( x , λ ) d µ ( x ) ≤ C " ϕ ( O λ , λ ) + λ − Z λ t ϕ ( { x ∈ X ; S L ( f ) ( x ) > t } , λ ) dt , which, together with the assumption ν ∈ (1 , ∞ ), Lemma 2.3 and the uniformly upper type 1 of ϕ ,we further get that Z X ϕ (cid:0) x , S L ,ν ( f ) ( x ) (cid:1) d µ ( x ) ≤ C Z ∞ λ − ϕ (cid:0)(cid:8) x ∈ X ; S L ,ν ( f ) ( x ) > λ (cid:9) , λ (cid:1) d λ ≤ C "Z ∞ λ − ϕ ( O λ , λ ) d λ + Z ∞ λ − Z λ t ϕ ( { x ∈ X ; S L ( f ) ( x ) > t } , λ ) dtd λ ≤ C "Z ∞ λ − ϕ ( { x ∈ X ; S L ( f ) ( x ) > λ } , λ ) d λ + Z ∞ λ − Z λ t ϕ ( { x ∈ X ; S L ( f ) ( x ) > t } , λ ) dtd λ ≤ C "Z X ϕ ( x , S L ( f ) ( x )) d µ ( x ) + Z ∞ ϕ ( { x ∈ X ; S L ( f ) ( x ) > t } , t ) Z ∞ t λ − d λ dt ≤ C Z X ϕ ( x , S L ( f ) ( x )) d µ ( x ) . This finishes the proof of Lemma 3.3. (cid:3)
Moreover, we also need the following Lemma, whose standard proof can be found in [7], weomit the details. And in what follows, we recall that the Hardy-Littlewood maximal operator M on ( X , µ, d ) is defined by M ( f ) ( x ) : = sup B ∋ x µ ( B ) Z B | f ( y ) | d µ ( y ) . where the supremum is taken over all balls B ∋ x .12 emma 3.4. Suppose < q ≤ and N > n / q, where n is the doubling dimension of the space in (5) . Fix an integer k, and let n s Q k α o α ∈ I k be as in Proposition 2.1, then for any subsequence I k ′ ⊂ I k and each x ∈ X, X α ∈ I k ′ (cid:12)(cid:12)(cid:12) s Q k α (cid:12)(cid:12)(cid:12)h + ℓ ( Q k α ) − d (cid:0) x , y k α (cid:1)i N ≤ C M X α ∈ I k ′ (cid:12)(cid:12)(cid:12) s Q k α (cid:12)(cid:12)(cid:12) q X ( · ) / q ( x ) . where y k α denotes the center of Q k α .Proof of theorem 3.1 For any fixed f ∈ H ϕ, L ( X ) ∩ L ( X ), we let λ = k f k H ϕ, L ( X ) and λ = (cid:13)(cid:13)(cid:13) W f (cid:13)(cid:13)(cid:13) L ϕ ( X ) . It su ffi ces to show that for all λ ∈ (0 , ∞ ), Z X ϕ x , S L ( f ) ( x ) λ ! d µ ( x ) (cid:27) Z X ϕ x , (cid:12)(cid:12)(cid:12) W f ( x ) (cid:12)(cid:12)(cid:12) λ d µ ( x ) . (16)In fact, if (16) holds for all λ ∈ (0 , ∞ ), then there exists a constant C such that Z X ϕ x , S L ( f ) ( x ) λ ! d µ ( x ) ≤ C Z X ϕ x , (cid:12)(cid:12)(cid:12) W f ( x ) (cid:12)(cid:12)(cid:12) λ d µ ( x ) ≤ C , which, together with (8), implies that Z X ϕ x , G L ( f ) ( x ) C λ ! d µ ( x ) ≤ C , and hence we have λ ≤ C λ . In a similar fashion, one can prove λ ≤ C λ for some constants C , and get the desired property.Let f be a function in H ϕ, L ( X ) ∩ L ( X ). In the view of Lemma 2.1, for any fixed ( x , k ) ∈ X × Z there exists a unique α ∈ I k , such that x ∈ Q k α . Let Q kx denote the such Q k α and we write W f ( x ) = X k ∈ Z X α ∈ I k (cid:20) µ (cid:16) Q k α (cid:17) − / (cid:12)(cid:12)(cid:12) s Q k α (cid:12)(cid:12)(cid:12) X Q k α ( x ) (cid:21) / = X k ∈ Z µ (cid:16) Q kx (cid:17) − (cid:12)(cid:12)(cid:12) s Q kx (cid:12)(cid:12)(cid:12) / = X k ∈ Z Z δ k δ k + µ (cid:16) Q kx (cid:17) − Z Q kx | t Le − t L f ( y ) | d µ ( y ) dtt / , (17)where δ ∈ (0 ,
1) is a constant as in Lemma 2.1 and the last quantity follows from Proposition 2.1.Moreover, by ( iv ) and ( v ) of Lemma 2.1, we know that for any fixed ( x , k ) ∈ X × Z there exists z kx ∈ Q kx and constants C ∈ (0 , C = δ − such that diam (cid:16) Q k α (cid:17) ≤ C δ k : = ℓ (cid:16) Q k α (cid:17) and B (cid:16) z kx , C δ k (cid:17) ⊂ Q kx ⊂ B (cid:16) x , C δ k (cid:17) ⊂ B (cid:16) x , C δ − t (cid:17) t ∈ (cid:16) δ k + , δ k (cid:17) . It follows immediately from (6) that µ (cid:16) Q k α (cid:17) − ≤ V (cid:16) z kx , C δ k (cid:17) − ≤ C + d (cid:16) x , z kx (cid:17) C δ k m V (cid:16) x , C δ k (cid:17) − ≤ CV (cid:16) x , C δ k (cid:17) − ≤ CV (cid:16) x , δ k (cid:17) − ≤ CV ( x , t ) − , where the last but one inequality follows from the fact that V (cid:16) x , δ k (cid:17) ≤ C D C − m V (cid:16) x , C δ k (cid:17) with C ∈ (0 , W f ( x ) ≤ C X k ∈ Z Z δ k δ k + V ( x , t ) − Z B ( x , C δ − t ) | t Le − t L f ( y ) | d µ ( y ) dtt / = C (Z ∞ Z d ( x , y ) <δ − t | t Le − t L f ( y ) | d µ ( y ) V ( x , t ) dtt ) / = CS L ,δ − ( f ) ( x ) , which, together with Lemma 3.3, we deduce the ≥ inequality of (16).It remains to establish the reverse inequality. In the view of Proposition 2.1, we write f = P k ∈ Z P α ∈ I k s Q k α a Q k α . Let δ be as in Lemma 2.1 we get S L ( f ) ( x ) = Z ∞ Z d ( x , y ) < t | t Le − t L ( f ) ( y ) | d µ ( y ) V ( x , t ) dtt ! / = Z ∞ Z d ( x , y ) < t | t Le − t L ( X k ∈ Z X α ∈ I k s Q k α a Q k α ) ( y ) | d µ ( y ) V ( x , t ) dtt / = X j ∈ Z Z δ j − δ j Z d ( x , y ) < t | t Le − t L ( X k ∈ Z X α ∈ I k s Q k α a Q k α ) ( y ) | d µ ( y ) V ( x , t ) dtt / ≤ X j ∈ Z Z δ j − δ j Z d ( x , y ) < t | t Le − t L ( X k > j X α ∈ I k s Q k α a Q k α ) ( y ) | d µ ( y ) V ( x , t ) dtt / + X j ∈ Z Z δ j − δ j Z d ( x , y ) < t | t Le − t L ( X k ≤ j X α ∈ I k s Q k α a Q k α ) ( y ) | d µ ( y ) V ( x , t ) dtt / . (18)We now estimate the first part of (18). For any k > j and α ∈ I k , noting that a Q k α = L M b Q k α , wewrite (cid:12)(cid:12)(cid:12)(cid:12) t Le − t L (cid:16) a Q k α (cid:17) ( y ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) t L M + e − t L (cid:16) b Q k α (cid:17) ( y ) (cid:12)(cid:12)(cid:12)(cid:12) = t − M (cid:12)(cid:12)(cid:12)(cid:12)(cid:16) t L (cid:17) M + e − t L (cid:16) b Q k α (cid:17) ( y ) (cid:12)(cid:12)(cid:12)(cid:12) . n be as in (5), since M > nq ( ϕ ) / (2 p ), we can choose some q = r with r be as in Corollary 2.1such that 2 M > n / q . We then let N be some positive number such that 2 M > N > n / q . Then byDefinition 2.3, the upper bound of the kernel of (cid:16) t L (cid:17) M + e − t L and (7), we get (cid:12)(cid:12)(cid:12)(cid:12) t Le − t L (cid:16) a Q k α (cid:17) ( y ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C V ( y , t ) t − M ℓ (cid:16) Q k α (cid:17) M µ (cid:16) Q k α (cid:17) − / Z Q k α exp − d ( y , z ) C t ! d µ ( z ) ≤ Ct − M ℓ (cid:16) Q k α (cid:17) M µ (cid:16) Q k α (cid:17) − / tt + d ( y , z k α ) ! N , where we denote by z k α the center of Q k α . By the fact that d ( x , y ) < t , we further obtain Z d ( x , y ) < t | t Le − t L ( a Q k α ) ( y ) | d µ ( y ) V ( x , t ) ! / ≤ Ct − M ℓ (cid:16) Q k α (cid:17) M µ (cid:16) Q k α (cid:17) − / Z d ( x , y ) < t t + d ( x , y ) t + d (cid:0) x , z k α (cid:1) ! N d µ ( y ) V ( x , t ) / ≤ Ct − M ℓ (cid:16) Q k α (cid:17) M µ (cid:16) Q k α (cid:17) − / (cid:16) + t − d ( x , z k α ) (cid:17) − N . Hence, we have Z d ( x , y ) < t | t Le − t L ( X k > j X α ∈ I k s Q k α a Q k α ) ( y ) | d µ ( y ) V ( x , t ) / ≤ C X k > j X α ∈ I k t − M ℓ (cid:16) Q k α (cid:17) M µ (cid:16) Q k α (cid:17) − / | s Q k α | (cid:16) + t − d ( x , z k α ) (cid:17) − N ≤ C X k > j δ (2 M − N )( k − j ) X α ∈ I k µ (cid:16) Q k α (cid:17) − / | s Q k α | h + ℓ ( Q k α ) − d ( x , z k α ) i N ≤ C X k > j δ (2 M − N )( k − j ) M X α ∈ I k | s Q k α | q µ (cid:16) Q k α (cid:17) − q / X Q k α ( · ) ( x ) / q , (19)where the last inequality follows from Lemma 3.4.Estimate of the second part of (18). For any k ≤ j and α ∈ I k , we write (cid:12)(cid:12)(cid:12)(cid:12) t Le − t L (cid:16) a Q k α (cid:17) ( y ) (cid:12)(cid:12)(cid:12)(cid:12) = t (cid:12)(cid:12)(cid:12)(cid:12) e − t L (cid:16) L (cid:16) a Q k α (cid:17)(cid:17) ( y ) (cid:12)(cid:12)(cid:12)(cid:12) . Then by Definition 2.3, the Gaussian estimate (1) and inequality (7), we get (cid:12)(cid:12)(cid:12)(cid:12) t Le − t L (cid:16) a Q k α (cid:17) ( y ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C V ( y , t ) t ℓ (cid:16) Q k α (cid:17) − µ (cid:16) Q k α (cid:17) − / Z Q k α exp − d ( y , z ) C t ! d µ ( z ) . ≤ Ct ℓ (cid:16) Q k α (cid:17) − µ (cid:16) Q k α (cid:17) − / (cid:16) + ℓ ( Q k α ) − d ( y , z k α ) (cid:17) − N , d ( x , y ) < t ≤ ℓ ( Q k α ) further implies that Z d ( x , y ) < t | t Le − t L ( a Q k α ) ( y ) | d µ ( y ) V ( x , t ) ! / ≤ Ct ℓ (cid:16) Q k α (cid:17) − µ (cid:16) Q k α (cid:17) − / Z d ( x , y ) < t ℓ ( Q k α ) + d ( x , y ) ℓ ( Q k α ) + d ( x , z k α ) ! N d µ ( y ) V ( x , t ) / ≤ Ct ℓ (cid:16) Q k α (cid:17) − µ (cid:16) Q k α (cid:17) − / (cid:16) + ℓ ( Q k α ) − d ( x , z k α ) (cid:17) − N . Therefore, by employing Lemma 3.4 once again, we get Z d ( x , y ) < t | t Le − t L ( X k ≤ j X α ∈ I k s Q k α a Q k α ) ( y ) | d µ ( y ) V ( x , t ) / ≤ C X k ≤ j X α ∈ I k t ℓ (cid:16) Q k α (cid:17) − µ (cid:16) Q k α (cid:17) − / | s Q k α | (cid:16) + ℓ ( Q k α ) − d ( x , z k α ) (cid:17) − N ≤ C X k ≤ j δ j − k ) X α ∈ I k µ (cid:16) Q k α (cid:17) − / | s Q k α | h + ℓ ( Q k α ) − d ( x , z k α ) i N ≤ C X k ≤ j δ j − k ) M X α ∈ I k | s Q k α | q µ (cid:16) Q k α (cid:17) − q / X Q k α ( · ) ( x ) / q . (20)Set F k ( x ) : = M (cid:16)P α ∈ I k | s Q k α | q µ ( Q k α ) − q / X Q k α ( · ) (cid:17) ( x ). For any fix j ∈ Z , we let β > τ = k > j and τ = − k ≤ j . We now turn to estimate (cid:16)P k δ βτ ( k − j ) F k ( x ) / q (cid:17) for the case either k > j or k ≤ j . Since δ βτ ( k − j ) = βδ β − δ β Z δ τ ( k − j ) − δ τ ( k − j ) s β − ds , we let E k : = h δ τ ( k − j ) , δ τ ( k − j ) − i and it follows that (cid:16)X k δ βτ ( k − j ) F k ( x ) / q (cid:17) = C β X k Z δ τ ( k − j ) − δ τ ( k − j ) F k ( x ) / q s β − ds = C β Z X k χ E k ( s ) F k ( x ) / q s β − ds ≤ C β Z s β − ds ! Z (cid:16)X k χ E k ( s ) F k ( x ) / q (cid:17) s β − ds ! ≤ C β Z (cid:16)X k χ E k ( s ) F k ( x ) / q (cid:17) s β − ds = C β X k Z δ τ ( k − j ) − δ τ ( k − j ) F k ( x ) / q s β − ds C β X k δ βτ ( k − j ) F k ( x ) / q . (21)Combining now (18)-(21), we get the following estimate for S L ( f ), S L ( f ) ( x ) ≤ X j ∈ Z Z δ j − δ j Z d ( x , y ) < t | t Le − t L ( X k > j X α ∈ I k s Q k α a Q k α ) ( x ) | d µ ( y ) V ( x , t ) dtt / + X j ∈ Z Z δ j − δ j Z d ( x , y ) < t | t Le − t L ( X k ≤ j X α ∈ I k s Q k α a Q k α ) ( x ) | d µ ( y ) V ( x , t ) dtt / ≤ C X j ∈ Z Z δ j − δ j | X k > j δ (2 M − N )( k − j ) F k ( x ) / q | dtt + X j ∈ Z Z δ j − δ j | X k ≤ j δ j − k ) F k ( x ) / q | dtt / ≤ C X j ∈ Z Z δ j − δ j X k > j δ (2 M − N )( k − j ) F k ( x ) / q dtt + X j ∈ Z Z δ j − δ j X k ≤ j δ j − k ) F k ( x ) / q dtt / = C X k ∈ Z F k ( x ) / q ( X j > k δ (2 M − N )( k − j ) + X j ≥ k δ j − k ) ) / ≤ C X k ∈ Z F k ( x ) / q / . where we used (21) with β = M − N , τ = β = τ = − Z X ϕ ( x , S L ( f ) ( x ) / λ ) d µ ( x ) ≤ C Z X ϕ (cid:18) x , λ − (cid:16)X k ∈ Z F k ( x ) / q (cid:17) / (cid:19) d µ ( x ) ≤ C Z X ϕ x , λ − X k ∈ Z (cid:18)X α ∈ I k | s Q k α | q µ (cid:16) Q k α (cid:17) − q / X Q k α ( x ) (cid:19) / q ! / d µ ( x ) = C Z X ϕ x , λ − (cid:18)X k ∈ Z X α ∈ I k | s Q k α | µ (cid:16) Q k α (cid:17) − X Q k α ( x ) (cid:19) / ! d µ ( x ) = C Z X ϕ (cid:16) x , W f ( x ) . λ (cid:17) d µ ( x ) , which proves the ≤ inequality in (16) and completes the proof of the theorem. (cid:3)
17e now turn to characterize the Musielark-Orlicz Hardy space H L , G ,ϕ and have the followingresult. Theorem 3.2.
Suppose L is an operator that satisfies ( H1 ) and ( H2 ). Let ϕ be a growth functionwith uniformly lower type p and f ∈ H L , G ,ϕ ( X ) ∩ L ( X ) , then for all natural number M > nq ( ϕ ) / (2 p ) , f has an AT L , M -expansion such that k f k H L , G ,ϕ ( X ) (cid:27) (cid:13)(cid:13)(cid:13) W f (cid:13)(cid:13)(cid:13) L ϕ ( X ) . We prove Theorem 3.2 by borrowing some ideas from Duong et al. [6, Proof of Theorem 3.2].To this end, we start with listing some known facts as follows.Given f ∈ L ( X ), a > x , t ) ∈ X × (0 , ∞ ), the Fe ff erman-Stein-type maximal function isdefined as M ∗ a , L ( f ) ( x , t ) = esssup y ∈ X | t Le − t L f ( y ) | (cid:2) + t − d ( x , y ) (cid:3) a , and we have the following Lemma 3.5 which was established in [10]. Lemma 3.5.
Assume that L satisfies ( H1 ) and ( H2 ). Let m be as in (6) . Then, for any β > ,r > and a > m / , there exist a positive constant C such that for all f ∈ L ( X ) , l ∈ Z , x ∈ X andt ∈ [1 , , (cid:12)(cid:12)(cid:12)(cid:12) M ∗ a , L ( f ) (cid:16) x , − l t (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) r ≤ C ∞ X j = l − ( j − l ) β r Z X | (2 − j t ) Le − (2 − j t ) L f ( z ) | r V ( z , − l )[1 + l d ( x , z )] ar d µ ( z ) . Moreover, we also need the following Lemma, whose proof is standard, we omit the details.And in what follows, we recall that the Hardy-Littlewood maximal operator M on ( X , µ, d ) isdefined by M ( f ) ( x ) : = sup B ∋ x µ ( B ) Z B | f ( y ) | d µ ( y ) , where the supremum is taken over all balls B ∋ x . Lemma 3.6.
Let n and m be as in (5) and (6) , and suppose that N > n + m. Then there exists aconstant C > such that for all measurable functions f on ( X , µ, d ) , t > and each y ∈ X, Z X | f ( x ) | V ( x , t ) (cid:2) + t − d ( x , y ) (cid:3) N d µ ( x ) ≤ C M ( f ) ( y ) . Proof of theorem 3.2
For any fixed f ∈ H L , G ,ϕ ( X ) ∩ L ( X ), we let λ = k f k H L , G ,ϕ ( X ) and λ = (cid:13)(cid:13)(cid:13) W f (cid:13)(cid:13)(cid:13) L ϕ ( X ) . It su ffi ces to show that for all λ ∈ (0 , ∞ ), Z X ϕ x , G L ( f ) ( x ) λ ! d µ ( x ) (cid:27) Z X ϕ x , (cid:12)(cid:12)(cid:12) W f ( x ) (cid:12)(cid:12)(cid:12) λ d µ ( x ) . (22)In fact, if (22) holds for all λ ∈ (0 , ∞ ), then there exists a constant C such that Z X ϕ x , G L ( f ) ( x ) λ ! d µ ( x ) ≤ C Z X ϕ x , (cid:12)(cid:12)(cid:12) W f ( x ) (cid:12)(cid:12)(cid:12) λ d µ ( x ) ≤ C , Z X ϕ x , G L ( f ) ( x ) C λ ! d µ ( x ) ≤ C , and hence we have λ ≤ C λ . In a similar fashion, one can prove λ ≤ C λ for some constants C , and get the desired property.Now we fix arbitrary λ ∈ (0 , ∞ ) and prove (22). In the view of Lemma 2.1, for any fixed( x , k ) ∈ X × Z there exists a unique α ∈ I k , such that x ∈ Q k α . Let Q kx denote the such Q k α and wewrite W f ( x ) = X k ∈ Z X α ∈ I k (cid:20) µ (cid:16) Q k α (cid:17) − / (cid:12)(cid:12)(cid:12) s Q k α (cid:12)(cid:12)(cid:12) X Q k α ( x ) (cid:21) / = X k ∈ Z µ (cid:16) Q kx (cid:17) − (cid:12)(cid:12)(cid:12) s Q kx (cid:12)(cid:12)(cid:12) / = X k ∈ Z Z δ k δ k + µ (cid:16) Q kx (cid:17) − Z Q kx (cid:12)(cid:12)(cid:12)(cid:12) t Le − t L f ( y ) (cid:12)(cid:12)(cid:12)(cid:12) d µ ( y ) dtt / , (23)where δ ∈ (0 ,
1) is a constant as in Lemma 2.1 and the last quantity follows from Proposition 2.1.Moreover, by ( iv ) and ( v ) of Lemma 2.1, we know that for any fixed ( x , k ) ∈ X × Z there exists z kx ∈ Q kx and constants C ∈ (0 , C > (cid:16) Q k α (cid:17) ≤ C δ k and B (cid:16) z kx , C δ k (cid:17) ⊂ Q kx ⊂ B (cid:16) x , C δ k (cid:17) ⊂ B (cid:16) x , C δ − t (cid:17) : = B x for all t ∈ (cid:16) δ k + , δ k (cid:17) . Then for each k ∈ Z we compute by (5) and (6), Z δ k δ k + µ (cid:16) Q kx (cid:17) − Z Q kx (cid:12)(cid:12)(cid:12)(cid:12) t Le − t L f ( y ) (cid:12)(cid:12)(cid:12)(cid:12) d µ ( y ) dtt ≤ Z δ k δ k + µ (cid:16) B (cid:16) z kx , C δ k (cid:17)(cid:17) − Z B ( x , C δ k ) (cid:12)(cid:12)(cid:12)(cid:12) t Le − t L f ( y ) (cid:12)(cid:12)(cid:12)(cid:12) d µ ( y ) dtt ≤ C Z δ k δ k + µ (cid:16) B (cid:16) x , C δ k (cid:17)(cid:17) µ (cid:0) B (cid:0) z kx , C δ k (cid:1)(cid:1) esssup y ∈ B ( x , C δ k ) (cid:12)(cid:12)(cid:12)(cid:12) t Le − t L f ( y ) (cid:12)(cid:12)(cid:12)(cid:12) dtt ≤ C Z δ k δ k + esssup y ∈ B x (cid:12)(cid:12)(cid:12)(cid:12) t Le − t L f ( y ) (cid:12)(cid:12)(cid:12)(cid:12) dtt ≤ C Z δ k δ k + h M ∗ a , L ( f ) ( x , t ) i dtt (24)for some appropriate constant C , where M ∗ a , L ( f ) ( x , t ) is the Fe ff erman-Stein-type maximal func-tion, with some large enough constants a to be selected. And the last inequality follows from19sssup y ∈ B x (cid:12)(cid:12)(cid:12)(cid:12) t Le − t L f ( y ) (cid:12)(cid:12)(cid:12)(cid:12) = esssup y ∈ B x | t Le − t L f ( y ) | [1 + t − d ( x , y )] a h + t − d ( x , y ) i a ≤ (cid:16) + C δ − (cid:17) h M ∗ a , L ( f ) ( x , t ) i . Combining now (23)-(24), we have the following estimate of W f ( x ), W f ( x ) ≤ C (Z ∞ h M ∗ a , L ( f ) ( x , t ) i dtt ) / = C X k ∈ Z Z − k + − k h M ∗ a , L ( f ) ( x , t ) i dtt / = C X k ∈ Z Z h M ∗ a , L ( f ) (cid:16) x , − k t (cid:17)i dtt / . In the view of Lemma 3.5, we see that for any β > , r > a > m /
2, there exists a constant C such that for all f ∈ L ( X ) , k ∈ Z , x ∈ X and t ∈ [1 , (cid:12)(cid:12)(cid:12)(cid:12) M ∗ a , L ( f ) (cid:16) x , − k t (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) r ≤ C ∞ X j = k − ( j − k ) β r Z X | (2 − j t ) Le − (2 − j t ) L f ( z ) | r V ( z , − k )[1 + k d ( x , z )] ar d µ ( z ) . (25)Let r ∈ (0 ,
1) be as in Corollary 2.1, with p = / r >
1. Fix some β > a > m / ar > m + n . We then take the norm (cid:20)R |·| / r dtt (cid:21) r / on the both sides of (25), and employ theMinkowski’s inequality to get "Z (cid:12)(cid:12)(cid:12)(cid:12) M ∗ a , L ( f ) (cid:16) x , − k t (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) dtt r / ≤ C Z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X j = k − ( j − k ) β r Z X | (2 − j t ) Le − (2 − j t ) L f ( z ) | r V ( z , − k )[1 + k d ( x , z )] ar d µ ( z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) / r dtt r / = C ∞ X j = k − ( j − k ) β r Z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z X | (2 − j t ) Le − (2 − j t ) L f ( z ) | r V ( z , − k )[1 + k d ( x , z )] ar d µ ( z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) / r dtt r / ≤ C ∞ X j = k − ( j − k ) β r Z X [ R | (2 − j t ) Le − (2 − j t ) L f ( z ) | dt / t ] r / V ( z , − k )[1 + k d ( x , z )] ar d µ ( z ) = C Z X P ∞ j = k − ( j − k ) β r [ R | (2 − j t ) Le − (2 − j t ) L f ( z ) | dt / t ] r / V ( z , − k )[1 + k d ( x , z )] ar d µ ( z ) ≤ C M X ∞ j = k − ( j − k ) β r "Z | (2 − j t ) Le − (2 − j t ) L f ( · ) | dt / t r / ( x ): = C M ( F k ) ( x ) , F k ( x ) : = P ∞ j = k − ( j − k ) β r [ R | (2 − j t ) Le − (2 − j t ) L f ( x ) | dt / t ] r / , and the last inequality follows fromLemma 3.6. It follows that Z X ϕ x , (cid:12)(cid:12)(cid:12) W f (cid:12)(cid:12)(cid:12) λ d µ ( x ) ≤ Z X ϕ x , C X k ∈ Z Z h M ∗ a , L ( f ) (cid:16) x , − k t (cid:17)i dt / t / , λ d µ ( x ) = Z X ϕ x , C X k ∈ Z Z h M ∗ a , L ( f ) (cid:16) x , − k t (cid:17)i dt / t ! r / / r / , λ d µ ( x ) ≤ Z X ϕ x , C X k ∈ Z [ M ( F k ) ( x )] / r / , λ d µ ( x ) ≤ C Z X ϕ x , X k ∈ Z F k ( x ) / r / , λ d µ ( x ) , (26)where we used the Corollary 2.1 in the last inequality.We now turn to estimate F k ( x ) / r . For any k ∈ Z , we recall F k ( x ) = X ∞ j = k − ( j − k ) β r "Z | (2 − j t ) Le − (2 − j t ) L f ( x ) | dt / t r / . Since 2 − ( j − k ) β r = β r − − β r Z j − k + j − k s − β r − ds , we let E j : = h j − k , j − k + i and it follows that F k ( x ) = C X ∞ j = k Z j − k + j − k "Z | (2 − j t ) Le − (2 − j t ) L f ( x ) | dtt r / dss β r + = C X ∞ j = k Z ∞ "Z | (2 − j t ) Le − (2 − j t ) L f ( x ) | dtt r / X E j ( s ) dss β r + = C Z ∞ X ∞ j = k "Z | (2 − j t ) Le − (2 − j t ) L f ( x ) | dtt r / X E j ( s ) dss β r + .
21e then apply the H¨older’s inequality to obtain F k ( x ) / r = C Z ∞ X ∞ j = k "Z | (2 − j t ) Le − (2 − j t ) L f ( x ) | dtt r / X E j ( s ) dss β r + / r ≤ C Z ∞ s − β r − ds ! / r (2 − r ) × Z ∞ X ∞ j = k "Z | (2 − j t ) Le − (2 − j t ) L f ( x ) | dtt r / X E j ( s ) / r dss β r + = C Z ∞ X ∞ j = k Z | (2 − j t ) Le − (2 − j t ) L f ( x ) | dtt ! X E j ( s ) ! dss β r + = C X ∞ j = k Z j − k + j − k dss β r + Z | (2 − j t ) Le − (2 − j t ) L f ( x ) | dtt = C X ∞ j = k − ( j − k ) β r Z | (2 − j t ) Le − (2 − j t ) L f ( x ) | dtt ! . (27)Summation by all k ∈ Z , we have X k ∈ Z F k ( x ) / r ≤ C X k ∈ Z X j ≥ k − ( j − k ) β r Z | (2 − j t ) Le − (2 − j t ) L f ( x ) | dtt ! = C X j ∈ Z X k ≤ j − ( j − k ) β r Z | (2 − j t ) Le − (2 − j t ) L f ( x ) | dtt ! = C (1 − − β r ) − X j ∈ Z Z | (2 − j t ) Le − (2 − j t ) L f ( x ) | dtt = C X j ∈ Z Z − j + − j | t Le − t L f ( x ) | dtt = C Z ∞ | t Le − t L f ( x ) | dtt = CG L ( f ) ( x ) , which, together with (26) and (8), yields the ≥ inequality in (22).It remains to establish the reverse inequality. In the view of Proposition 2.1, we write f = k ∈ Z P α ∈ I k s Q k α a Q k α . Let δ be as in Lemma 2.1 we get G L ( f ) ( x ) = Z ∞ | t Le − t L ( f ) ( x ) | dtt ! / = Z ∞ | t Le − t L ( X k ∈ Z X α ∈ I k s Q k α a Q k α ) ( x ) | dtt / = X j ∈ Z Z δ j − δ j | t Le − t L ( X k ∈ Z X α ∈ I k s Q k α a Q k α ) ( x ) | dtt / ≤ X j ∈ Z Z δ j − δ j | t Le − t L ( X k > j X α ∈ I k s Q k α a Q k α ) ( x ) | dtt / + X j ∈ Z Z δ j − δ j | t Le − t L ( X k ≤ j X α ∈ I k s Q k α a Q k α ) ( x ) | dtt / . (28)We now estimate the first part of (28). For any k > j and α ∈ I k , noting that a Q k α = L M b Q k α , wewrite (cid:12)(cid:12)(cid:12)(cid:12) t Le − t L (cid:16) a Q k α (cid:17) ( x ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) t L M + e − t L (cid:16) b Q k α (cid:17) ( x ) (cid:12)(cid:12)(cid:12)(cid:12) = t − M (cid:12)(cid:12)(cid:12)(cid:12)(cid:16) t L (cid:17) M + e − t L (cid:16) b Q k α (cid:17) ( x ) (cid:12)(cid:12)(cid:12)(cid:12) . Let n be as in (5), since M > nq ( ϕ ) / (2 p ), we can choose some q = r with r be as in Corollary2.1 such that 2 M > n / q . We then let N be some positive number such that 2 M > N > n / q . Thenby Definition 2.3, the upper bound of the kernel of (cid:16) t L (cid:17) M + e − t L and (7), we get (cid:12)(cid:12)(cid:12)(cid:12) t Le − t L (cid:16) a Q k α (cid:17) ( x ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C V ( x , t ) t − M ℓ (cid:16) Q k α (cid:17) M µ (cid:16) Q k α (cid:17) − / Z Q k α exp − d ( x , y ) C t ! d µ ( y ) ≤ Ct − M ℓ (cid:16) Q k α (cid:17) M µ (cid:16) Q k α (cid:17) − / tt + d ( x , y k α ) ! N , where we denote by y k α the center of Q k α . Hence, | t Le − t L ( X k > j X α ∈ I k s Q k α a Q k α ) ( x ) |≤ C X k > j X α ∈ I k t − M ℓ (cid:16) Q k α (cid:17) M µ (cid:16) Q k α (cid:17) − / | s Q k α | (cid:2) + t − d ( x , y k α ) (cid:3) N ≤ C X k > j δ (2 M − N )( k − j ) X α ∈ I k µ (cid:16) Q k α (cid:17) − / | s Q k α | h + ℓ (cid:0) Q k α (cid:1) − d ( x , y k α ) i N ≤ C X k > j δ (2 M − N )( k − j ) M X α ∈ I k | s Q k α | q µ (cid:16) Q k α (cid:17) − q / X Q k α ( · ) ( x ) / q , (29)23here the last inequality follows from Lemma 3.4.Estimate of the second part of (28). For any k ≤ j and α ∈ I k , we write (cid:12)(cid:12)(cid:12)(cid:12) t Le − t L (cid:16) a Q k α (cid:17) ( x ) (cid:12)(cid:12)(cid:12)(cid:12) = t (cid:12)(cid:12)(cid:12)(cid:12) e − t L (cid:16) L (cid:16) a Q k α (cid:17)(cid:17) ( x ) (cid:12)(cid:12)(cid:12)(cid:12) . Then by Definition 2.3, the Gaussian estimate (1) and inequality (7), we get (cid:12)(cid:12)(cid:12)(cid:12) t Le − t L (cid:16) a Q k α (cid:17) ( x ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C V ( x , t ) t ℓ (cid:16) Q k α (cid:17) − µ (cid:16) Q k α (cid:17) − / Z Q k α exp − d ( x , y ) C t ! d µ ( y ) . ≤ Ct ℓ (cid:16) Q k α (cid:17) − µ (cid:16) Q k α (cid:17) − / (cid:16) + ℓ ( Q k α ) − d (cid:16) x , y k α (cid:17)(cid:17) − N , which implies that | t Le − t L ( X k ≤ j X α ∈ I k s Q k α a Q k α ) ( x ) |≤ C X k ≤ j X α ∈ I k t ℓ (cid:16) Q k α (cid:17) − µ (cid:16) Q k α (cid:17) − / | s Q k α | h + ℓ ( Q k α ) − d ( x , y k α ) i N ≤ C X k ≤ j δ j − k ) X α ∈ I k µ (cid:16) Q k α (cid:17) − / | s Q k α | h + ℓ (cid:0) Q k α (cid:1) − d ( x , y k α ) i N ≤ C X k ≤ j δ j − k ) M X α ∈ I k | s Q k α | q µ (cid:16) Q k α (cid:17) − q / X Q k α ( · ) ( x ) / q . (30)By the same technique we used in (27), we combine now (28)-(30), and get the following es-timate of G L ( f ), G L ( f ) ( x ) ≤ X j ∈ Z Z δ j − δ j | t Le − t L ( X k > j X α ∈ I k s Q k α a Q k α ) ( x ) | dtt / + X j ∈ Z Z δ j − δ j | t Le − t L ( X k ≤ j X α ∈ I k s Q k α a Q k α ) ( x ) | dtt / ≤ C X j ∈ Z Z δ j − δ j | X k > j δ (2 M − N )( k − j ) G k ( x ) / q | dtt + X j ∈ Z Z δ j − δ j | X k ≤ j δ j − k ) G k ( x ) / q | dtt / C X j ∈ Z Z δ j − δ j X k > j δ (2 M − N )( k − j ) G k ( x ) / q dtt + X j ∈ Z Z δ j − δ j X k ≤ j δ j − k ) G k ( x ) / q dtt / = C X k ∈ Z G k ( x ) / q ( X j > k δ (2 M − N )( k − j ) + X j ≥ k δ j − k ) ) / ≤ C X k ∈ Z G k ( x ) / q / , where G k ( x ) = M (cid:18)P α ∈ I k | s Q k α | q µ (cid:16) Q k α (cid:17) − q / X Q k α ( · ) (cid:19) ( x ), for k ∈ Z . Hence, by employing Corollary2.1, we have Z X ϕ ( x , G L ( f ) ( x ) / λ ) d µ ( x ) ≤ C Z X ϕ (cid:18) x , λ − (cid:16)X k ∈ Z G k ( x ) / q (cid:17) / (cid:19) d µ ( x ) ≤ C Z X ϕ x , λ − X k ∈ Z (cid:18)X α ∈ I k | s Q k α | q µ (cid:16) Q k α (cid:17) − q / X Q k α ( x ) (cid:19) / q ! / d µ ( x ) = C Z X ϕ x , λ − (cid:18)X k ∈ Z X α ∈ I k | s Q k α | µ (cid:16) Q k α (cid:17) − X Q k α ( x ) (cid:19) / ! d µ ( x ) = C Z X ϕ (cid:16) x , W f ( x ) . λ (cid:17) d µ ( x ) , which proves the reverse inequality in (22) and completes the proof of the theorem. (cid:3) Using Theorem 3.1 and Theorem 3.2, we immediately obtain the following result.
Theorem 3.3.
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