A Complete Real-Variable Theory of Hardy Spaces on Spaces of Homogeneous Type
Ziyi He, Yongsheng Han, Ji Li, Liguang Liu, Dachun Yang, Wen Yuan
aa r X i v : . [ m a t h . C A ] A p r A Complete Real-Variable Theory of Hardy Spaces on Spaces ofHomogeneous Type
Ziyi He, Yongsheng Han, Ji Li, Liguang Liu, Dachun Yang ∗ and Wen Yuan Abstract
Let ( X , d , µ ) be a space of homogeneous type, with the upper dimension ω , in thesense of R. R. Coifman and G. Weiss. Assume that η is the smoothness index of the waveletson X constructed by P. Auscher and T. Hyt¨onen. In this article, when p ∈ ( ω/ ( ω + η ) , H p cw ( X ) introduced by Coifman and Weiss, the authors estab-lish their various real-variable characterizations, respectively, in terms of the grand maxi-mal function, the radial maximal function, the non-tangential maximal functions, the variousLittlewood-Paley functions and wavelet functions. This completely answers the question ofR. R. Coifman and G. Weiss by showing that no any additional (geometrical) condition isnecessary to guarantee the radial maximal function characterization of H ( X ) and even of H p cw ( X ) with p as above. As applications, the authors obtain the finite atomic characterizationsof H p cw ( X ), which further induce some criteria for the boundedness of sublinear operators on H p cw ( X ). Compared with the known results, the novelty of this article is that µ is not assumedto satisfy the reverse doubling condition and d is only a quasi-metric, moreover, the range p ∈ ( ω/ ( ω + η ) ,
1] is natural and optimal.
The real-variable theory of Hardy spaces plays a fundamental role in harmonic analysis. Theclassical Hardy space on the n -dimensional Euclidean space R n was initially developed by Steinand Weiss [48] and later by Fe ff erman and Stein [11]. Hardy spaces H p ( R n ) have been proved tobe a suitable substitute of Lebesgue spaces L p ( R n ) with p ∈ (0 ,
1] in the study of the boundednessof operators. Indeed, any element in the Hardy space can be decomposed into a sum of some basicelements (which are called atoms ); see Coifman [5] for n = n ∈ N .Characterizations of Hardy spaces via Littlewood-Paley functions were due to Uchiyama [49]. Formore study on classical Hardy spaces on R n , we refer the reader to the well-known monographs[47, 41, 16, 17, 19]. Modern developments regarding the real-variable theory of Hardy spaces areso deep and vast that we can only list a few literatures here, for example, the theory of Hardyspaces associated with operators (see [2, 3, 30, 10]), Hardy spaces with variable exponents (see Mathematics Subject Classification . Primary 42B30; Secondary 42B25, 42B20, 30L99.
Key words and phrases. space of homogeneous type, Hardy space, maximal function, atom, Littlewood-Paley func-tion, wavelet.This project is supported by the National Natural Science Foundation of China (Grant Nos. 11771446, 11571039,11726621 and 11761131002). Ji Li is supported by ARC DP 160100153. ∗ Corresponding author / April 1, 2018 / newest version. Z iyi H e , Y ongsheng H an , J i L i , L iguang L iu , D achun Y ang and W en Y uan [44]), the real-variable theory of Musielak-Orlicz Hardy spaces (see [35, 51]), and also Hardyspaces for ball quasi-Banach spaces (see [46]).In this article, we focus on the real-variable theory of Hardy spaces on spaces of homogeneoustype. It is known that the space of homogeneous type introduced by Coifman and Weiss [6, 7]provides a natural setting for the study of both functions spaces and the boundedness of operators.A quasi-metric space ( X , d ) is a non-empty set X equipped with a quasi-metric d , that is, a non-negative function defined on X × X , satisfying that, for any x , y , z ∈ X ,(i) d ( x , y ) = x = y ;(ii) d ( x , y ) = d ( y , x );(iii) there exists a constant A ∈ [1 , ∞ ) such that d ( x , z ) ≤ A [ d ( x , y ) + d ( y , z )].The ball B on X centered at x ∈ X with radius r ∈ (0 , ∞ ) is defined by setting B : = B ( x , r ) : = { x ∈ X : d ( x , x ) < r } . For any ball B and τ ∈ (0 , ∞ ), denote by τ B the ball with the same center as that of B but of radius τ times that of B . Given a quasi-metric space ( X , d ) and a non-negative measure µ , we call ( X , d , µ )a space of homogeneous type if µ satisfies the doubling condition : there exists a positive constant C ( µ ) ∈ [1 , ∞ ) such that, for any ball B ⊂ X , µ (2 B ) ≤ C ( µ ) µ ( B ) . The above doubling condition is equivalent to that, for any ball B and λ ∈ [1 , ∞ ),(1.1) µ ( λ B ) ≤ C ( µ ) λ ω µ ( B ) , where ω : = log C ( µ ) is called the upper dimension of X . If A =
1, we call ( X , d , µ ) a doublingmetric measure space .According to [7, pp. 587–588], we always make the following assumptions throughout this arti-cle. For any point x ∈ X , assume that the balls { B ( x , r ) } r ∈ (0 , ∞ ) form a basis of open neighborhoodsof x ; assume that µ is Borel regular, which means that open sets are measurable and every set A ⊂ X is contained in a Borel set E satisfying that µ ( A ) = µ ( E ); we also assume that µ ( B ( x , r )) ∈ (0 , ∞ )for any x ∈ X and r ∈ (0 , ∞ ). For the presentation concision, we always assume that ( X , d , µ ) isnon-atomic [namely, µ ( { x } ) = x ∈ X ] and diam( X ) : = sup { d ( x , y ) : x , y ∈ X } = ∞ . It isknown that diam( X ) = ∞ implies that µ ( X ) = ∞ (see, for example, [1, Lemma 8.1]).Let us recall the notion of the atomic Hardy space on spaces of homogeneous type introducedby Coifman and Weiss [7]. For any α ∈ (0 , ∞ ), the Lipschitz space L α ( X ) is defined to be thecollection of all measurable functions f such that k f k L α ( X ) : = sup x , y | f ( x ) − f ( y ) | [ µ ( B ( x , d ( x , y )))] α < ∞ . Denote by ( L α ( X )) ′ the dual space of L α ( X ) equipped with the weak- ∗ topology. Definition 1.1.
Let p ∈ (0 ,
1] and q ∈ ( p , ∞ ] ∩ [1 , ∞ ]. A function a is called a ( p , q ) -atom if ardy S paces on S paces of H omogenous T ype a : = { x ∈ X : a ( x ) , } ⊂ B ( x , r ) for some x ∈ X and r ∈ (0 , ∞ );(ii) [ R X | a ( x ) | q d µ ( x )] q ≤ [ µ ( B ( x , r ))] q − p ;(iii) R X a ( x ) d µ ( x ) = atomic Hardy space H p , q cw ( X ) is defined as the subspace of ( L / p − ( X )) ′ when p ∈ (0 ,
1) or of L ( X ) when p =
1, which consists of all the elements f admitting an atomic decomposition(1.2) f = ∞ X j = λ j a j , where { a j } ∞ j = are ( p , q )-atoms, { λ j } ∞ j = ⊂ C satisfies P ∞ j = | λ j | p < ∞ and the series in (1.2) con-verges in ( L / p − ( X )) ′ when p ∈ (0 ,
1) or in L ( X ) when p =
1. Define k f k H p , q cw ( X ) : = inf ∞ X j = | λ j | p p , where the infimum is taken over all the representations of f as in (1.2).It was proved in [7] that the atomic Hardy space H p , q cw ( X ) is independent of the choice of q and hence we sometimes write H p cw ( X ) for short. It was also proved in [7] that the dual space of H p cw ( X ) is the Lipschitz space L / p − ( X ) when p ∈ (0 , X ) of bounded meanoscillation when p = H ( R n ) and BMO( R n ) from Carleson [4] can be extendedto the general setting of spaces of homogeneous type provided a certain additional geometricalassumption is added, from which one can then obtain a radial maximal function characterization of H ( X ). Coifman and Weiss [7, p. 642] then asked that to what extent their geometrical conditionis necessary for the validity of the radial maximal characterization of H ( X ). Since then, lotsof e ff orts are made to build various real-variable characterizations of the atomic Hardy spaces onspaces of homogeneous type with few geometrical assumptions. In this article, we completelyanswer the aforementioned question of Coifman and Weiss by showing that no any additional(geometrical) condition is necessary to guarantee the radial maximal function characterization of H ( X ) and even of H p cw ( X ) with p ≤ X , d , µ ) is said to be Ahlfors-n regular if µ ( B ( x , r )) ∼ r n for any x ∈ X and r ∈ (0 , diam X ) with equivalent positive constants independent of x and r . When ( X , d , µ ) isAhlfors- n regular, upon assuming the quasi-metric d satisfying that there exists θ ∈ (0 ,
1) suchthat, for any x , x ′ , y ∈ X ,(1.3) | d ( x , y ) − d ( x ′ , y ) | . [ d ( x , x ′ )] θ [ d ( x , y ) + d ( x ′ , y )] − θ , Mac´ıas and Segovia [43] characterized Hardy spaces via the grand maximal functions, and Li [37]obtained another grand maximal function characterization via test functions introduced in [28]. Z iyi H e , Y ongsheng H an , J i L i , L iguang L iu , D achun Y ang and W en Y uan Also, Duong and Yan [9] characterized Hardy spaces via the Lusin area function associated withcertain semigroup.Recall that an RD- space ( X , d , µ ) is a doubling metric measure space with the measure µ furthersatisfying the reverse doubling condition , that is, there exist a positive constant e C ∈ (0 ,
1] and κ ∈ (0 , ω ] such that, for any ball B ( x , r ) with x ∈ X , r ∈ (0 , diam X /
2) and λ ∈ [1 , diam X / [2 r ]), e C λ κ µ ( B ( x , r )) ≤ µ ( B ( x , λ r )) . Indeed, any path connected doubling metric measure space is an RD-space (see [27, 55]). Charac-terizations of Hardy spaces on RD-spaces via various Littlewood-Paley functions were establishedin [26, 27]. Also, characterizations of Hardy spaces on RD-spaces via various maximal functionscan be found in [20, 21, 54]. It should be mentioned that local Hardy spaces can be used to charac-terize more general scale of function spaces like Besov and Triebel-Lizorkin spaces on RD-spaces(see [55]). For a systematic study of Besov and Triebel-Lizorkin spaces on RD-spaces, we referthe reader to [27]. More on analysis over Ahlfors- n regular metric measure spaces or RD-spacescan be found in [18, 22, 33, 34, 52, 32, 55, 8, 56].The main motivation of studying the real-variable theory of function spaces and the bounded-ness of operators on spaces of homogeneous type comes from the celebrated work of Auscher andHyt¨onen [1], in which they constructed an orthonormal wavelet basis { ψ k α : k ∈ Z , α ∈ G k } of L ( X ) with H ¨older continuity exponent η ∈ (0 ,
1) and exponential decay by using the system ofrandom dyadic cubes. The first creative attempt of using the idea of [1] to investigate the real-variable theory of Hardy spaces on spaces of homogeneous type was due to Han et al. [23] (seealso Han et al. [24]). Indeed, in [23], Hardy spaces via wavelets on spaces of homogeneous typewere introduced and then these spaces were proved to have atomic decompositions. The methodused in [23] is based on a new Calder´on reproducing formula on spaces of homogeneous type(see [23, Proposition 2.5]). But there exists an error in the proof of [23, Proposition 2.5], namely,since the regularity exponent of the approximations of the identity in [23, p. 3438] is θ [indeed, θ is from the regularity of the quasi-metric d in (1.3)], it follows that the regularity exponent in[23, (2.6)] should be min { θ, η } and hence the correct range of p in [23, Proposition 2.5] (indeed,all results of [23]) seems to be ( ω/ [ ω + min { θ, η } ] ,
1] which is not optimal. Moreover, the criteriaof the boundedness of Calder´on-Zygmund operators on the dual of Hardy spaces were establishedin [23]. Also, Fu and Yang [14] obtained an unconditional basis of H ( X ) and several equivalentcharacterizations of H ( X ) in terms of wavelets.Another motivation of this article comes from the Calder´on reproducing formulae established in[29]. Indeed, the work of [29] was partly motivated by the wavelet theory of Auscher and Hyt¨onenin [1] and a corresponding wavelet reproducing formula (which can converge in the distributionspace) in [29]. The already existing works (see [26, 27, 20, 54, 55]) regarding Hardy spaces onRD-spaces show the feasibility of establishing various real-variable characterizations of the atomicHardy spaces on spaces of homogeneous type via the Calder´on reproducing formulae. It should bementioned that a characterization of the atomic Hardy spaces via the Littlewood-Paley functionswas established in [25] via the aforementioned wavelet reproducing formula; see also [25] forsome corresponding conclusions of product Hardy spaces on spaces of homogeneous type.In this article, motivated by [23, 29], for the atomic Hardy spaces H p cw ( X ) with any p ∈ ( ω/ [ ω + η ] , ardy S paces on S paces of H omogenous T ype H p cw ( X ) with p ∈ ( ω/ [ ω + η ] ,
1] and X being any space of homogeneous type without any ad-ditional (geometrical) conditions , which completely answers the aforementioned question askedby Coifman and Weiss [7, p. 642]. As an application, we obtain the finite atomic characterizationsof Hardy spaces, which further induce some criteria for the boundedness of sublinear operatorson Hardy spaces. Compared with the known results, the novelty of this article is that µ is not as-sumed to satisfy the reverse doubling condition and d is only a quasi-metric. Moreover, the rangeof p ∈ ( ω/ ( ω + η ) ,
1] for the various maximal function characterizations and the Littlewood-Paleyfunction characterizations of the atomic Hardy spaces H p cw ( X ) is natural and optimal. The key toolused through this article is those Calder´on reproducing formulae from [29].In addition, we point out that, when X is a doubling metric measure space, the finite atomiccharacterizations of Hardy spaces are also useful in establishing the bilinear decomposition of theproduct space H ( X ) × BMO( X ) and H p cw ( X ) × L / p − ( X ), with p ∈ ( ω/ [ ω + η ] ,
1) in [15, 40, 13,14], and also in the study of the endpoint boundedness of commutators generated by Calder´on-Zygmund operators and BMO( X ) functions in [38, 39].The organization of this article is as follows.In Section 2, we recall the notions of the space of test functions and the space of distributionsintroduced in [26], as well as the random dyadic cubes in [1] and the approximation of the identitywith exponential decay introduced in [29]. Then we restate the Calder´on reproducing formulaeestablished in [29].Section 3 concerns Hardy spaces defined via the grand maximal function, the radial maximalfunction and the non-tangential maximal function. We show that these Hardy spaces are all equiv-alent to the Lebesgue space L p ( X ) when p ∈ (1 , ∞ ] (see Section 3.1), and they are all mutuallyequivalent when p ∈ ( ω/ ( ω + η ) ,
1] (see Section 3.2), all in the sense of equivalent (quasi-)norms.The proof for the latter borrows some ideas from [54] and uses the Calder´on reproducing for-mulae built in [29]. Moreover, we prove that the Hardy space H ∗ , p ( X ) defined via the grandmaximal function is independent of the choices of the distribution space ( G η ( β, γ )) ′ whenever β, γ ∈ ( ω [1 / p − , η ); see Proposition 3.8 below.Section 4 is devoted to the atomic characterization of H ∗ , p ( X ). Notice that, if a distributionhas an atomic decomposition, then it belongs to H ∗ , p ( X ) obviously by the definition of atoms;see Section 4.1. All we remain to do is to establish the converse relationship. In Section 4.2,by modifying the definition of the grand maximal function f ∗ to f ⋆ so that the level set { x ∈ X : f ⋆ ( x ) > λ } with λ ∈ (0 , ∞ ) is open, we then apply the partition of unity to the open set Ω λ andobtain a Calder´on-Zygmund decomposition of f ∈ H ∗ , p ( X ). This is further used in Section 4.3to construct an atomic decomposition of f . In Section 4.4, we compare the atomic Hardy spaces H p , q at ( X ) with H p , q cw ( X ) and prove that they are exactly the same space in the sense of equivalent(quasi-)norms.Section 5 deals with the Littlewood-Paley theory of Hardy spaces. In Section 5.1, we showthat the Hardy space H p ( X ), defined via the Lusin area function, is independent of the choicesof exp-ATIs. In Section 5.2, we use the homogeneous continuous Calder´on reproducing formulaand the molecular characterizations of the atomic Hardy spaces (see [39]) to establish the atomicdecompositions of elements in H p ( X ), and then we connect H p ( X ) with H ∗ , p ( X ). In Section 5.3,we characterize Hardy spaces H p ( X ) via the Lusin area function with aperture, the Littlewood-Paley g -function and the Littlewood-Paley g ∗ λ -function. Z iyi H e , Y ongsheng H an , J i L i , L iguang L iu , D achun Y ang and W en Y uan In Section 6, we consider the Hardy space H pw ( X ) defined via wavelets, which was introducedin [23]. We improve the result of [25, Theorem 4.3] and prove that H pw ( X ) coincides with H p ( X )in the sense of equivalent (quasi-)norms.In Section 7, as an application, we obtain criteria of the boundedness of the sublinear operatorsfrom Hardy spaces to quasi-Banach spaces. To this end, we first establish the finite atomic charac-terizations, namely, we show that, if q ∈ ( p , ∞ ) ∩ [1 , ∞ ), then k·k H p , q fin ( X ) and k·k H p , q at ( X ) are equivalent(quasi)-norms on a dense subspace H p , q fin ( X ) of H p , q at ( X ); the above equivalence also holds true ona dense subspace H p , ∞ fin ( X ) ∩ UC( X ) of H p , ∞ at ( X ), where UC( X ) denotes the space of all uniformlycontinuous functions on X .At the end of this section, we make some conventions on notation. We always assume that ω is as in (1.1) and η is the smoothness index of wavelets (see [1, Theorem 7.1] or Definition 2.4below). We assume that δ is a very small positive number, for example, δ ≤ (2 A ) − in order toconstruct the dyadic cube system and the wavelet system on X (see [31, Theorem 2.2] or Lemma2.3 below). For any x , y ∈ X and r ∈ (0 , ∞ ), let V r ( x ) : = µ ( B ( x , r )) and V ( x , y ) : = µ ( B ( x , d ( x , y ))) , where B ( x , r ) : = { y ∈ X : d ( x , y ) < r } . We always let N : = { , , . . . } and Z + : = N ∪ { } . Forany p ∈ [1 , ∞ ], we use p ′ to denote its conjugate index , namely, 1 / p + / p ′ =
1. The symbol C denotes a positive constant which is independent of the main parameters, but it may vary from lineto line. We also use C ( α,β,... ) to denote a positive constant depending on the indicated parameters α , β , . . . . The symbol A . B means that there exists a positive constant C such that A ≤ CB . Thesymbol A ∼ B is used as an abbreviation of A . B . A . We also use A . α,β,... B to indicate thathere the implicit positive constant depends on α , β , . . . and, similarly, A ∼ α,β,... B . For any s , t ∈ R ,denote the minimum of s and t by s ∧ t . For any finite set J , we use J to denote its cardinality .Also, for any set E of X , we use χ E to denote its characteristic function and E ∁ the set X \ E . This section is devoted to recalling Calder´on reproducing formulae obtained in [29]. To thisend, we first recall the notions of both the space of test functions and the distribution space.
Definition 2.1.
Let x ∈ X , r ∈ (0 , ∞ ), β ∈ (0 ,
1] and γ ∈ (0 , ∞ ). A function f defined on X is called a test function of type ( x , r , β, γ ), denoted by f ∈ G ( x , r , β, γ ), if there exists a positiveconstant C such that(i) (the size condition ) for any x ∈ X , | f ( x ) | ≤ C V r ( x ) + V ( x , x ) " rr + d ( x , x ) γ ;(ii) (the regularity condition ) for any x , y ∈ X satisfying d ( x , y ) ≤ (2 A ) − [ r + d ( x , x )], | f ( x ) − f ( y ) | ≤ C " d ( x , y ) r + d ( x , x ) β V r ( x ) + V ( x , x ) " rr + d ( x , x ) γ . ardy S paces on S paces of H omogenous T ype f ∈ G ( x , r , β, γ ), define the norm k f k G ( x , r ,β,γ ) : = inf { C ∈ (0 , ∞ ) : C satisfies (i) and (ii) } . Define ˚ G ( x , r , β, γ ) : = ( f ∈ G ( x , r , β, γ ) : Z X f ( x ) d µ ( x ) = ) equipped with the norm k · k ˚ G ( x , r ,β,γ ) : = k · k G ( x , r ,β,γ ) .Observe that the above version of G ( x , r , β, γ ) was originally introduced by Han et al. [27] (seealso [26]).Fix x ∈ X . For any x ∈ X and r ∈ (0 , ∞ ), we know that G ( x , r , β, γ ) = G ( x , , β, γ ) withequivalent norms, but the equivalent positive constants depend on x and r . Obviously, G ( x , , β, γ )is a Banach space. In what follows, we simply write G ( β, γ ) : = G ( x , , β, γ ) and ˚ G ( β, γ ) : = ˚ G ( x , , β, γ ).Fix ǫ ∈ (0 ,
1] and β, γ ∈ (0 , ǫ ). Let G ǫ ( β, γ ) [resp., ˚ G ǫ ( β, γ )] be the completion of the set G ( ǫ, ǫ ) [resp., ˚ G ( ǫ, ǫ )] in G ( β, γ ), that is, if f ∈ G ǫ ( β, γ ) [resp., f ∈ ˚ G ǫ ( β, γ )], then there exists { φ j } ∞ j = ⊂ G ( ǫ, ǫ ) [resp., { φ j } ∞ j = ⊂ ˚ G ( ǫ, ǫ )] such that k φ j − f k G ( β,γ ) → j → ∞ . If f ∈ G ǫ ( β, γ )[resp., f ∈ ˚ G ǫ ( β, γ )], we then let k f k G ǫ ( β,γ ) : = k f k G ( β,γ ) [resp., k f k ˚ G ǫ ( β,γ ) : = k f k G ( β,γ ) ] . The dual space ( G ǫ ( β, γ )) ′ [resp., ( ˚ G ǫ ( β, γ )) ′ ] is defined to be the set of all continuous linear func-tionals on G ǫ ( β, γ ) [resp., ˚ G ǫ ( β, γ )] and equipped with the weak- ∗ topology. The spaces ( G ǫ ( β, γ )) ′ and ( ˚ G ǫ ( β, γ )) ′ are called the spaces of distributions .Let L ( X ) be the space of all locally integrable functions on X . Denote by M the Hardy-Littlewood maximal operator , that is, for any f ∈ L ( X ) and 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 of X that contain x . For any p ∈ (0 , ∞ ], the Lebesguespace L p ( X ) is defined to be the set of all µ -measurable functions f such that k f k L p ( X ) : = "Z X | f ( x ) | p d µ ( x ) p < ∞ with the usual modification made when p = ∞ ; the weak Lebesgue space L p , ∞ ( X ) is defined to bethe set of all µ -measurable functions f such that k f k L p , ∞ ( X ) : = sup λ ∈ (0 , ∞ ) λ [ µ ( { x ∈ X : | f ( x ) | > λ } )] p < ∞ . It is known (see [7]) that M is bounded on L p ( X ) when p ∈ (1 , ∞ ] and bounded from L ( X ) to L , ∞ ( X ). Then we state some estimates from [27, Lemma 2.1], which are proved by using (1.1). Z iyi H e , Y ongsheng H an , J i L i , L iguang L iu , D achun Y ang and W en Y uan Lemma 2.2.
Let β, γ ∈ (0 , ∞ ) . (i) For any x , y ∈ X and r ∈ (0 , ∞ ) , V ( x , y ) ∼ V ( y , x ) andV r ( x ) + V r ( y ) + V ( x , y ) ∼ V r ( x ) + V ( x , y ) ∼ V r ( y ) + V ( x , y ) ∼ µ ( B ( x , r + d ( x , y ))) , where the equivalent positive constants are independent of x, y and r. (ii) There exists a positive constant C such that, for any x ∈ X and r ∈ (0 , ∞ ) , Z X V r ( x ) + V ( x , x ) " rr + d ( x , x ) γ d µ ( x ) ≤ C . (iii) There exists a positive constant C such that, for any x ∈ X and R ∈ (0 , ∞ ) , Z d ( x , y ) ≤ R V ( x , y ) " d ( x , y ) R β d µ ( y ) ≤ C and Z d ( x , y ) ≥ R V ( x , y ) " Rd ( x , y ) β d µ ( y ) ≤ C . (iv) There exists a positive constant C such that, for any x ∈ X and R , r ∈ (0 , ∞ ) , Z d ( x , x ) ≥ R V r ( x ) + V ( x , x ) " rr + d ( x , x ) γ d µ ( x ) ≤ C (cid:18) rr + R (cid:19) γ . (v) There exists a positive constant C such that, for any r ∈ (0 , ∞ ) , f ∈ L ( X ) and x ∈ X, Z X V r ( x ) + V ( x , y ) " rr + d ( x , y ) γ | f ( y ) | d µ ( y ) ≤ C M ( f )( x ) . Next we recall the system of dyadic cubes established in [31, Theorem 2.2] (see also [1]), whichis restated in the following version.
Lemma 2.3.
Fix constants < c ≤ C < ∞ and δ ∈ (0 , such that A C δ ≤ c . Assume thata set of points, { z k α : k ∈ Z , α ∈ A k } ⊂ X with A k for any k ∈ Z being a countable set of indices,has the following properties: for any k ∈ Z , (i) d ( z k α , z k β ) ≥ c δ k if α , β ; (ii) min α ∈A k d ( x , z k α ) ≤ C δ k for any x ∈ X.Then there exists a family of sets, { Q k α : k ∈ Z , α ∈ A k } , satisfying (iii) for any k ∈ Z , S α ∈A k Q k α = X and { Q k α : α ∈ A k } is disjoint; (iv) if k , l ∈ Z and l ≥ k, then either Q l β ⊂ Q k α or Q l β ∩ Q k α = ∅ ; (v) for any k ∈ Z and α ∈ A k , B ( z k α , c ♮ δ k ) ⊂ Q k α ⊂ B ( z k α , C ♮ δ k ) , where c ♮ : = (3 A ) − c , C ♮ : = A C and z k α is called “the center” of Q k α . ardy S paces on S paces of H omogenous T ype k ∈ Z , let X k : = { z k α } α ∈A k , G k : = A k + \ A k and Y k : = { z k + α } α ∈G k = : { y k α } α ∈G k . Next we recall the notion of approximations of the identity with exponential decay introduced in[29].
Definition 2.4.
A sequence { Q k } k ∈ Z of bounded linear integral operators on L ( X ) is called an approximation of the identity with exponential decay (for short, exp-ATI) if there exist constants C , ν ∈ (0 , ∞ ), a ∈ (0 ,
1] and η ∈ (0 ,
1) such that, for any k ∈ Z , the kernel of operator Q k , which isstill denoted by Q k , satisfying(i) (the identity condition ) P ∞ k = −∞ Q k = I in L ( X ), where I is the identity operator on L ( X );(ii) (the size condition ) for any x , y ∈ X , | Q k ( x , y ) | ≤ C p V δ k ( x ) V δ k ( y ) exp ( − ν " d ( x , y ) δ k a ) (2.1) × exp ( − ν " max { d ( x , Y k ) , d ( y , Y k ) } δ k a ) ;(iii) (the regularity condition ) for any x , x ′ , y ∈ X with d ( x , x ′ ) ≤ δ k , | Q k ( x , y ) − Q k ( x ′ , y ) | + | Q k ( y , x ) − Q k ( y , x ′ ) | (2.2) ≤ C " d ( x , x ′ ) δ k η p V δ k ( x ) V δ k ( y ) exp ( − ν " d ( x , y ) δ k a ) × exp ( − ν " max { d ( x , Y k ) , d ( y , Y k ) } δ k a ) ;(iv) (the second di ff erence regularity condition ) for any x , x ′ , y , y ′ ∈ X with d ( x , x ′ ) ≤ δ k and d ( y , y ′ ) ≤ δ k , then | [ Q k ( x , y ) − Q k ( x ′ , y )] − [ Q k ( x , y ′ ) − Q k ( x ′ , y ′ )] | (2.3) ≤ C " d ( x , x ′ ) δ k η " d ( y , y ′ ) δ k η p V δ k ( x ) V δ k ( y ) exp ( − ν " d ( x , y ) δ k a ) × exp ( − ν " max { d ( x , Y k ) , d ( y , Y k ) } δ k a ) ;(v) (the cancelation condition ) for any x , y ∈ X , Z X Q k ( x , y ′ ) d µ ( y ′ ) = = Z X Q k ( x ′ , y ) d µ ( x ′ ) . Z iyi H e , Y ongsheng H an , J i L i , L iguang L iu , D achun Y ang and W en Y uan Remark 2.5.
By [29, Remark 2.8], we know that the factor √ V δ k ( x ) V δ k ( y ) in (2.1), (2.2) and(2.3) can be replaced by V δ k ( x ) or V δ k ( y ) , and max { d ( x , Y k ) , d ( y , Y k ) } by d ( x , Y k ) or by d ( y , Y k ),with exp {− ν [ d ( x , y ) δ k ] a } replaced by exp {− ν ′ [ d ( x , y ) δ k ] a } , where ν ′ ∈ (0 , ν ) only depends on a and A . Moreover, the condition in Definition 2.4(iii) [resp., (iv)] can be replaced by d ( x , x ′ ) ≤ (2 A ) − [ δ k + d ( x , y )] (resp., d ( x , x ′ ) ≤ (2 A ) − [ δ k + d ( x , y )] and d ( y , y ′ ) ≤ (2 A ) − [ δ k + d ( x , y )]).For their proofs, see [29, Proposition 2.9].With the above exp-ATI, we have the following homogeneous continuous Calder´on reproducingformula established in [29]. Theorem 2.6.
Let { Q k } k ∈ Z be an exp - ATI and β, γ ∈ (0 , η ) . Then there exists a sequence { e Q k } k ∈ Z of bounded linear operators on L ( X ) such that, for any f ∈ ( ˚ G η ( β, γ )) ′ ,f = ∞ X k = −∞ e Q k Q k f , where the series converges in ( ˚ G η ( β, γ )) ′ . Moreover, there exists a positive constant C such that,for any k ∈ Z , the kernel of e Q k satisfies the following conditions: (i) for any x , y ∈ X, (cid:12)(cid:12)(cid:12) e Q k ( x , y ) (cid:12)(cid:12)(cid:12) ≤ C V δ k ( x ) + V ( x , y ) " δ k δ k + d ( x , y ) γ ;(ii) for any x , x ′ , y ∈ X with d ( x , x ′ ) ≤ (2 A ) − [ δ k + d ( x , y )] , (cid:12)(cid:12)(cid:12) e Q k ( x , y ) − e Q k ( x ′ , y ) (cid:12)(cid:12)(cid:12) ≤ C " d ( x , x ′ ) δ k + d ( x , y ) β V δ k ( x ) + V ( x , y ) " δ k δ k + d ( x , y ) γ ;(iii) for any x ∈ X, Z X e Q k ( x , y ) d µ ( y ) = = Z X e Q k ( y , x ) d µ ( y ) . Next, we recall the homogeneous discrete Calder´on reproducing formulae established in [29].To this end, let j ∈ N be a su ffi ciently large integer such that δ j ≤ (2 A ) − C ♮ , where C ♮ is as inLemma 2.3. Based on Lemma 2.3, for any k ∈ Z and α ∈ A k , we let N ( k , α ) : = { τ ∈ A k + j : Q k + j τ ⊂ Q k α } and N ( k , α ) be the cardinality of the set N ( k , α ). For any k ∈ Z and α ∈ A k , we rearrange the set { Q k + j τ : τ ∈ N ( k , α ) } as { Q k , m α } N ( k ,α ) m = , whose centers are denoted, respectively, by { z k , m α } N ( k ,α ) m = . Theorem 2.7.
Let { Q k } k ∈ Z be an exp - ATI and β, γ ∈ (0 , η ) . For any k ∈ Z , α ∈ A k and m ∈{ , . . . , N ( k , α ) } , suppose that y k , m α is an arbitrary point in Q k , m α . Then, for any i ∈ { , } , there existsa sequence { e Q ( i ) k } ∞ k = −∞ of bounded linear operators on L ( X ) such that, for any f ∈ ( ˚ G η ( β, γ )) ′ ,f ( · ) = ∞ X k = −∞ X α ∈A k N ( k ,α ) X m = e Q (1) k (cid:16) · , y k , m α (cid:17) Z Q k , m α Q k f ( y ) d µ ( y ) ardy S paces on S paces of H omogenous T ype = ∞ X k = −∞ X α ∈A k N ( k ,α ) X m = µ (cid:16) Q k , m α (cid:17) e Q (2) k (cid:16) · , y k , m α (cid:17) Q k f (cid:16) y k , m α (cid:17) , where the equalities converge in ( ˚ G η ( β, γ )) ′ . Moreover, for any k ∈ Z , the kernels of e Q (1) k and e Q (2) k satisfy (i), (ii) and (iii) of Theorem 2.6. To recall the inhomogeneous discrete Calder´on reproducing formulae established in [29], weintroduce the following 1-exp-ATI and exp-IATI.
Definition 2.8.
A sequence { P k } ∞ k = −∞ of bounded linear operators on L ( X ) is called an approxi-mation of the identity with exponential decay and integration { P k } ∞ k = −∞ has the following properties:(i) for any k ∈ Z , P k satisfies (ii), (iii) and (iv) of Definition 2.4 but without the exponentialdecay factor exp ( − ν " max { d ( x , Y k ) , d ( y , Y k ) } δ k a ) ;(ii) for any k ∈ Z and x ∈ X , R X P k ( x , y ) d µ ( y ) = = R X P k ( y , x ) d µ ( y );(iii) for any k ∈ Z , letting Q k : = P k − P k − , then { Q k } k ∈ Z is an exp-ATI. Remark 2.9.
The existence of the 1-exp-ATI is guaranteed by [1, Lemma 10.1]. Moreover, bythe proofs of [29, Proposition 2.9] and [27, Proposition 2.7(iv)], we know that, for any f ∈ L ( X ),lim k →∞ P k f = f in L ( X ). Definition 2.10.
A sequence { Q k } ∞ k = of bounded linear operators on L ( X ) is called an inhomoge-neous approximation of the identity with exponential decay (for short, exp-IATI) if there exists a1-exp-ATI { P k } ∞ k = −∞ such that Q = P and Q k = P k − P k − for any k ∈ N .Next we recall the following inhomogeneous discrete reproducing formula established in [29]. Theorem 2.11.
Let { Q k } ∞ k = be an exp-IATI and β, γ ∈ (0 , η ) . Then there exists a sequence { e Q k } ∞ k = of bounded linear operators on L ( X ) such that, for any f ∈ ( G η ( β, γ )) ′ ,f ( · ) = N X k = X α ∈A k N (0 ,α ) X m = Z Q k , m α e Q k ( · , y ) d µ ( y ) Q k , m α, ( f ) + ∞ X k = X α ∈A k N ( k ,α ) X m = µ (cid:16) Q k , m α (cid:17) e Q k (cid:16) · , y k , m α (cid:17) Q k f (cid:16) y k , m α (cid:17) , where the equality converges in ( G η ( β, γ )) ′ , every y k , m α is an arbitrary point in Q k , m α and, for anyk ∈ { , . . . , N } , Q k , m α, ( f ) : = µ ( Q k , m α ) Z Q k , m α Q k f ( u ) d µ ( u ) . Moreover, for any k ∈ Z + , e Q k satisfies (i) and (ii) of Theorem 2.6 and, for any x ∈ X, Z X e Q k ( x , y ) d µ ( y ) = Z X e Q k ( y , x ) d µ ( y ) = if k ∈ { , . . . , N } , if k ∈ { N + , N + , . . . } , where N ∈ N is some fixed constant independent of f and y k , m α . Z iyi H e , Y ongsheng H an , J i L i , L iguang L iu , D achun Y ang and W en Y uan Let β, γ ∈ (0 , η ) and f ∈ ( G η ( β, γ )) ′ . Let { P k } k ∈ Z be a 1-exp-ATI as in Definition 2.8. Definethe radial maximal function M + ( f ) of f by setting M + ( f )( x ) : = sup k ∈ Z | P k f ( x ) | , ∀ x ∈ X . Define the non-tangential maximal function M θ ( f ) of f with aperture θ ∈ (0 , ∞ ) by setting M θ ( f )( x ) : = sup k ∈ Z sup y ∈ B ( x ,θδ k ) | P k f ( y ) | , ∀ x ∈ X . Also, define the grand maximal function f ∗ of f by setting f ∗ ( x ) : = sup n |h f , ϕ i| : ϕ ∈ G η ( β, γ ) and k ϕ k G ( x , r ,β,γ ) ≤ r ∈ (0 , ∞ ) o , ∀ x ∈ X . Correspondingly, for any p ∈ (0 , ∞ ], the Hardy spaces H + , p ( X ), H p θ ( X ) with θ ∈ (0 , ∞ ) and H ∗ , p ( X ) are defined, respectively, by setting H + , p ( X ) : = n f ∈ (cid:16) G η ( β, γ ) (cid:17) ′ : k f k H + , p ( X ) : = kM + ( f ) k L p ( X ) < ∞ o , H p θ ( X ) : = n f ∈ (cid:16) G η ( β, γ ) (cid:17) ′ : k f k H p θ ( X ) : = kM θ ( f ) k L p ( X ) < ∞ o and H ∗ , p ( X ) : = n f ∈ (cid:16) G η ( β, γ ) (cid:17) ′ : k f k H ∗ , p ( X ) : = k f ∗ k L p ( X ) < ∞ o . Based on [20, Remark 2.9(ii)], we easily observe that, for any f ∈ ( G η ( β, γ )) ′ and x ∈ X ,(3.1) M + f ( x ) ≤ M θ ( f )( x ) ≤ C f ∗ ( x ) , where C is a positive constant only depending on θ .The aim of this section is to prove that the Hardy spaces H + , p ( X ), H p θ ( X ) and H ∗ , p ( X ) aremutually equivalent when p ∈ ( ω/ ( ω + η ) , ∞ ] in the sense of equivalent (quasi-)norms (see Section3.2); in particular, they all are equivalent to the Lebesgue space L p ( X ) when p ∈ (1 , ∞ ] in thesense of equivalent norms (see Section 3.1). Moreover, we prove that H ∗ , p ( X ) is independent ofthe choices of the distribution space ( G η ( β, γ )) ′ whenever β, γ ∈ ( ω (1 / p − , η ); see Proposition3.8 below. L p ( X ) when p ∈ (1 , ∞ ] In this section, we show that the Hardy spaces H + , p ( X ), H p θ ( X ) and H ∗ , p ( X ) are all equiva-lent to the Lebesgue space L p ( X ), when p ∈ (1 , ∞ ], in the sense of both representing the samedistributions and equivalent norms. First we give some basic properties of H ∗ , p ( X ). Proposition 3.1.
Let p ∈ (0 , ∞ ] . Then H ∗ , p ( X ) is a (quasi-)Banach space, which is continuouslyembedded into ( G η ( β, γ )) ′ , where β, γ ∈ (0 , η ) . ardy S paces on S paces of H omogenous T ype Proof.
Let f ∈ H ∗ , p ( X ) and ϕ ∈ G η ( β, γ ) with k ϕ k G ( β,γ ) ≤
1. For any x ∈ B ( x , k ϕ k G ( x , ,β,γ ) . x andhence |h f , ϕ i| . f ∗ ( x ). Therefore, for any ϕ ∈ G η ( β, γ ) with β, γ ∈ (0 , η ), we have |h f , ϕ i| p . V ( x ) Z B ( x , [ f ∗ ( x )] p d µ ( x ) . k f ∗ k pL p ( X ) ∼ k f k pH ∗ , p ( X ) . This implies that H ∗ , p ( X ) is continuously embedded into ( G η ( β, γ )) ′ .To see that H ∗ , p ( X ) is a (quasi-)Banach space, we only prove its completeness. Indeed, supposethat { f k } ∞ k = in H ∗ , p ( X ) is a Cauchy sequence, which is also a Cauchy sequence in ( G η ( β, γ )) ′ with β, γ ∈ (0 , η ). By the completeness of ( G η ( β, γ )) ′ , the sequence { f k } ∞ k = converges to some element f ∈ ( G η ( β, γ )) ′ as k → ∞ . If ϕ ∈ G η ( β, γ ) satisfies k ϕ k G ( x , r ,β,γ ) ≤ x ∈ X and r ∈ (0 , ∞ ),then |h f k + l − f k , ϕ i| ≤ ( f k + l − f k ) ∗ ( x ) for any k , l ∈ N . Letting l → ∞ , we obtain |h f − f k , ϕ i| ≤ lim inf l →∞ ( f k + l − f k ) ∗ ( x ) , which further implies that, for any x ∈ X ,( f − f k ) ∗ ( x ) ≤ lim inf l →∞ ( f k + l − f k ) ∗ ( x ) . By the Fatou lemma, we conclude that k ( f − f k ) ∗ k L p ( X ) ≤ lim inf l →∞ k ( f k + l − f k ) ∗ k L p ( X ) → k → ∞ , which, together with the sublinearity of k · k H ∗ , p ( X ) , further implies that f ∈ H ∗ , p ( X ) andlim k →∞ k f − f k k H ∗ , p ( X ) =
0. Therefore, H ∗ , p ( X ) is complete. This finishes the proof of Proposition3.1. (cid:3) To show the equivalence of H + , p ( X ), H p θ ( X ) and H ∗ , p ( X ) to the Lebesgue space L p ( X ) when p ∈ (1 , ∞ ] in the sense of both representing the same distributions and equivalent norms, we needthe following technical lemma. Lemma 3.2.
Let { P k } k ∈ Z be a as in Definition 2.8. Assume that β, γ ∈ (0 , η ) . Then thefollowing statements hold true: (i) there exists a positive constant C such that, for any k ∈ Z and ϕ ∈ G ( β, γ ) , k P k ϕ k G ( β,γ ) ≤ C k ϕ k G ( β,γ ) ; (ii) for any f ∈ G ( β, γ ) and β ′ ∈ (0 , β ) , lim k →∞ P k f = f in G ( β ′ , γ ) ; (iii) if f ∈ G η ( β, γ ) [resp., f ∈ ( G η ( β, γ )) ′ ], then lim k →∞ P k f = f in G η ( β, γ ) [resp., ( G η ( β, γ )) ′ ].Proof. The proof of (i) can be obtained by the method used in the proof of [29, Lemma 4.14].The proof of (ii) is given in [20, Lemma 3.6], whose proof does not rely on the reverse doublingcondition of µ and the metric d . We obtain (iii) directly by (i), (ii) and a standard duality argument.This finishes the proof of Lemma 3.2. (cid:3) Then we have the following proposition.4 Z iyi H e , Y ongsheng H an , J i L i , L iguang L iu , D achun Y ang and W en Y uan Proposition 3.3.
Let p ∈ [1 , ∞ ] , β, γ ∈ (0 , η ) and { P k } k ∈ Z be a . If f ∈ ( G η ( β, γ )) ′ belongs to H + , p ( X ) , then there exists e f ∈ L p ( X ) such that, for any ϕ ∈ G η ( β, γ ) , h f , ϕ i = Z X e f ( x ) ϕ ( x ) d µ ( x )(3.2) and k e f k L p ( X ) ≤ kM + ( f ) k L p ( X ) ; moreover, if p ∈ [1 , ∞ ) , then, for almost every x ∈ X, | e f ( x ) | ≤M + ( f )( x ) .Proof. Let f ∈ ( G η ( β, γ )) ′ and M + ( f ) = sup k ∈ Z | P k f | ∈ L p ( X ), where { P k } k ∈ Z is a 1-exp-ATI asin Definition 2.8. Then { P k f } k ∈ Z is uniformly bounded in L p ( X ). If p ∈ (1 , ∞ ], then p ′ ∈ [1 , ∞ )and L p ′ ( X ) is separable. Thus, by the Banach-Alaoglu theorem (see, for example, [45, Theorem3.17]), we find a function e f ∈ L p ( X ) and a sequence { k j } ∞ j = ⊂ Z such that k j → ∞ and P k j f → e f as j → ∞ in the weak- ∗ topology of L p ( X ). By this and the H ¨older inequality, for any g ∈ L p ′ ( X ),we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z X e f ( x ) g ( x ) d µ ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = lim j →∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z X P k j f ( x ) g ( x ) d µ ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ kM + ( f ) k L p ( X ) k g k L p ′ ( X ) , which further implies that k e f k L p ( X ) ≤ kM + ( f ) k L p ( X ) .If p =
1, notice that k sup k ∈ Z | P k f |k L ( X ) = kM + ( f ) k L ( X ) < ∞ . Then, by the proof of [50,Theorem III.C.12], { P k f } k ∈ Z is relatively compact in L ( X ). Therefore, by the Eberlin- ˘Smuliantheorem (see [50, II.C]), we know that { P k f } k ∈ Z is weakly sequentially compact, that is, there exista function e f ∈ L ( X ) and a subsequence { P k j f } ∞ j = such that { P k j f } ∞ j = converges to e f weakly in L ( X ). As the arguments for the case p ∈ (1 , ∞ ], we still have k e f k L ( X ) ≤ kM + ( f ) k L ( X ) .Moreover, for any ϕ ∈ G η ( β, γ ), by the fact G η ( β, γ ) ⊂ L p ( X ) for any p ∈ [1 , ∞ ] and Lemma3.2(iii), we conclude that(3.3) h f , ϕ i = lim k →∞ h P k j f , ϕ i = lim j →∞ Z X P k j f ( x ) ϕ ( x ) d µ ( x ) = Z X e f ( x ) ϕ ( x ) d µ ( x ) . Let p ∈ [1 , ∞ ). For any j ∈ N and x ∈ X , we have P k j ( x , · ) ∈ G ( η, η ) (see the proof of [29,Proposition 2.10]), which, together with (3.3), implies that P k j f ( x ) = h f , P k j ( x , · ) i = Z X P k j ( x , y ) e f ( y ) d µ ( y ) = P k j e f ( x ) . From this and [27, Proposition 2.7(iv)], we deduce that { P k j f } j ∈ N converges to e f in the senseof k · k L p ( X ) . Then, by the Riesz theorem, we find a subsequence of { P k j f } j ∈ N , still denoted by { P k j f } j ∈ N , such that P k j f ( x ) → e f ( x ) as k j → ∞ for almost every x ∈ X . Therefore, | e f ( x ) | ≤M + ( f )( x ) for almost every x ∈ X . This finishes the proof of Proposition 3.3. (cid:3) Finally, we show the following main result of this section.
Theorem 3.4.
Let p ∈ (1 , ∞ ] and β, γ ∈ (0 , η ) . Then the following hold true: (i) if f ∈ ( G η ( β, γ )) ′ belongs to H + , p ( X ) , then there exists e f ∈ L p ( X ) such that (3.2) holds trueand k e f k L p ( X ) ≤ k f k H + , p ( X ) ; ardy S paces on S paces of H omogenous T ype any f ∈ L p ( X ) induces a distribution on G η ( β, γ ) as in (3.2) , still denoted by f , such thatf ∈ H ∗ , p ( X ) and k f k H ∗ , p ( X ) ≤ C k f k L p ( X ) , where C is a positive constant independent of f .Consequently, for any fixed θ ∈ (0 , ∞ ) , H + , p ( X ) = H p θ ( X ) = H ∗ , p ( X ) = L p ( X ) in the sense of bothrepresenting the same distributions and equivalent norms.Proof. We obtain (i) directly by Proposition 3.3. Now we prove (ii). Suppose that p ∈ (1 , ∞ ]and f ∈ L p ( X ). Clearly, f induces a distribution on G η ( β, γ ) as in (3.2). By [20, Proposition3.9], we find that, for almost every x ∈ X , f ∗ ( x ) . M ( f )( x ), with the implicit positive con-stant independent of f and x . Therefore, from the boundedness of M on L p ( X ), we deduce that k f ∗ k L p ( X ) . kM ( f ) k L p ( X ) . k f k L p ( X ) . This finishes the proof of (ii).By (i), (ii) and (3.1), we obtain H + , p ( X ) = H p θ ( X ) = H ∗ , p ( X ) = L p ( X ), which completes theproof of Theorem 3.4. (cid:3) The main aim of this section concerns the equivalence of Hardy spaces defined via variousmaximal functions for the case p ∈ ( ω/ ( ω + η ) , Theorem 3.5.
Assume that p ∈ ( ω/ ( ω + η ) , and θ ∈ (0 , ∞ ) . Then, for any f ∈ ( G η ( β, γ )) ′ with β, γ ∈ ( ω (1 / p − , η ) , k f k H + , p ( X ) ∼ k f k H p θ ( X ) ∼ k f k H ∗ , p ( X ) , with equivalent positive constants independent of f . In other words, H + , p ( X ) = H p θ ( X ) = H ∗ , p ( X ) with equivalent (quasi-)norms. To prove Theorem 3.5, we borrow some ideas from [54]. To this end, we need the followingtwo technical lemmas.
Lemma 3.6.
Assume that φ ∈ G η ( β, γ ) with β, γ ∈ (0 , η ) . Let σ : = R X φ ( x ) d µ ( x ) . If ψ ∈ G ( η, η ) with R X ψ ( x ) d µ ( x ) = , then φ − σψ ∈ ˚ G η ( β, γ ) .Proof. Since φ ∈ G η ( β, γ ) with β, γ ∈ (0 , η ), it follows that there exists { φ n } ∞ n = ⊂ G ( η, η ) suchthat lim n →∞ k φ − φ n k G ( β,γ ) =
0. Letting σ n : = R X φ n ( x ) d µ ( x ) for any n ∈ N , by Definition 2.1and Lemma 2.2(ii), we conclude that lim n →∞ | σ − σ n | =
0, where σ : = R X φ ( x ) d µ ( x ). Let ϕ n : = φ n − σ n ψ for any n ∈ N . Then ϕ n ∈ ˚ G ( η, η ) and k φ − σψ − ϕ n k G ( β,γ ) ≤ k φ − φ n k G ( β,γ ) + | σ − σ n |k ψ k G ( β,γ ) → n → ∞ . Thus, φ − σψ ∈ ˚ G η ( β, γ ). This finishes the proof of Lemma 3.6. (cid:3) The next lemma comes from [27, Lemma 5.3], whose proof remains true for a quasi-metric d and also does not rely on the reverse doubling condition of µ .6 Z iyi H e , Y ongsheng H an , J i L i , L iguang L iu , D achun Y ang and W en Y uan Lemma 3.7.
Let all the notation be as in Theorem 2.7. Let k , k ′ ∈ Z , { a k , m α } k ∈ Z , α ∈A k , m ∈{ ,..., N ( k ,α ) } ⊂ C , γ ∈ (0 , η ) and r ∈ ( ω/ ( ω + γ ) , . Then there exists a positive constant C, independent of k, k ′ ,y k , m α ∈ Q k , m α and a k , m α with k ∈ Z , α ∈ A k and m ∈ { , . . . , N ( k , α ) } , such that, for any x ∈ X, X α ∈A k N ( k ,α ) X m = µ (cid:16) Q k , m α (cid:17) V δ k ∧ k ′ ( x ) + V ( x , y k , m α ) δ k ∧ k ′ δ k ∧ k ′ + d ( x , y k , m α ) γ (cid:12)(cid:12)(cid:12) a k , m α (cid:12)(cid:12)(cid:12) ≤ C δ [ k − ( k ∧ k ′ )] ω (1 − r ) M X α ∈A k N ( k ,α ) X m = (cid:12)(cid:12)(cid:12) a k , m α (cid:12)(cid:12)(cid:12) r χ Q k , m α ( x ) r . Now we show Theorem 3.5 by using the above two technical lemmas. In what follows, the symbol ǫ → + means that ǫ ∈ (0 , ∞ ) and ǫ → Proof of Theorem 3.5.
Let f ∈ ( G η ( β, γ )) ′ with β, γ ∈ ( ω (1 / p − , η ). Fix θ ∈ (0 , ∞ ). By (3.1),we have kM + ( f ) k L p ( X ) ≤ kM θ ( f ) k L p ( X ) . k f ∗ k L p ( X ) . Thus, the proof of Theorem 3.5 is reduced to showing k f ∗ k L p ( X ) . kM + ( f ) k L p ( X ) . (3.4)To obtain (3.4), it su ffi ces to prove that, for some r ∈ (0 , p ) and any x ∈ X ,(3.5) f ∗ ( x ) . M + ( f )( x ) + n M (cid:16)(cid:2) M + ( f ) (cid:3) r (cid:17) ( x ) o r . If (3.5) holds true, then, by the boundedness of M on L p / r ( X ), we conclude that k f ∗ k L p ( X ) . kM + ( f ) k L p ( X ) + (cid:13)(cid:13)(cid:13)(cid:13) M (cid:16)(cid:2) M + ( f ) (cid:3) r (cid:17)(cid:13)(cid:13)(cid:13)(cid:13) r L p / r ( X ) ∼ kM + ( f ) k L p ( X ) , which proves (3.4).We now fix x ∈ X and show (3.5). Let { P k } k ∈ Z be a 1-exp-ATI. For any k ∈ Z , define Q k : = P k − P k − . Then { Q k } k ∈ Z is an exp-ATI. Assume for the moment that, for any ϕ ∈ ˚ G η ( β, γ ) with k ϕ k G ( x ,δ l ,β,γ ) ≤ l ∈ Z ,(3.6) |h f , ϕ i| . n M (cid:16)(cid:2) M + ( f ) (cid:3) r (cid:17) ( x ) o r . We now use (3.6) to show (3.5). For any φ ∈ G η ( β, γ ) with k φ k G ( x , r ,β,γ ) ≤ r ∈ (0 , ∞ ),choose l ∈ Z such that δ l + ≤ r < δ l . Clearly, k φ k G ( x ,δ l ,β,γ ) .
1. Let σ : = R X φ ( y ) d µ ( y ) and ϕ : = φ − σ P l ( x , · ). Notice that R X P l ( x , y ) d µ ( y ) = P l ( x , · ) ∈ G ( η, η ) (see the proof of [29,Proposition 2.10]). From Lemma 3.6, it follows that ϕ ∈ ˚ G η ( β, γ ). Moreover, k ϕ k G ( x ,δ l ,β,γ ) . k φ k G ( x ,δ l ,β,γ ) + | σ |k P l ( x , · ) k G ( x ,δ l ,β,γ ) .
1. By (3.6), we know that |h f , φ i| ≤ |h f , ϕ i| + | σ ||h f , P l ( x , · ) i| . n M (cid:16)(cid:2) M + ( f ) (cid:3) r (cid:17) ( x ) o r + | P l f ( x ) | . n M (cid:16)(cid:2) M + ( f ) (cid:3) r (cid:17) ( x ) o r + M + ( f )( x ) , ardy S paces on S paces of H omogenous T ype ǫ ∈ (0 , ∞ ), choose y k , m α ∈ Q k , m α such that (cid:12)(cid:12)(cid:12)(cid:12) Q k f (cid:16) y k , m α (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) ≤ inf z ∈ Q k , m α | Q k f ( z ) | + ǫ ≤ z ∈ Q k , m α M + ( f )( z ) + ǫ. Let g : = f | ˚ G η ( β,γ ) be the restriction of f on ˚ G η ( β, γ ). Obviously, g ∈ ( ˚ G η ( β, γ )) ′ and k g k ( ˚ G η ( β,γ )) ′ ≤k f k ( G η ( β,γ )) ′ . By Theorem 2.7, we conclude that h f , ϕ i = h g , ϕ i = ∞ X k = −∞ X α ∈A k N ( k ,α ) X m = µ (cid:16) Q k , m α (cid:17) e Q ∗ k ϕ (cid:16) y k , m α (cid:17) Q k g (cid:16) y k , m α (cid:17) = ∞ X k = −∞ X α ∈A k N ( k ,α ) X m = µ (cid:16) Q k , m α (cid:17) e Q ∗ k ϕ (cid:16) y k , m α (cid:17) Q k f (cid:16) y k , m α (cid:17) , where e Q ∗ k denotes the dual operator of e Q k . By the proof of [27, (3.2)], which remains true for aquasi-metric d and does not rely on the reverse doubling condition of µ , we find that, for any fixed β ′ ∈ (0 , β ∧ γ ) and any k ∈ Z ,(3.7) (cid:12)(cid:12)(cid:12)(cid:12) e Q ∗ k ϕ (cid:16) y k , m α (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) . δ | k − l | β ′ V δ k ∧ l ( x ) + V ( x , y k , m α ) δ k ∧ l δ k ∧ l + d ( x , y k , m α ) γ . Choose β ′ ∈ (0 , β ∧ γ ) such that ω/ ( ω + β ′ ) < p . From this and Lemma 3.7, we deduce that, forany fixed r ∈ ( ω/ ( ω + β ′ ) , p ), |h f , ϕ i| . ∞ X k = −∞ δ | k − l | β ′ X α ∈A k N ( k ,α ) X m = µ (cid:16) Q k , m α (cid:17) inf z ∈ Q k , m α M + ( f )( z ) + ǫ V δ k ∧ l ( x ) + V ( x , y k , m α ) δ k ∧ l δ k ∧ l + d ( x , y k , m α ) γ (3.8) . ∞ X k = −∞ δ | k − l | β ′ δ [ k − ( k ∧ l )] ω (1 − r ) M X α ∈A k N ( k ,α ) X m = inf z ∈ Q k , m α M + ( f )( z ) + ǫ r χ Q k , m α ( x ) r . ∞ X k = −∞ δ | k − l | β ′ δ [ k − ( k ∧ l )] ω (1 − r ) n M (cid:16)(cid:2) M + ( f ) + ǫ (cid:3) r (cid:17) ( x ) o r . n M (cid:16)(cid:2) M + ( f ) (cid:3) r (cid:17) ( x ) + ǫ r o r → n M (cid:16)(cid:2) M + ( f ) (cid:3) r (cid:17) ( x ) o r as ǫ → + . This proves (3.6) and hence finishes the proof of Theorem 3.5. (cid:3)
To conclude this section, we show that the Hardy space H ∗ , p ( X ) is independent of the choicesof ( G η ( β, γ )) ′ whenever β, γ ∈ ( ω (1 / p − , η ). Proposition 3.8.
Let p ∈ ( ω/ ( ω + η ) , and β , β , γ , γ ∈ ( ω (1 / p − , η ) . If f ∈ ( G η ( β , γ )) ′ and f ∈ H ∗ , p ( X ) , then f ∈ ( G η ( β , γ )) ′ . Z iyi H e , Y ongsheng H an , J i L i , L iguang L iu , D achun Y ang and W en Y uan Proof.
Let f ∈ ( G η ( β , γ )) ′ with k f k H ∗ , p ( X ) < ∞ . We first prove that there exists θ ∈ (0 , ∞ ) suchthat, for any ϕ ∈ G ( η, η ) with k ϕ k G ( β ,γ ) ≤ |h f , ϕ i| . kM θ ( f ) k L p ( X ) . Notice that ϕ ∈ G ( η, η ) ⊂ G η ( β , γ ) and f ∈ ( G η ( β , γ )) ′ . With all the notation involved as inTheorem 2.11, we have h f , ϕ i = N X k = X α ∈A N ( k ,α ) X m = Z Q k , m α e Q ∗ k ϕ ( y ) d µ ( y ) Q k , m α, ( f ) + ∞ X k = N + X α ∈A k N ( k ,α ) X m = µ (cid:16) Q k , m α (cid:17) e Q ∗ k ϕ (cid:16) y k , m α (cid:17) Q k f (cid:16) y k , m α (cid:17) = : Z + Z . Choose θ : = A C ♮ with C ♮ as in Lemma 2.3(v). By the definition of Q k , m α and Lemma 2.3(v), wehave Q k , m α ⊂ B ( z k , m α , C ♮ δ k + j ) ⊂ B ( z , A C ♮ δ k ) = B ( z , θδ k ) for any z ∈ Q k , m α .Fix x ∈ B ( x , k ϕ k G ( x , ,β ,γ ) ∼ k ϕ k G ( x , ,β ,γ ) .
1. If k ∈ { , . . . , N } , then we have k ϕ k G ( x ,δ k ,β ,γ ) ∼ k ϕ k G ( x , ,β ,γ ) .
1, where the implicit constants are independent of x but candepend on N . Let β − : = min { β , γ , β , γ } . By [27, (3.2)], we conclude that, for any y ∈ Q k , m α , (cid:12)(cid:12)(cid:12) e Q ∗ k ϕ ( y ) (cid:12)(cid:12)(cid:12) . V ( x ) + V ( x , y ) " + d ( x , y ) β − ∼ V ( x ) + V ( x , y k , m α ) + d ( x , y k , m α ) β − . Moreover, for any k ∈ { , . . . , N } and z ∈ Q k , m α , we have (cid:12)(cid:12)(cid:12)(cid:12) Q k , m α, ( f ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ µ ( Q k , m α ) Z Q k , m α [ | P k f ( y ) | + | P k − f ( y ) | ] d µ ( y ) ≤ M θ ( f )( z ) . Thus, we obtain | Z | . N X k = X α ∈A k N ( k ,α ) X m = V ( x ) + V ( x , y k , m α ) + d ( x , y k , m α ) β − inf z ∈ Q k , m α M θ ( f )( z ) . (3.10)If k ∈ { N + , N + , . . . } , then | Q k f ( y k , m α ) | ≤ z ∈ Q k , m α M θ ( f )( z ). Again, by k ϕ k G ( x , ,β ,γ ) . β ′ ∈ (0 , β − ), (cid:12)(cid:12)(cid:12)(cid:12) e Q ∗ k ϕ (cid:16) y k , m α (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) . δ k β ′ V ( x ) + V ( x , y k , m α ) + d ( x , y k , m α ) β − , because now k ∈ Z + and we do not need the cancelation of ϕ . Therefore, we have | Z | . ∞ X k = N + δ k β ′ X α ∈A k N ( k ,α ) X m = V ( x ) + V ( x , y k , m α ) + d ( x , y k , m α ) β − inf z ∈ Q k , m α M θ ( f )( z ) . (3.11) ardy S paces on S paces of H omogenous T ype r ∈ ( ω/ ( ω + η ) , p ), |h f , ϕ i| . (cid:8) M (cid:0)(cid:2) M θ ( f ) (cid:3) r (cid:1) ( x ) (cid:9) r . Notice that the above inequality holds true for any x ∈ B ( x , M on L p / r ( X ), we further conclude that |h f , ϕ i| p . V ( x ) Z X (cid:8) M (cid:0)(cid:2) M θ ( f ) (cid:3) r (cid:1) ( x ) (cid:9) pr d µ ( x ) . kM θ ( f ) k pL p ( X ) , which is exactly (3.9).Combining (3.9) and (3.1), we find that, for any ϕ ∈ G ( η, η ),(3.12) |h f , ϕ i| . kM θ ( f ) k L p ( X ) k ϕ k G ( β ,γ ) . k f k H ∗ , p ( X ) k ϕ k G ( β ,γ ) . Now let g ∈ G η ( β , γ ). By the definition of G η ( β , γ ), we know that there exist { ϕ j } ∞ j = ⊂ G ( η, η )such that k g − ϕ j k G ( β ,γ ) → j → ∞ , which implies that { ϕ j } ∞ j = is a Cauchy sequence in G ( β , γ ). By (3.12), we find that, for any j , k ∈ N , |h f , ϕ j − ϕ k i| . k f k H ∗ , p ( X ) k ϕ j − ϕ k k G ( β ,γ ) . Therefore, lim j →∞ h f , ϕ j i exists and the limit is independent of the choice of { ϕ j } ∞ j = . Thus, it isreasonable to define h f , g i : = lim j →∞ h f , ϕ j i . Moreover, by (3.12), we conclude that |h f , g i| = lim j →∞ |h f , ϕ j i| . k f k H ∗ , p ( X ) lim inf j →∞ k ϕ j k G ( β ,γ ) ∼ k f k H ∗ , p ( X ) k g k G η ( β ,γ ) . This implies f ∈ ( G η ( β , γ )) ′ and k f k ( G η ( β ,γ )) ′ . k f k H ∗ , p ( X ) , which completes the proof of Propo-sition 3.8. (cid:3) In this section, we establish the atomic characterizations of H ∗ , p ( X ) with p ∈ ( ω/ ( ω + η ) , Definition 4.1.
Let p ∈ ( ω/ ( ω + η ) , q ∈ ( p , ∞ ] ∩ [1 , ∞ ] and β, γ ∈ ( ω (1 / p − , η ). The atomicHardy space H p , q at ( X ) is defined to be the set of all f ∈ ( G η ( β, γ )) ′ such that f = P ∞ j = λ j a j , where { a j } ∞ j = is a sequence of ( p , q )-atoms and { λ j } ∞ j = ⊂ C satisfies P ∞ j = | λ j | p < ∞ . Moreover, let k f k H p , q at ( X ) : = inf ∞ X j = | λ j | p p , where the infimum is taken over all the decompositions of f as above.Observe that the di ff erence between H p , q cw ( X ) and H p , q at ( X ) mainly lies on the choices of distribu-tion spaces. When ( X , d , µ ) is a doubling metric measure space, it was proved in [40, Theorem 4.4]that H p , q cw ( X ) and H p , q at ( X ) coincide with equivalent (quasi-)norms. Since now d is a quasi-metric,for the completeness of this article, we include a proof of their equivalence in Section 4.4 below.The main aim in this section is to prove the following conclusion.0 Z iyi H e , Y ongsheng H an , J i L i , L iguang L iu , D achun Y ang and W en Y uan Theorem 4.2.
Let p ∈ ( ω/ ( ω + η ) , , q ∈ ( p , ∞ ] ∩ [1 , ∞ ] and β, γ ∈ ( ω (1 / p − , η ) . As subspacesof ( G η ( β, γ )) ′ , H ∗ , p ( X ) = H p , q at ( X ) with equivalent (quasi-)norms. We divide the proof of Theorem 4.2 into three sections. In Section 4.1, we prove that H p , q at ( X ) ⊂ H ∗ , p ( X ) directly by the definition of H p , q at ( X ). The next two sections mainly deal with the proof of H ∗ , p ( X ) ⊂ H p , q at ( X ). In Section 4.2, we obtain a Calder´on-Zygmund decomposition for any f ∈ H ∗ , p ( X ). Then, in Section 4.3, we show that any f ∈ H ∗ , p ( X ) has a ( p , ∞ )-atomic decomposition.In Section 4.4, we reveal the equivalent relationship between H p , q at ( X ) and H p , q cw ( X ). H p , q at ( X ) ⊂ H ∗ , p ( X ) In this section, we prove H p , q at ( X ) ⊂ H ∗ , p ( X ), as subspaces of ( G η ( β, γ )) ′ with β, γ ∈ ( ω (1 / p − , η ). To do this, we need the following technical lemma. Lemma 4.3.
Let p ∈ ( ω/ ( ω + η ) , and q ∈ ( p , ∞ ] ∩ [1 , ∞ ] . Then there exists a positive constantC such that, for any ( p , q ) -atom a supported on B : = B ( x B , r B ) , with x B ∈ X and r B ∈ (0 , ∞ ) , andany x ∈ X, a ∗ ( x ) ≤ C M ( a )( x ) χ B ( x B , A r B ) ( x ) + C " r B d ( x B , x ) β [ µ ( B )] − p V ( x B , x ) χ [ B ( x B , A r B )] ∁ ( x )(4.1) and k a ∗ k L p ( X ) ≤ C , (4.2) where the atom a is viewed as a distribution on G η ( β, γ ) with β, γ ∈ ( ω (1 / p − , η ) .Proof. First, we show (4.1). Let ϕ ∈ G η ( β, γ ) be such that k ϕ k G ( x , r ,β,γ ) ≤ r ∈ (0 , ∞ ),where β, γ ∈ ( ω (1 / p − , η ). When x ∈ B ( x B , A r B ), by Lemma 2.2(v), we find that |h a , ϕ i| = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z X a ( y ) ϕ ( y ) d µ ( y ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Z X | a ( y ) | V r ( x ) + V ( x , y ) " rr + d ( x , y ) γ d µ ( y ) . M ( a )( x ) , which consequently implies that a ∗ ( x ) . M ( a )( x ).Let x < B ( x B , A r B ). Then, for any y ∈ B , we have d ( x , x B ) ≥ A r B > A d ( x B , y ). Therefore,by the definition of ( p , q )-atoms and Definition 2.1(ii), we conclude that |h a , ϕ i| = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z B a ( y ) ϕ ( y ) d µ ( y ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Z B | a ( y ) || ϕ ( y ) − ϕ ( x B ) | d µ ( y ) ≤ Z B | a ( y ) | " d ( x B , y ) r + d ( x , x B ) β V r ( x ) + V ( x , x B ) " rr + d ( x , x B ) γ d µ ( y ) ≤ " r B d ( x B , x ) β V ( x , x B ) k a k L ( X ) . " r B d ( x B , x ) β [ µ ( B )] − p V ( x B , x ) . Taking the supremum over all such ϕ ∈ G η ( β, γ ) satisfying k ϕ k G ( x , r ,β,γ ) ≤ r ∈ (0 , ∞ ),we obtain (4.1). ardy S paces on S paces of H omogenous T ype q ∈ (1 , ∞ ], from the H ¨older inequality and the bound-edness of M on L q ( X ), we deduce that Z B ( x B , A r B ) [ M ( a )( x )] p d µ ( x ) ≤ [ µ ( B ( x B , A r B ))] − p / q kM ( a ) k pL q ( X ) . [ µ ( B )] − p / q k a k pL q ( X ) . . If q =
1, then, by p ∈ ( ω/ ( ω + η ) ,
1) and the boundedness of M from L ( X ) to L , ∞ ( X ), weconclude that Z B ( x B , A r B ) [ M ( a )( x )] p d µ ( x ) = Z ∞ µ ( { x ∈ B ( x B , A r B ) : M ( a )( x ) > λ } ) d λ p . Z ∞ min ( µ ( B ) , k a k L ( X ) λ ) d λ p . Z k a k L X ) /µ ( B )0 µ ( B ) d λ p + Z ∞k a k L X ) /µ ( B ) k a k L ( X ) λ − d λ p . k a k pL ( X ) [ µ ( B )] − p . . By the fact β > ω (1 / p −
1) and the doubling condition (1.1), we have Z d ( x , x B ) ≥ A r B " r B d ( x B , x ) β p " µ ( B ) − p " V ( x B , x ) p d µ ( x ) . ∞ X k = − k β p k ω (1 − p ) Z k A r B ≤ d ( x , x B ) < k + A r B V ( x B , x ) d µ ( x ) . . Combining the last three formulae with (4.1), we obtain (4.2), which then completes the proof ofLemma 4.3. (cid:3)
Proof of H p , q at ( X ) ⊂ H ∗ , p ( X ) . Assume that f ∈ ( G η ( β, γ )) ′ is non-zero and it belongs to H p , q at ( X )with β, γ ∈ ( ω (1 / p − , η ). Then f = P ∞ j = λ j a j , where { a j } ∞ j = are ( p , q )-atoms and { λ j } ∞ j = ⊂ C satisfy P ∞ j = | λ j | p ∼ k f k pH p , q at ( X ) . By the definition of the grand maximal function, we conclude that,for any x ∈ X , f ∗ ( x ) ≤ ∞ X j = | λ j | a ∗ j ( x ) . From this and (4.2), we deduce that k f ∗ k pL p ( X ) . ∞ X j = | λ j | p (cid:13)(cid:13)(cid:13) a ∗ j (cid:13)(cid:13)(cid:13) L p ( X ) . ∞ X j = | λ j | p ∼ k f k pH p , q at ( X ) . This finishes the proof of H p , q at ( X ) ⊂ H ∗ , p ( X ). (cid:3) H ∗ , p ( X ) In this section, we obtain a Calder´on-Zygmund decomposition of any f ∈ H ∗ , p ( X ). First weestablish a partition of unity for an open set Ω with µ ( Ω ) < ∞ .2 Z iyi H e , Y ongsheng H an , J i L i , L iguang L iu , D achun Y ang and W en Y uan Proposition 4.4.
Suppose Ω ⊂ X is a proper open set with µ ( Ω ) ∈ (0 , ∞ ) and A ∈ [1 , ∞ ) . For anyx ∈ Ω , let r ( x ) : = d ( x , Ω ∁ )2 AA ∈ (0 , ∞ ) . Then there exist L ∈ N and a sequence { x k } k ∈ I ⊂ Ω , where I is a countable index set, such that (i) { B ( x k , r k / (5 A )) } k ∈ I is disjoint. Here and hereafter, r k : = r ( x k ) for any k ∈ I; (ii) S k ∈ I B ( x k , r k ) = Ω and B ( x k , Ar k ) ⊂ Ω ; (iii) for any x ∈ Ω , Ar k ≤ d ( x , Ω ∁ ) ≤ AA r k whenever x ∈ B ( x k , r k ) and k ∈ I; (iv) for any k ∈ I, there exists y k < Ω such that d ( x k , y k ) < AA r k ; (v) for any given k ∈ I, the number of balls B ( x j , Ar j ) that intersect B ( x k , Ar k ) is at most L ; (vi) if, in addition, Ω is bounded, then, for any σ ∈ (0 , ∞ ) , the set { k ∈ I : r k > σ } is finite.Proof. We show this proposition by borrowing some ideas from [47, pp. 15–16]. Let ǫ : = (5 A ) − and { B ( x , ǫ r ( x )) } x ∈ Ω be a covering of Ω . Now we pick the maximal disjoint subcollection of { B ( x , ǫ r ( x )) } x ∈ Ω , denoted by { B k } k ∈ I , which is at most countable, because of (1.1) and µ ( Ω ) ∈ (0 , ∞ ). For any k ∈ I , denote the center of B k by x k and r ( x k ) by r k . Then we obtain (i).Properties (iii) and (iv) can be shown by the definition of r k , the details being omitted. Now weshow (ii). Obviously, B ( x k , Ar k ) ⊂ Ω for any k ∈ I . It su ffi ces to prove that Ω ⊂ S k ∈ I B ( x k , r k ). Forany x ∈ Ω , since { B k } k ∈ I is maximal, it then follows that there exists k ∈ I such that B ( x k , ǫ r k ) ∩ B ( x , ǫ r ( x )) , ∅ . We claim that r k ≥ r ( x ) / (4 A ). If not, then r k < r ( x ) / (4 A ). Suppose that x ∈ B ( x k , ǫ r k ) ∩ B ( x , ǫ r ( x )). Then, for any y ∈ B ( x k , AA r k ), we have d ( y , x ) ≤ A [ d ( y , x ) + d ( x , x )] ≤ A [ d ( y , x k ) + d ( x k , x )] + A d ( x , x ) ≤ AA r k + A ǫ r ( x ) ≤ AA r ( x ) + AA r ( x ) = AA r ( x )and hence B ( x k , AA r k ) ⊂ B ( x , AA r ( x )) ⊂ Ω , which contradicts to (iv). This proves the claim.Further, by the fact that r ( x ) ≤ A r k , we have d ( x , x k ) ≤ A [ d ( x , x ) + d ( x , x k )] < A ǫ r ( x ) + A ǫ r k ≤ A ǫ r k = r k , that is, x ∈ B ( x k , r k ). This finishes the proof of (ii).Now we prove (v). Fix k ∈ I . Suppose that B ( x j , Ar j ) ∩ B ( x k , Ar k ) , ∅ . We claim that r j ≤ A r k . If not, then r j > A r k . Choose y ∈ B ( x j , Ar j ) ∩ B ( x k , Ar k ). For any y ∈ B ( x k , AA r k ),we have d ( y , x j ) ≤ A [ d ( y , y ) + d ( y , x j )] ≤ A [ d ( y , x k ) + d ( x k , y )] + A d ( y , x j ) ≤ AA r k + AA r k + AA r j ≤ AA r j , ardy S paces on S paces of H omogenous T ype y ∈ B ( x j , AA r j ). Therefore, B ( x k , AA r k ) ⊂ B ( x j , AA r j ) ⊂ Ω ,which contradicts to (iv), Thus, we have r j ≤ A r k . By symmetry, we also have r k ≤ A r j . Let J : = { j ∈ I : B ( x j , Ar j ) ∩ B ( x k , Ar k ) , ∅} . Then, for any j ∈ J , d ( x j , x k ) < AA ( r j + r k ) ≤ AA r k , which further implies that B (cid:16) x j , (5 A ) − r j (cid:17) ⊂ B (cid:16) x k , A h d ( x j , x k ) + (5 A ) − r j i(cid:17) ⊂ B ( x k , AA r k ) . Then, from the fact d ( x j , x k ) . min { r j , r k } and (1.1), we deduce that µ (cid:16) B (cid:16) x j , (5 A ) − r j (cid:17)(cid:17) ∼ µ ( B ( x j , r j )) ∼ µ ( B ( x k , r k )) ∼ µ ( B ( x k , AA r k ))with the equivalent positive constants depending on A . Thus, we obtain (v) by (i).Finally we prove (vi). Since Ω is bounded, it follows that there exist x ∈ X and R ∈ (0 , ∞ ) suchthat Ω ⊂ B ( x , R ). If (vi) fails, then there exists σ ∈ (0 , ∞ ) such that K : = { k ∈ I : r k > σ R } isinfinite. Then, for any k ∈ K , µ ( B ( x k , r k / (5 A ))) ∼ µ ( B ( x k , ǫ R )) & µ ( B ( x , R )) & µ ( Ω ) > . By this and (i), we have µ ( Ω ) ≥ P k ∈K µ ( B ( x k , r k / (5 A ))) = ∞ . That is a contradiction. This proves(vi) and hence finishes the proof of Proposition 4.4. (cid:3) Proposition 4.5.
Let Ω ⊂ X be an open set and µ ( Ω ) < ∞ . Suppose that sequences { x k } k ∈ I and { r k } k ∈ I are as in Proposition 4.4 with A : = A . Then there exist non-negative functions { φ k } k ∈ I such that (i) for any k ∈ I, ≤ φ k ≤ and supp φ k ⊂ B ( x k , A r k ) ; (ii) P k ∈ I φ k = χ Ω ; (iii) for any k ∈ I, φ k ≥ L − in B ( x k , r k ) , where L is as in Proposition 4.4; (iv) there exists a positive constant C such that, for any k ∈ I, k φ k k G ( x k , r k ,η,η ) ≤ CV r k ( x k ) .Proof. By [1, Corollary 4.2], for any k ∈ I , we find a function ψ k such that χ B ( x k , r k ) ≤ ψ k ≤ χ B ( x k , A r k ) and k ψ k k ˙ C η ( X ) . r − η k . Here and hereafter, for any s ∈ (0 , η ] and a measurable function f , define k f k ˙ C s ( X ) : = sup x , y | f ( x ) − f ( y ) | [ d ( x , y )] β . Since A ≥ A , from (ii) and (v) of Proposition 4.4, it follows that, for any x ∈ Ω , 1 ≤ P k ∈ I ψ k ( x ) ≤ L . For any k ∈ I and x ∈ X , let φ k ( x ) : = ψ k ( x ) X j ∈ I ψ j ( x ) − when x ∈ Ω , , when x < Ω . Z iyi H e , Y ongsheng H an , J i L i , L iguang L iu , D achun Y ang and W en Y uan Then, for any k ∈ I , we have 0 ≤ φ k ≤
1, supp φ k ⊂ B ( x k , A r k ) and P k ∈ I φ k ( x ) = x ∈ Ω .Moreover, for any k ∈ I , we have φ k ≥ L − in B ( x k , r k ). Thus, we prove (i), (ii) and (iii).It remains to prove (iv). Fix k ∈ I . For any y ∈ X , we have | φ k ( y ) | ≤ χ B ( x k , A r k ) ( y ) . V r k ( x k ) 1 V r k ( x k ) + V ( x k , y ) " r k r k + d ( x k , y ) η . Now we prove that φ k satisfies the regularity condition. Suppose that d ( y , y ′ ) ≤ (2 A ) − [ r k + d ( x k , y )]. If | φ k ( y ) − φ k ( y ′ ) | ,
0, then d ( x k , y ) < (3 A ) r k . If not, then d ( x k , y ) ≥ (3 A ) r k , so that φ k ( y ) = d ( y ′ , x k ) ≥ A − d ( x k , y ) − d ( y , y ′ ) ≥ (2 A ) − d ( x k , y ) − (2 A ) − r k > A r k and hence φ k ( y ′ ) =
0, which contradicts to | φ k ( y ) − φ k ( y ′ ) | ,
0. Notice that ψ k ( y ′ ) | ψ j ( y ) − ψ j ( y ′ ) | , y ′ ∈ B ( x k , A r k ) and also y or y ′ belongs to B ( x j , A r j ), which further implies that B ( x k , Ar k ) ∩ B ( x j , Ar j ) , ∅ . Then, by the proof of Proposition 4.4(v), the number of j satisfying ψ k ( y ′ ) | ψ j ( y ) − ψ j ( y ′ ) | , L and r j ∼ r k . Therefore, | φ k ( y ) − φ k ( y ′ ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ψ k ( y ) P j ∈ I ψ j ( y ) − ψ k ( y ′ ) P j ∈ I ψ j ( y ′ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ | ψ k ( y ) − ψ k ( y ′ ) | P j ∈ I ψ j ( y ) + ψ k ( y ′ ) P j ∈ I | ψ j ( y ) − ψ j ( y ′ ) | [ P j ∈ I ψ j ( y )][ P j ∈ I ψ j ( y ′ )] . " d ( y , y ′ ) r k η + X { j ∈ I : B ( x k , Ar k ) ∩ B ( x j , Ar j ) , ∅} " d ( y , y ′ ) r j η . " d ( y , y ′ ) r k η ∼ V r k ( x k ) " d ( y , y ′ ) r k + d ( x k , y ) η V r k ( x k ) + V ( x k , y ) " r k r k + d ( x k , y ) η . Then we obtain the desired regularity condition of φ k . This finishes the proof of (iv) and hence ofProposition 4.5. (cid:3) Assume that f ∈ ( G η ( β, γ )) ′ belongs to f ∈ H ∗ , p ( X ), where p ∈ ( ω/ ( ω + η ) ,
1] and β, γ ∈ ( ω (1 / p − , η ). To obtain the Calder´on-Zygmund decomposition of f , we apply Propositions 4.4and 4.5 to the level set { x ∈ X : f ∗ ( x ) > λ } with λ ∈ (0 , ∞ ). The encountering problem is that sucha level set may not be open even in the case that d is a metric. To solve this problem in the casethat d is a metric, a variant of the notion of the space of test functions is adopted in [20, Definition2.5] so that to ensure that the level set is open (see [20, Remark 2.9]). Here, we borrow some ideafrom [20].By the proof of [42, Theorem 2], we know that there exist θ ∈ (0 ,
1) and a metric d ′ such that d ′ ∼ d θ . For any x ∈ X and r ∈ (0 , ∞ ), define the d ′ -ball B ′ ( x , r ) : = { y ∈ X : d ′ ( x , y ) < r } . Then( X , d ′ , µ ) is a doubling metric measure space. Moreover, for any x , y ∈ X and r ∈ (0 , ∞ ), we have µ ( B ( y , r + d ( x , y ))) ∼ µ (cid:16) B ′ (cid:16) y , (cid:2) r + d ( x , y ) (cid:3) θ (cid:17)(cid:17) ∼ µ (cid:16) B ′ (cid:16) y , r θ + d ′ ( x , y ) (cid:17)(cid:17) , where the equivalent positive constants are independent of x and r . Using the metric d ′ , weintroduce a variant of the space of test functions as follows. ardy S paces on S paces of H omogenous T ype Definition 4.6.
For any x ∈ X , ρ ∈ (0 , ∞ ) and β ′ , γ ′ ∈ (0 , ∞ ), define G ( x , ρ, β ′ , γ ′ ) to be the set ofall functions f satisfying that there exists a positive constant C such that(i) (the size condition ) for any y ∈ X , | f ( y ) | ≤ C µ ( B ′ ( y , ρ + d ′ ( x , y ))) " ρρ + d ′ ( x , y ) γ ′ ;(ii) (the regularity condition ) for any y , y ′ ∈ X satisfying d ( y , y ′ ) ≤ [ ρ + d ′ ( x , y )] /
2, then | f ( y ) − f ( y ′ ) | ≤ C " d ′ ( y , y ′ ) ρ + d ′ ( y , y ′ ) β ′ µ ( B ′ ( y , ρ + d ′ ( x , y ))) " ρρ + d ′ ( x , y ) γ ′ . Also, define k f k G ( x ,ρ,β ′ ,γ ′ ) : = inf { C ∈ (0 , ∞ ) : (i) and (ii) hold true } . By the previous argument, we find that G ( x , r , β, γ ) = G ( x , r θ , β/θ, γ/θ ) with equivalent norms,where the equivalent positive constants are independent of x and r . For any β, γ ∈ (0 , η ) and f ∈ ( G η ( β, γ )) ′ , define the modified grand maximal function of f by setting, for any x ∈ X , f ⋆ ( x ) : = sup n h f , ϕ i : ϕ ∈ G η ( β, γ ) with k ϕ k G ( x , r θ ,β/θ,γ/θ ) ≤ r ∈ (0 , ∞ ) o . Then f ⋆ ∼ f ∗ pointwisely on X . For any λ ∈ (0 , ∞ ) and j ∈ Z , define Ω λ : = { x ∈ X : f ⋆ ( x ) > λ } and Ω j : = Ω j . By the argument used in [20, Remark 2.9(ii)], we find that Ω λ is open under the topology inducedby d ′ , so is it under the topology induced by d .Now suppose that p ∈ ( ω/ ( ω + η ) , β, γ ∈ ( ω (1 / p − , η ) and f ∈ H ∗ , p ( X ). Then f ⋆ ∈ L p ( X )and every Ω j with j ∈ Z has finite measure. Consequently, there exist { x jk } k ∈ I j ⊂ X with I j beinga countable index set, { r jk } k ∈ I j ⊂ (0 , ∞ ), L ∈ N and a sequence { φ jk } k ∈ I j of non-negative functionssatisfying all the conclusions of Propositions 4.4 and 4.5. For any j ∈ Z and k ∈ I j , define Φ jk bysetting, for any ϕ ∈ G η ( β, γ ) and x ∈ X , Φ jk ( ϕ )( x ) : = φ jk ( x ) "Z X φ jk ( z ) d µ ( z ) − Z X [ ϕ ( x ) − ϕ ( z )] φ jk ( z ) d µ ( z ) . It can be seen that Φ jk is bounded on G η ( β, γ ) with operator norm depending on j and k ; see[20, Lemma 4.9]. Thus, it makes sense to define a distribution b jk on G η ( β, γ ) by setting, for any ϕ ∈ G η ( β, γ ),(4.3) D b jk , ϕ E : = D f , Φ jk ( ϕ ) E . To estimate ( b jk ) ∗ , we have the following result. For its proof, see, for example, [37, Lemma 3.7].6 Z iyi H e , Y ongsheng H an , J i L i , L iguang L iu , D achun Y ang and W en Y uan Proposition 4.7.
For any j ∈ Z and k ∈ I j , b jk is defined as in (4.3) . Then there exists a positiveconstant C such that, for any j ∈ Z , k ∈ I j and x ∈ X, (cid:16) b jk (cid:17) ∗ ( x ) ≤ C j µ ( B ( x jk , r jk )) µ ( B ( x jk , r jk )) + V ( x jk , x ) r jk r jk + d ( x jk , x ) β χ [ B ( x jk , A r jk )] ∁ ( x ) + C f ∗ ( x ) χ B ( x jk , A r jk ) ( x ) . The next lemma is exactly [20, Lemma 4.10]. The proof remains true if d is a quasi-metric and µ does not satisfy the reverse doubling condition. Lemma 4.8.
Let β ∈ (0 , ∞ ) , p ∈ ( ω/ ( ω + β ) , ∞ ) , L ∈ N and I be a countable index set. Thenthere exists a positive constant C such that, for any sequences { x k } k ∈ I ⊂ X and { r k } k ∈ I ⊂ (0 , ∞ ) satisfying P k ∈ I χ B ( x k , r k ) ≤ L , Z X X k ∈ I V r k ( x k ) V r k ( x k ) + V ( x k , x ) " r k r k + d ( x k , x ) β p d µ ( x ) ≤ C µ [ k ∈ I B ( x k , r k ) . Then, by Proposition 4.7 and Lemma 4.8, we have the following result.
Proposition 4.9.
Let p ∈ ( ω/ ( ω + η ) , . For any j ∈ Z and k ∈ I j , let b jk be as in (4.3) . Then thereexists a positive constant C such that, for any j ∈ Z , (4.4) Z X X k ∈ I j h(cid:16) b jk (cid:17) ∗ ( x ) i p d µ ( x ) ≤ C (cid:13)(cid:13)(cid:13) f ∗ χ Ω j (cid:13)(cid:13)(cid:13) pL p ( X ) ; moreover, there exists b j ∈ H ∗ , p ( X ) such that b j = P k ∈ I j b jk in H ∗ , p ( X ) and, for any x ∈ X, (4.5) ( b j ) ∗ ( x ) ≤ C j X k ∈ I j µ ( B ( x jk , r jk )) µ ( B ( x jk , r jk )) + V ( x jk , x ) r jk r jk + d ( x jk , x ) β + C f ∗ ( x ) χ Ω j ( x ); if g j : = f − b j for any j ∈ Z , then, for any x ∈ X, (4.6) ( g j ) ∗ ( x ) ≤ C j X k ∈ I j µ ( B ( x jk , r jk )) µ ( B ( x jk , r jk )) + V ( x jk , x ) r jk r jk + d ( x jk , x ) β + C f ∗ ( x ) χ ( Ω j ) ∁ ( x ) . Proof.
Fix j ∈ Z . We first prove (4.4). Indeed, by Proposition 4.7, we find that Z X X k ∈ I j h(cid:16) b jk (cid:17) ∗ ( x ) i p d µ ( x ) . jp Z X X k ∈ I j µ ( B ( x jk , r jk )) µ ( B ( x jk , r jk )) + V ( x jk , x ) r jk r jk + d ( x jk , x ) β p d µ ( x ) + Z S k ∈ Ij B ( x jk , A r jk ) [ f ∗ ( x )] p d µ ( x ) . By Proposition 4.4(ii), we have Ω j = S k ∈ I j B ( x jk , A r jk ) . Applying this and Lemma 4.8, the firstterm in the right-hand side of the above formula is bounded by a harmlessly positive constantmultiple of 2 jp µ ( Ω j ). Combining this with f ∗ ∼ f ⋆ implies that Z X X k ∈ I j h(cid:16) b jk (cid:17) ∗ ( x ) i p d µ ( x ) . jp µ (cid:16) Ω j (cid:17) + Z Ω j [ f ∗ ( x )] p d µ ( x ) . (cid:13)(cid:13)(cid:13) f ∗ χ Ω j (cid:13)(cid:13)(cid:13) pL p ( X ) , ardy S paces on S paces of H omogenous T ype H ∗ , p ( X ) (see Proposition 3.1), we know that there exists b j ∈ H ∗ , p ( X ) such that b j = P k ∈ I j b jk in H ∗ , p ( X ). Moreover, from Proposition 4.7 and Ω j = S k ∈ I j B ( x jk , A r jk ), we deduce that, for any x ∈ X ,( b j ) ∗ ( x ) ≤ X k ∈ I j (cid:16) b jk (cid:17) ∗ ( x ) . j X k ∈ I j µ ( B ( x jk , r jk )) µ ( B ( x jk , r jk )) + V ( x jk , x ) r jk r jk + d ( x jk , x ) β + f ∗ ( x ) χ Ω j ( x ) . This finishes the proof of (4.5).It remains to prove (4.6). If x ∈ ( Ω j ) ∁ , then, by (4.5), we conclude that( g j ) ∗ ( x ) ≤ f ∗ ( x ) + ( b j ) ∗ ( x ) . j X k ∈ I j µ ( B ( x jk , r jk )) µ ( B ( x jk , r jk )) + V ( x jk , x ) r jk r jk + d ( x jk , x ) β + f ∗ ( x ) , as desired.Now we consider the case x ∈ Ω j . According to Proposition 4.4(v), for any n ∈ I j , we choosea point y jn < Ω j satisfying 32 A r jn ≤ d ( x jn , y jn ) < A r jn . Since x ∈ Ω j , it follows that thereexists k ∈ I j such that x ∈ B ( x jk , r jk ). Let J be the set of all n ∈ I j such that B ( x jn , A r jn ) ∩ B ( x jk , A r jk ) , ∅ . Then, by the proof of Proposition 4.4(v), J ≤ L and r jn ∼ r jk whenever n ∈ J .Suppose that ϕ ∈ G η ( β, γ ) with k ϕ k G ( x , r ,β,γ ) ≤ r ∈ (0 , ∞ ). We then estimate h g j , ϕ i by considering the cases r ≤ r jk and r > r jk , respectively. Case 1) r ≤ r jk . In this case, we write h g j , ϕ i = h f , ϕ i − X n ∈ I j h b jn , ϕ i = h f , ϕ i − X n ∈J h b jn , ϕ i − X n < J h b jn , ϕ i = h f , e ϕ i − X n ∈J h f , e ϕ n i − X n < J h b jn , ϕ i , where e ϕ : = (1 − P n ∈J φ jn ) ϕ and, for any n ∈ J , e ϕ n : = φ jn "Z X φ jn ( z ) d µ ( z ) − Z X ϕ ( z ) φ jn ( z ) d µ ( z ) . We first consider the term P n < J h b jn , ϕ i . Indeed, from x ∈ B ( x jk , r jk ), it follows that x < B ( x jn , A x jn ) when n < J . Applying Proposition 4.7 implies that (cid:12)(cid:12)(cid:12)(cid:12)D b jn , ϕ E(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:16) b jn (cid:17) ∗ ( x ) (cid:12)(cid:12)(cid:12)(cid:12) . j µ ( B ( x jn , r jn )) µ ( B ( x jn , r jn )) + V ( x jn , x ) r jn r jn + d ( x jn , x ) β , and hence X n < J (cid:12)(cid:12)(cid:12)(cid:12)D b jn , ϕ E(cid:12)(cid:12)(cid:12)(cid:12) . j X n < J µ ( B ( x jn , r jn )) µ ( B ( x jn , r jn )) + V ( x jn , x ) r jn r jn + d ( x jn , x ) β , Z iyi H e , Y ongsheng H an , J i L i , L iguang L iu , D achun Y ang and W en Y uan as desired.Next we consider the term P n ∈J h f , e ϕ n i . Notice that k e ϕ n k G ( x jn , r jn ,β,γ ) .
1. By d ( x jn , y jn ) ∼ r jn , wethen have k e ϕ n k G ( y jn , r jn ,β,γ ) .
1. Therefore, |h f , e ϕ n i| . f ∗ (cid:16) y jn (cid:17) ∼ f ⋆ (cid:16) y jn (cid:17) . j ∼ j µ ( B ( x jn , r jn )) µ ( B ( x jn , r jn )) + V ( x jn , x ) r jn r jn + d ( x jn , x ) β , where, in the last step, we used the facts that x ∈ B ( x jk , r jk ) and d ( x jn , x jk ) . r jn + r jk ∼ r jn whenever n ∈ J . Then, summing all n ∈ J , we obtain the desired estimate.Finally, we consider the term h f , e ϕ i . Since ϕ ∈ G η ( β, γ ), it is easy to see that e ϕ ∈ G η ( β, γ ). Oncewe have proved that(4.7) k e ϕ k G ( y jk , r jk ,β,γ ) . , then |h f , e ϕ i| . f ∗ (cid:16) y jk (cid:17) ∼ f ⋆ (cid:16) y jk (cid:17) . j ∼ j µ ( B ( x jk , r jk )) µ ( B ( x jk , r jk )) + V ( x jk , x ) r jk r jk + d ( x jk , x ) β , as desired.To prove (4.7), we first consider the size condition. For any z ∈ B ( x jk , A r jk ), by Proposition4.5, we have P n ∈J φ jn ( z ) = P n ∈ I j φ jn ( z ) = e ϕ ( z ) =
0. When d ( z , x jk ) ≥ A r jk , by thefact d ( x jk , z ) ≥ A d ( x , x jk ), we have r jk + d (cid:16) z , y jk (cid:17) ≤ r jk + A h d (cid:16) z , x jk (cid:17) + d (cid:16) x jk , y jk (cid:17)i ≤ (2 A ) h r jk + d (cid:16) z , x jk (cid:17)i (4.8) ≤ (2 A ) d (cid:16) z , x jk (cid:17) ≤ (2 A ) d ( x , z ) ≤ (2 A ) [ r + d ( x , z )]and hence µ ( B ( y jk , r jk )) + V ( y jk , z ) . V r ( x ) + V ( x , z ), which, together with the size condition of ϕ and the fact that r ≤ r jk , further implies that | e ϕ ( z ) | ≤ | ϕ ( z ) | ≤ V r ( x ) + V ( x , z ) " rr + d ( x , z ) γ . µ ( B ( y jk , r jk )) + V ( y jk , z ) r jk r jk + d ( y jk , z ) γ . This finishes the proof of the size condition.Now we consider the regularity of e ϕ . Suppose that z , z ′ ∈ X with d ( z , z ′ ) ≤ (2 A ) − [ r jk + d ( z , y jk )]. Due to the size condition, we only need to consider the case d ( z , z ′ ) ≤ (2 A ) − [ r jk + d ( z , y jk )]. If e ϕ ( z ) − e ϕ ( z ′ ) ,
0, then either d ( z , x jk ) ≥ A r jk or d ( z ′ , x jk ) ≥ A r jk , which alwaysimplies that d ( z , x jk ) ≥ A r jk .Indeed, if d ( z , x jk ) < A r jk , then d ( z , y jk ) ≤ A [ d ( z , x jk ) + d ( x jk , y jk )] < (2 A ) r jk and hence d ( z , z ′ ) ≤ (2 A ) r jk , which further implies that d ( z ′ , x jk ) ≤ A [ d ( z ′ , z ) + d ( z , x jk )] < A r jk and itis a contraction. ardy S paces on S paces of H omogenous T ype d ( z , x jk ) ≥ A r jk , which, together with an argument as in the estimation of (4.8),implies r jk + d ( z , y jk ) ≤ (2 A ) [ r + d ( z , x )], so that d ( z , z ′ ) ≤ (2 A ) − [ r + d ( z , x )]. By the definitionof e ϕ , we find that (cid:12)(cid:12)(cid:12)e ϕ ( z ) − e ϕ ( z ′ ) (cid:12)(cid:12)(cid:12) ≤ − X n ∈J φ jn ( z ) | ϕ ( z ) − ϕ ( z ′ ) | + | ϕ ( z ′ ) | X n ∈J (cid:12)(cid:12)(cid:12)(cid:12) φ jn ( z ) − φ jn ( z ′ ) (cid:12)(cid:12)(cid:12)(cid:12) . Using the regularity condition of ϕ and the fact d ( z , z ′ ) ≤ (2 A ) − [ r + d ( z , x )], we obtain − X n ∈J φ jn ( z ) | ϕ ( z ) − ϕ ( z ′ ) | . " d ( z , z ′ ) r + d ( z , x ) β V r ( x ) + V ( x , z ) " rr + d ( x , z ) γ . d ( z , z ′ ) r jk + d ( z , y jk ) β µ ( B ( y jk , r jk )) + V ( y jk , z ) r jk r jk + d ( y jk , z ) γ , where, in the last step, we used r jk + d ( z , y jk ) . r + d ( z , x ), r ≤ r jk , x ∈ B ( x jk , r jk ) and d ( y jk , x jk ) ∼ r jk .We now estimate | ϕ ( z ′ ) | P n ∈J | φ jn ( z ) − φ jn ( z ′ ) | . If ϕ ( z ′ ) | φ jn ( z ) − φ jn ( z ′ ) | ,
0, then z ′ < B ( x jk , A r jk )and z or z ′ belongs to B ( x jn , A r jn ). When n ∈ J , we have r jn ∼ r jk ∼ r jk + d ( y jk , z ). Also, r jk + d ( z , y jk ) . r + d ( z , x ) ∼ r + d ( z ′ , x ). By these, J ≤ L and r ≤ r jk , we conclude that | ϕ ( z ′ ) | X n ∈J (cid:12)(cid:12)(cid:12)(cid:12) φ jn ( z ) − φ jn ( z ′ ) (cid:12)(cid:12)(cid:12)(cid:12) . V r ( x ) + V ( x , z ′ ) " rr + d ( z , x ) γ X n ∈J d ( z , z ′ ) r jn η . d ( z , z ′ ) r jk + d ( y jk , z ) β µ ( B ( y jk , r jk )) + V ( y jk , z ) r jk r jk + d ( y jk , z ) γ . This finishes the proof of the regularity condition and hence of (4.7). Thus, we complete the proofof Case 1).
Case 2) r > r jk . In this case, we write (cid:12)(cid:12)(cid:12)(cid:12)D g j , ϕ E(cid:12)(cid:12)(cid:12)(cid:12) ≤ |h f , ϕ i| + X n ∈J (cid:12)(cid:12)(cid:12)(cid:12)D b jn , ϕ E(cid:12)(cid:12)(cid:12)(cid:12) + X n < J (cid:12)(cid:12)(cid:12)(cid:12)D b jn , ϕ E(cid:12)(cid:12)(cid:12)(cid:12) . The estimation of P n < J |h b jn , ϕ i| has already been given in Case 1).From x ∈ B ( x jk , r jk ) and d ( y jk , x jk ) ∼ r jk . r , it follows that k ϕ k G ( y jk , r ,β,γ ) . |h f , ϕ i| . f ∗ (cid:16) y jk (cid:17) . j ∼ j µ ( B ( x jk , r jk )) µ ( B ( x jk , r jk )) + V ( x jk , x ) r jk r jk + d ( x jk , x ) β . If n ∈ J , then r jn ∼ r jk and hence d ( y jn , x jk ) . r jk . This, together with the fact r jk < r and x ∈ B ( x jk , r jk ), implies that k ϕ k G ( y jn , r ,β,γ ) .
1. Thus, by Proposition 4.7, we have X n ∈J (cid:12)(cid:12)(cid:12)(cid:12)D b jn , ϕ E(cid:12)(cid:12)(cid:12)(cid:12) . X n ∈J (cid:16) b jn (cid:17) ∗ (cid:16) y jn (cid:17) . j X n ∈J µ ( B ( x jn , r jn )) µ ( B ( x jn , r jn )) + V ( x jn , x ) r jn r jn + d ( x jn , x ) β . Z iyi H e , Y ongsheng H an , J i L i , L iguang L iu , D achun Y ang and W en Y uan Then we obtain the desired estimate for h g j , ϕ i in the case r > r jk .Combining the two cases above, we find that, for any x ∈ Ω j ,( g j ) ∗ ( x ) . j X k ∈ I j µ ( B ( x jk , r jk )) µ ( B ( x jk , r jk )) + V ( x jk , x ) r jk r jk + d ( x jk , x ) β . Thus, (4.6) holds true. This finishes the proof of Proposition 4.9. (cid:3) H ∗ , p ( X ) In this section, we prove H ∗ , p ( X ) ⊂ H p , q at ( X ) and complete the proof of Theorem 4.2. First, weobtain dense subspaces of H ∗ , p ( X ). Lemma 4.10 ([20, Proposition 4.12]) . Let p ∈ ( ω/ ( ω + η ) , , β, γ ∈ ( ω (1 / p − , η ) and q ∈ [1 , ∞ ) .If regard H ∗ , p ( X ) as a subspace of ( G η ( β, γ )) ′ , then L q ( X ) ∩ H ∗ , p ( X ) is dense in H ∗ , p ( X ) . In the next two lemmas, we suppose that f ∈ L ( X ) ∩ H ∗ , p ( X ). Based on Proposition 3.3 and(3.1), we may follow [20, Remark 4.14] and assume that there exists a positive constant C suchthat, for any x ∈ X , | f ( x ) | ≤ C f ∗ ( x ). With all the notation as in the previous section, for any j ∈ Z and k ∈ I j , define(4.9) m jk : = k φ jk k L ( X ) Z X f ( ξ ) φ jk ( ξ ) d µ ( ξ ) and b jk : = (cid:16) f − m jk (cid:17) φ jk . Then we have the following technical lemma.
Lemma 4.11 ([20, Proposition 4.13]) . For any j ∈ Z and k ∈ I j , let m jk and b jk be as in (4.9) . Then (i) there exists a positive constant C, independent of j and k ∈ I j , such that | m jk | ≤ C j ; (ii) b jk induces the same distribution as defined in (4.3) ; (iii) P k ∈ I j b jk converges to some function b j in L ( X ) , which induces a distribution that coincideswith b j as in Proposition 4.9; (iv) let g j : = f − b j . Then g j = f χ ( Ω j ) ∁ + P k ∈ I j m jk φ jk . Moreover, there exists a positive constantC, independent of j, such that, for any x ∈ X, | g j ( x ) | ≤ C j . For any j ∈ Z , k ∈ I j and l ∈ I j + , define(4.10) L j + k , l : = k φ j + l k L ( X ) Z X h f ( ξ ) − m j + l i φ jk ( ξ ) φ j + l ( ξ ) d µ ( ξ )Then L j + k , l has the following properties. Lemma 4.12.
For any j ∈ Z , k ∈ I j and l ∈ I j + , let L j + k , l be as in (4.10) . Then ardy S paces on S paces of H omogenous T ype there exists a positive constant C, independent of j, k and l, such that sup x ∈ X (cid:12)(cid:12)(cid:12)(cid:12) L j + k , l φ j + l ( x ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C j ;(ii) P k ∈ I j P l ∈ I j + L j + k , l φ j + l = both in ( G η ( β, γ )) ′ and everywhere.Proof. We first show (i). Indeed, for any j ∈ Z , k ∈ I j , l ∈ I j + and x ∈ X , (cid:12)(cid:12)(cid:12)(cid:12) L j + k , l φ j + l ( x ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12) m j + l (cid:12)(cid:12)(cid:12)(cid:12) φ j + l ( x ) + φ j + l ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z X f ( ξ ) φ jk ( ξ ) φ j + l ( ξ ) k φ j + l k L ( X ) d µ ( ξ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = : Y + Y . By Lemma 4.11(i) and the definition of φ j + l , it is easy to obtain Y . j .Now we consider Y . If φ jk φ j + l is a non-zero function, then B ( x jk , A r jk ) ∩ B ( x j + l , A r j + l ) , ∅ , which further implies that r j + l ≤ A r jk . Otherwise, if r j + l > A r jk , then, for any y ∈ B ( x jk , A r jk ), d (cid:16) y , x j + l (cid:17) ≤ A h d (cid:16) y , x jk (cid:17) + d (cid:16) x jk , x j + l (cid:17)i < A r jk + A (cid:16) A r jk + A r j + l (cid:17) < A r j + l + A r j + l + A r j + l < A r j + l , which implies that B ( x jk , A r jk ) ⊂ B ( x j + l , A r j + l ) ⊂ Ω j + ⊂ Ω j and hence contradicts toProposition 4.4(v).Define ϕ : = φ jk φ j + l / k φ j + l k L ( X ) . According to Proposition 4.4(iv) with A : = A , we canchoose y j + l ∈ ( Ω j + ) ∁ such that d ( y j + l , x j + l ) ≤ A r j + l . We now show ϕ ∈ G ( y j + l , r j + l , η, η )and k ϕ k G ( y j + l , r j + l ,η,η ) .
1. Notice that supp ϕ ⊂ B ( x j + l , A r j + l ). Moreover, by this and the choiceof y j + l , we conclude that, for any x ∈ B ( x j + l , A r j + l ), | ϕ ( x ) | . | φ j + l ( x ) | . µ ( B ( x j + l , r j + l )) + V ( x j + l , x ) r j + l r j + l + d ( x j + l , x ) η ∼ µ ( B ( y j + l , r j + l )) + V ( y j + l , x ) r j + l r j + l + d ( y j + l , x ) η . This shows the size condition of ϕ .To consider the regularity condition of ϕ , we suppose that x , x ′ ∈ X satisfying d ( x , x ′ ) ≤ (2 A ) − [ r j + l + d ( y j + l , x )]. Due to the size condition, we may assume d ( x , x ′ ) ≤ (2 A ) − [ r j + l + d ( y j + l , x )]. We claim that ϕ ( x ) − ϕ ( x ′ ) , d ( x , x j + l ) ≤ A r j + l .Indeed, if d ( x , x j + l ) > A r j + l , then ϕ ( x ) =
0. By d ( x j + l , y j + l ) ≤ A r j + l , we find that d ( x , y j + l ) > A r j + l and hence d ( x , x ′ ) ≤ (2 A ) − d ( x , y j + l ) ≤ (2 A ) − d ( x , x j + l ). Consequently, d ( x ′ , x j + l ) ≥ A − d ( x , x j + l ) − d ( x , x ′ ) > A r j + l and ϕ ( x ′ ) =
0. This contradicts to ϕ ( x ) − ϕ ( x ′ ) , r j + l ≤ A r jk and d ( y j + l , x j + j ) ∼ r j + l , we know that | ϕ ( x ) − ϕ ( x ′ ) | . µ ( B ( x j + l , r j + l )) (cid:20) φ jk ( x ) (cid:12)(cid:12)(cid:12)(cid:12) φ j + l ( x ) − φ j + l ( x ′ ) (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) φ jk ( x ) − φ jk ( x ′ ) (cid:12)(cid:12)(cid:12)(cid:12) φ j + l ( x ′ ) (cid:21) Z iyi H e , Y ongsheng H an , J i L i , L iguang L iu , D achun Y ang and W en Y uan . µ ( B ( x j + l , r j + l )) d ( x , x ′ ) r j + l η + d ( x , x ′ ) r jk η ∼ d ( x , x ′ ) r j + l + d ( y j + l , x ) η µ ( B ( y j + l , r j + l )) + V ( y j + l , x ) r j + l r j + l + d ( y j + l , x ) η . Thus, we obtain ϕ ∈ G ( y j + l , r j + l , η, η ) and k ϕ k G ( y j + l , r j + l ,η,η ) .
1, which further implies that k ϕ k G ( y j + l , r j + l ,β,γ ) . = |h f , ϕ i| . f ∗ (cid:16) y j + l (cid:17) . j . This finishes the proof of (i).Next we prove (ii). If L j + k , l ,
0, then the proof in (i) implies B ( x jk , A r jk ) ∩ B ( x j + l , A r j + l ) , ∅ and r j + l ≤ A r jk . Further, for any y ∈ B ( x j + l , A r j + l ), we have d (cid:16) y , x jk (cid:17) ≤ A h d (cid:16) y , x j + l (cid:17) + d (cid:16) x jk , x j + l (cid:17)i < A r j + l + A (cid:16) A r jk + A r j + l (cid:17) < A r jk + A r jk + A r jk ≤ A r jk < A r jk , which implies that B ( x j + l , A r j + l ) ⊂ B ( x jk , A r jk ) ⊂ Ω j by Proposition 4.4(v). Thus, for any k ∈ I j and x ∈ X , we find that(4.11) X l ∈ I j + (cid:12)(cid:12)(cid:12)(cid:12) L j + k , l φ j + l (cid:12)(cid:12)(cid:12)(cid:12) . j χ B ( x jk , A r jk ) ( x )and hence X k ∈ I j X l ∈ I j + (cid:12)(cid:12)(cid:12)(cid:12) L j + k , l φ j + l ( x ) (cid:12)(cid:12)(cid:12)(cid:12) . j X k ∈ I j χ B ( x jk , A r jk ) ( x ) . j χ Ω j ( x ) . Consequently, X k ∈ I j X l ∈ I j + L j + k , l φ j + l = X l ∈ I j + X k ∈ I j L j + k , l φ j + l = X l ∈ I j + φ j + l k φ j + l k L ( X ) Z X h f ( ξ ) − m j + l i φ j + l ( ξ ) X k ∈ I j φ jk ( ξ ) d µ ( ξ ) = X l ∈ I j + φ j + l k φ j + l k L ( X ) Z X h f ( ξ ) − m j + l i φ j + l ( ξ ) d µ ( ξ ) = X l ∈ I j + φ j + l k φ j + l k L ( X ) Z X b j + l ( ξ ) d µ ( ξ ) = . By the fact that P k ∈ I j P l ∈ I j + R X | L j + k , l φ j + l ( ξ ) | d µ ( ξ ) . j µ ( Ω j ) < ∞ and the dominated convergencetheorem, we find that P k ∈ I j P l ∈ I j + L j + k , l φ j + l = L ( X ) and hence in ( G η ( β, γ )) ′ . This finishesthe proof of Lemma 4.12. (cid:3) ardy S paces on S paces of H omogenous T ype Proof of H ∗ , p ( X ) ⊂ H p , q at ( X ) . By Lemma 4.10, we first suppose f ∈ L ( X ) ∩ H ∗ , p ( X ). We may alsoassume | f ( x ) | . f ∗ ( x ) for any x ∈ X . We use the same notation as in Lemmas 4.11 and 4.12.For any j ∈ N , let h j : = g j + − g j = b j − b j + . Then f − P mj = − m h j = b m + − g m . For any m ∈ Z , by Lemma 4.11, we conclude that k g − m k L ∞ ( X ) . − m . Moreover, by (4.5), we find that k ( b m + ) ∗ k L p ( X ) . k f ∗ χ ( Ω m + ) ∁ k L p ( X ) → m → ∞ . Thus, f = P ∞ j = −∞ h j in ( G η ( β, γ )) ′ . Besides, bythe definition of b mk , we know that supp b m + ⊂ Ω m + , which then implies that P ∞ j = −∞ h j convergesalmost everywhere. Notice that, by Lemma 4.12(ii), for any j ∈ Z , we have h j = b j − b j + = X k ∈ I j b jk − X l ∈ I j + b j + l + X k ∈ I j X l ∈ I j + L j + k , l φ j + l (4.12) = X k ∈ I j b jk − X l ∈ I j + (cid:16) b j + l φ jk − L j + k , l φ j + l (cid:17) = : X k ∈ I j h jk , which converges in ( G η ( β, γ )) ′ and almost everywhere. Moreover, for any j ∈ Z and k ∈ N , h jk = b jk − X l ∈ I j + (cid:16) b j + l φ jk − L j + k , l φ j + l (cid:17) = (cid:16) f − m jk (cid:17) φ jk − X l ∈ I j + h(cid:16) f − m j + l (cid:17) φ jk − L j + k , l i φ j + l = f φ jk χ ( Ω j + ) ∁ − m jk φ jk + φ jk X l ∈ I j + m j + l φ j + l + X l ∈ I j + L j + k , l φ j + l . The fourth term is supported on B jk : = B ( x jk , A r jk ), which is deduced from (4.11). Thus,supp h jk ⊂ B jk . Moreover, by Lemmas 4.11(i) and 4.12(i), we conclude that there exists a posi-tive constant C , independent of j and k , such that k h jk k L ∞ ( X ) ≤ C j . Now, let(4.13) λ jk : = C j h µ (cid:16) B jk (cid:17)i p and a jk : = (cid:16) λ jk (cid:17) − h jk . Then a jk is a ( p , ∞ )-atom supported on B jk and f = P ∞ j = −∞ P k ∈ I j λ jk a jk in ( G η ( β, γ )) ′ . Moreover, wehave ∞ X j = −∞ X k ∈ I j (cid:12)(cid:12)(cid:12)(cid:12) λ jk (cid:12)(cid:12)(cid:12)(cid:12) p . ∞ X j = −∞ − jp X k ∈ I j µ (cid:16) B jk (cid:17) . ∞ X j = −∞ − jp µ (cid:16) Ω j (cid:17) ∼ (cid:13)(cid:13)(cid:13) f ⋆ (cid:13)(cid:13)(cid:13) pL p ( X ) ∼ (cid:13)(cid:13)(cid:13) f ∗ (cid:13)(cid:13)(cid:13) pL p ( X ) , which further implies that k f k H p , ∞ at ( X ) . k f k H ∗ , p ( X ) .When f ∈ H ∗ , p ( X ), using Lemma 4.10 and a standard density argument and following the proofin [43, pp. 301–302], we obtain the atomic decomposition of f , the details being omitted. Thisfinishes the proof of H ∗ , p ( X ) ⊂ H p , q at ( X ) and hence of Theorem 4.2. (cid:3) Remark 4.13.
By the argument used in the proof of H ∗ , p ( X ) ⊂ H p , q at ( X ), we find that, if f ∈ L q ( X ) ∩ H ∗ , p ( X ) with q ∈ [1 , ∞ ], then f = P ∞ j = P k ∈ I j h jk in ( G η ( β, γ )) ′ and almost everywhere,where, for any j ∈ Z and k ∈ I j , h jk is as in (4.12).4 Z iyi H e , Y ongsheng H an , J i L i , L iguang L iu , D achun Y ang and W en Y uan H p , q at ( X ) and H p , q cw ( X ) In this section, we consider the relationship between H p , q at ( X ) and H p , q cw ( X ). To see this, we needthe following two technical lemmas. Lemma 4.14 ([7, p. 592]) . Let p ∈ (0 , , q ∈ ( p , ∞ ] ∩ [1 , ∞ ] and a be a ( p , q ) -atom. Then, forany ϕ ∈ L / p − ( X ) , |h a , ϕ i| ≤ k ϕ k L / p − ( X ) . Lemma 4.15.
Let β ∈ (0 , η ] and γ ∈ (0 , ∞ ) . If ϕ ∈ G ( β, γ ) , then ϕ ∈ L β/ω ( X ) and there exists apositive constant C, independent of ϕ , such that k ϕ k L β/ω ( X ) ≤ C k ϕ k G ( β,γ ) .Proof. Suppose that k ϕ k G ( β,γ ) ≤
1. If d ( x , y ) ≤ (2 A ) − [1 + d ( x , x )], then, by the regularitycondition of ϕ and (1.1), we have | ϕ ( x ) − ϕ ( y ) | ≤ " d ( x , y )1 + d ( x , x ) β V ( x ) + V ( x , x ) " + d ( x , x ) γ . " µ ( B ( x , d ( x , y ))) µ ( B ( x , + d ( x , x ))) β/ω . [ V ( x , y )] β/ω . If d ( x , y ) > (2 A ) − [1 + d ( x , x )], then, from the size condition of ϕ , we deduce that | ϕ ( x ) − ϕ ( y ) | . ∼ [ µ ( B ( x , β/ω . [ µ ( B ( x , + d ( x , x )))] β/ω ∼ [ µ ( B ( x , + d ( x , x )))] β/ω . [ V ( x , y )] β/ω . Thus, for any x , y ∈ X , we always have | ϕ ( x ) − ϕ ( y ) | . k ϕ k G ( β,γ ) [ V ( x , y )] β/ω . This implies ϕ ∈L β/ω ( X ) and k ϕ k L β/ω ( X ) . k ϕ k G ( β,γ ) , which completes the proof of Lemma 4.15. (cid:3) Now we establish the relationship between two kinds of atomic Hardy spaces.
Theorem 4.16.
Let p ∈ ( ω/ ( ω + η ) , , q ∈ ( p , ∞ ] ∩ [1 , ∞ ] and β, γ ∈ ( ω (1 / p − , η ) . If regardH p , q at ( X ) as a subspace of ( G η ( β, γ )) ′ , then H p , q cw ( X ) = H p , q at ( X ) with equal (quasi-)norms.Proof. We only consider the case p ∈ ( ω/ ( ω + η ) , p = H p , q cw ( X ) ⊂ H p , q at ( X ). By Lemma 4.15, we have G η ( β, γ ) ⊂ G ( ω (1 / p − , γ ) ⊂L / p − ( X ) and hence ( L / p − ( X )) ′ ⊂ ( G η ( β, γ )) ′ . For any f ∈ H p , q cw ( X ), by Definition 1.1, we knowthat there exist ( p , q )-atoms { a j } ∞ j = and { λ j } ∞ j = ⊂ C with P ∞ j = | λ j | p < ∞ such that f = P ∞ j = λ j a j in( L / p − ( X )) ′ and hence in ( G η ( β, γ )) ′ . Let g : = f | G η ( β,γ ) . Then, for any ϕ ∈ G η ( β, γ ) ⊂ L / p − ( X ),we have h g , ϕ i = h f , ϕ i = ∞ X j = λ j h a j , ϕ i . Thus, g = P ∞ j = λ j a j in ( G η ( β, γ )) ′ and k g k H p , q at ( X ) ≤ ( P ∞ j = | λ j | p ) p . If we take the infimum overall the atomic decompositions of f as above, we obtain k g k H p , q at ( X ) ≤ k f k H p , q cw ( X ) . Thus, H p , q cw ( X ) ⊂ H p , q at ( X ). ardy S paces on S paces of H omogenous T ype H p , q cw ( X ) ⊃ H p , q at ( X ), following the proof of [7, p. 593, Theorem B], we conclude thatthe dual space of H p , q at ( X ) is L / p − ( X ) in the following sense: each bounded linear functional on H p , q at ( X ) is a mapping of the form f ∞ X j = λ j Z X a j ( x ) g ( x ) d µ ( x ) , where g ∈ L / p − ( X ) and f has an atomic decomposition(4.14) f = ∞ X j = λ j a j in ( G η ( β, γ )) ′ with ( p , q )-atoms { a j } ∞ j = and { λ j } ∞ j = ⊂ C satisfying P ∞ j = | λ j | p < ∞ . Therefore, it isreasonable to define the pair h f , g i as follows: h f , g i : = ∞ X j = λ j Z X a j ( x ) g ( x ) d µ ( x ) . In this way, we find that (4.14) also converges in ( L / p − ( X )) ′ , and hence f ∈ H p , q cw ( X ) and k f k H p , q cw ( X ) ≤ ( P ∞ j = | λ j | p ) p . Taking the infimum over all the atomic decompositions of f as above,we obtain k f k H p , q cw ( X ) ≤ k f k H p , q at ( X ) . Thus, H p , q at ( X ) ⊂ H p , q cw ( X ), which completes the proof of Theo-rem 4.16. (cid:3) In this section, we consider the Littlewood-Paley function characterizations of Hardy spaces.Di ff erently from Sections 3 and 4, we use ( ˚ G η ( β, γ )) ′ as underlying spaces to introduce Hardyspaces. Let p ∈ ( ω/ ( ω + η ) , β, γ ∈ ( ω (1 / p − , η ), f ∈ ( ˚ G η ( β, γ )) ′ and { Q k } k ∈ Z be an exp-ATI.For any θ ∈ (0 , ∞ ), define the Lusin area function of f , with aperture θ , S θ ( f ) , by setting, for any x ∈ X ,(5.1) S θ ( f )( x ) : = ∞ X k = −∞ Z B ( x ,θδ k ) | Q k f ( y ) | d µ ( y ) V θδ k ( x ) . In particular, when θ =
1, we write S θ simply as S . Define the Hardy space H p ( X ) via the Lusinarea function by setting H p ( X ) : = n f ∈ (cid:16) ˚ G η ( β, γ ) (cid:17) ′ : k f k H p ( X ) : = kS ( f ) k L p ( X ) < ∞ o . In Section 5.1, we show that H p ( X ) is independent of the choices of exp-ATIs. In Section 5.2, weconnect H p ( X ) with H ∗ , p ( X ) by considering the molecular and the atomic characterizations of ele-ments in H p ( X ). Section 5.3 deals with equivalent characterizations of H p ( X ) via the Littlewood-Paley g-function (5.2) g ( f )( x ) : = ∞ X k = −∞ | Q k f ( x ) | Z iyi H e , Y ongsheng H an , J i L i , L iguang L iu , D achun Y ang and W en Y uan and the Littlewood-Paley g ∗ λ -function (5.3) g ∗ λ ( f )( x ) : = ∞ X k = −∞ Z X | Q k f ( y ) | " δ k δ k + d ( x , y ) λ d µ ( y ) V δ k ( x ) + V δ k ( y ) . where f ∈ ( ˚ G η ( β, γ )) ′ with β, γ ∈ ( ω (1 / p − , η ), x ∈ X and λ ∈ (0 , ∞ ). exp -ATIs In this section, we show that H p ( X ) is independent of the choices of exp-ATIs. If E : = { E k } k ∈ Z and Q : = { Q k } k ∈ Z are two exp-ATIs, then we denote by S E and S Q the Lusin area functions via E and Q , respectively. Theorem 5.1.
Let E : = { E k } k ∈ Z and Q : = { Q k } k ∈ Z be two exp-ATIs . Suppose that p ∈ ( ω/ ( ω + η ) , and β, γ ∈ ( ω (1 / p − , η ) . Then there exists a positive constant C such that, for any f ∈ ( ˚ G η ( β, γ )) ′ ,C − kS Q ( f ) k L p ( X ) ≤ kS E ( f ) k L p ( X ) ≤ C kS Q ( f ) k L p ( X ) . To show Theorem 5.1, the Fe ff erman-Stein vector-valued maximal inequality is necessary. Lemma 5.2 ([22, Theorem 1.2]) . Suppose that p ∈ (1 , ∞ ) and u ∈ (1 , ∞ ] . Then there exists apositive constant C such that, for any sequence { f j } ∞ j = of measurable functions, (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X j = [ M ( f j )] u u (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ( X ) ≤ C (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X j = | f j | u u (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ( X ) with the usual modification made when u = ∞ .Proof of Theorem 5.1. By symmetry, we only need to prove kS E ( f ) k L p ( X ) . kS Q ( f ) k L p ( X ) . For any k ∈ Z , f ∈ ( ˚ G η ( β, γ )) ′ with β, γ as in Theorem 5.1, and z ∈ X , define m k ( f )( z ) : = " V δ k ( z ) Z B ( z ,δ k ) | Q k f ( u ) | d µ ( u ) . Now suppose that l ∈ Z , x ∈ X and y ∈ B ( x , δ l ). By Theorem 2.7, we conclude that E l f ( y ) = ∞ X k = −∞ X α ∈A k N ( k ,α ) X m = E l e Q k (cid:16) y , y k , m α (cid:17) Z Q k , m α Q k f ( u ) d µ ( u ) , where all the notation is as in Theorem 2.7 and { e Q k } ∞ k = −∞ satisfy the conditions of Theorem 2.7.Notice that, if z ∈ Q k , m α , then Q k , m α ⊂ B ( z , δ k ) and µ ( Q k , m α ) ∼ V δ k ( z ). Therefore, we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) µ ( Q k , m α ) Z Q k , m α Q k f ( u ) d µ ( u ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . " V δ k ( z ) Z B ( z ,δ k ) | Q k f ( u ) | d µ ( y ) ∼ m k ( f )( z ) , ardy S paces on S paces of H omogenous T ype (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) µ ( Q k , m α ) Z Q k , m α Q k f ( u ) d µ ( u ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . inf z ∈ Q k , m α m k ( f )( z ) . Moreover, by the proof of (3.7), we find that, for any fixed β ′ ∈ (0 , β ), (cid:12)(cid:12)(cid:12)(cid:12) E l e Q k (cid:16) y , y k , m α (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) . δ | k − l | β ′ V δ k ∧ l ( y ) + V ( y , y k , m α ) δ k ∧ l δ k ∧ l + d ( y , y k , m α ) γ ∼ δ | k − l | β ′ V δ k ∧ l ( x ) + V ( x , y k , m α ) δ k ∧ l δ k ∧ l + d ( x , y k , m α ) γ , where only the regularity condition of e Q k on the first variable is used. Therefore, by Lemma 3.7,for any fixed r ∈ ( ω/ ( ω + γ ) , | E l f ( y ) | . ∞ X k = −∞ δ | k − l | β ′ X α ∈A k N ( k ,α ) X m = µ (cid:16) Q k , m α (cid:17) V δ k ∧ l ( x ) + V ( x , y k , m α ) δ k ∧ l δ k ∧ l + d ( x , y k , m α ) γ inf z ∈ Q k , m α m k ( f )( z ) . ∞ X k = −∞ δ | k − l | β ′ δ [ k − ( k ∧ l )] ω (1 − r ) M X α ∈A k N ( k ,α ) X m = inf z ∈ Q k , m α [ m k ( f )( z )] r χ Q k , m α ( x ) r . Choose β ′ and r such that r ∈ ( ω/ ( ω + β ′ ) , p ). Then, by the H ¨older inequality, we conclude that (cid:2) S E ( f )( x ) (cid:3) = ∞ X l = −∞ Z B ( x ,δ l ) | E l f ( y ) | dyV δ l ( x ) . ∞ X l = −∞ ∞ X k = −∞ δ | k − l | β ′ δ [ k − ( k ∧ l )] ω (1 − r ) M X α ∈A k N ( k ,α ) X m = inf z ∈ Q k , m α [ m k ( f )( z )] r χ Q k , m α ( x ) r . ∞ X l = −∞ ∞ X k = −∞ δ | k − l | β ′ δ [ k − ( k ∧ l )] ω (1 − r ) M X α ∈A k N ( k ,α ) X m = inf z ∈ Q k , m α [ m k ( f )( z )] r χ Q k , m α ( x ) r . ∞ X k = −∞ M X α ∈A k N ( k ,α ) X m = inf z ∈ Q k , m α [ m k ( f )( z )] r χ Q k , m α ( x ) r . ∞ X k = −∞ (cid:8) M (cid:0) [ m k ( f )] r (cid:1) ( x ) (cid:9) r . Therefore, from Lemma 5.2, we deduce that kS E ( f ) k L p ( X ) . (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X k = −∞ (cid:8) M (cid:0) [ m k ( f )] r (cid:1)(cid:9) r r (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) r L p / r ( X ) . (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X k = −∞ [ m k ( f )] (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ( X ) ∼ kS Q ( f ) k L p ( X ) . This finishes the proof of Theorem 5.1. (cid:3) Z iyi H e , Y ongsheng H an , J i L i , L iguang L iu , D achun Y ang and W en Y uan H p ( X ) The main aim of this section is to obtain the atomic characterizations of H p ( X ) when p ∈ ( ω/ ( ω + η ) , p ∈ ( ω/ ( ω + η ) , q ∈ ( p , ∞ ] ∩ [1 , ∞ ] and β, γ ∈ ( ω (1 / p − , η ), we define the homogeneous atomic Hardy space ˚ H p , q at ( X ) in the same way of H p , q at ( X ), but with the distributionspace ( G η ( β, γ )) ′ replaced by ( ˚ G η ( β, γ )) ′ . Then the following relationship between H p , q at ( X ) and˚ H p , q at ( X ) can be found in [20, Theorem 5.4]. Proposition 5.3.
Suppose p ∈ ( ω/ ( ω + η ) , , β, γ ∈ ( ω (1 / p − , η ) and q ∈ ( p , ∞ ] ∩ [1 , ∞ ] .Then ˚ H p , q at ( X ) = H p , q at ( X ) with equivalent (quasi)-norms. More precisely, if f ∈ H p , q at ( X ) , then therestriction of f on ˚ G η ( β, γ ) belongs to ˚ H p , q at ( X ) ; Conversely, if f ∈ ˚ H p , q at ( X ) , then there exists aunique e f ∈ H p , q at ( X ) such that e f = f in ( ˚ G η ( β, γ )) ′ . Due to the fact that the kernels e Q k in the homogeneous continuous Calder´on formula in Theo-rem 2.6 has no compact support, we can only use Theorem 2.6 to decompose an element of H p ( X )into a linear combination of the following molecules . Definition 5.4.
Suppose that p ∈ (0 , q ∈ ( p , ∞ ] ∩ [1 , ∞ ] and ~ǫ : = { ǫ m } ∞ m = ⊂ [0 , ∞ ) satisfying(5.4) ∞ X m = m [ ǫ m ] p < ∞ . A function M ∈ L q ( X ) is called a ( p , q , ~ǫ ) -molecule centered at a ball B : = B ( x , r ) for some x ∈ X and r ∈ (0 , ∞ ) if m has the following properties:(i) k M χ B k L q ( X ) ≤ [ µ ( B )] q − p ;(ii) for any m ∈ N , k M χ B ( x ,δ − m r ) \ B ( x ,δ − m + r ) k L q ( X ) ≤ ǫ m [ µ ( B ( x , δ − m r ))] q − p ;(iii) R X M ( x ) d µ ( x ) = p ∈ (0 , M satisfies (i) and (ii) of Definition 5.4, then M ∈ L ( X ) and hence Definition 5.4(iii) makessense.After carefully checking the proof of [39, Theorem 3.4], we obtain the following molecularcharacterization of the atomic Hardy space H p , q cw ( X ) of Coifman and Weiss [7], the details beingomitted. Proposition 5.5.
Suppose that p ∈ (0 , , q ∈ ( p , ∞ ] ∩ [1 , ∞ ] and ~ǫ : = { ǫ l } ∞ l = satisfying (5.4) .Then f ∈ H p , q cw ( X ) if and only if there exist ( p , q , ~ǫ ) -molecules { M j } ∞ j = and { λ j } ∞ j = ⊂ C , with P ∞ j = | λ j | p < ∞ , such that (5.5) f = ∞ X j = λ j M j ardy S paces on S paces of H omogenous T ype converges in ( L / p − ( X )) ′ when p ∈ (0 , or in L ( X ) when p = . Moreover, there exists apositive constant C, independent of f , such that, for any f ∈ H p , q cw ( X ) ,C − k f k H p , q cw ( X ) ≤ inf ∞ X j = | λ j | p p ≤ C k f k H p , q cw ( X ) , where the infimum is taken over all the molecular decompositions of f as in (5.5) . Let p ∈ ( ω/ ( ω + η ) ,
1] and q ∈ ( p , ∞ ] ∩ [1 , ∞ ]. By Proposition 5.3, ˚ H p , q at ( X ) = H p , q cw ( X ) and thealready known fact that H p , q cw ( X ) is independent of the choice of q ∈ ( p , ∞ ] ∩ [1 , ∞ ], we know that˚ H p , q at ( X ) = ˚ H p , ( X ). With this observation, we show ˚ H p , q at ( X ) ⊂ H p ( X ) as follows. Proposition 5.6.
Let p ∈ ( ω/ ( ω + η ) , , β, γ ∈ ( ω (1 / p − , η ) , q ∈ ( p , ∞ ] ∩ [1 , ∞ ] and { Q k } k ∈ Z be an exp-ATI . Let θ ∈ (0 , ∞ ) and S θ be as in (5.1) . Then there exists a positive constant C,independent of θ , such that, for any distribution f ∈ ( ˚ G η ( β, γ )) ′ belonging to ˚ H p , ( X ) , (5.6) kS θ ( f ) k L p ( X ) ≤ C max n θ − ω/ , θ ω/ p o k f k ˚ H p , ( X ) . In particular, ˚ H p , q at ( X ) = ˚ H p , ( X ) ⊂ H p ( X ) .Proof. Let β, γ ∈ ( ω (1 / p − , η ). It su ffi ces to show (5.6) for the case θ ∈ [1 , ∞ ), because both(5.6) with θ = S θ ( f ) . θ − ω/ S ( f ) for any f ∈ ( ˚ G η ( β, γ )) ′ whenever θ ∈ (0 ,
1) imply that(5.6) also holds true for any θ ∈ (0 , g -function as in (5.2) is boundedon L ( X ). Indeed, for any h ∈ L ( X ), we write k g ( h ) k L ( X ) = ∞ X k = −∞ Z X | Q k h ( z ) | d µ ( z ) = ∞ X k = −∞ D Q ∗ k Q k h , h E . By Theorem 2.6 and the proof of [27, (3.2)], we find that, for any fixed β ′ ∈ (0 , β ∧ γ ), any k , k ∈ Z and x , y ∈ X , we have(5.7) (cid:12)(cid:12)(cid:12) Q k Q ∗ k ( x , y ) (cid:12)(cid:12)(cid:12) . δ | k − k | β ′ V δ k ∧ k ( x ) + V ( x , y ) " δ k ∧ k δ k ∧ k + V ( x , y ) γ . Notice that, in (5.7), only the regularity of Q k with respect to the second variable is used. Thus,by Lemma 2.2(v) and the boundedness of M on L ( X ), we conclude that, for any k , k ∈ Z , (cid:13)(cid:13)(cid:13)(cid:13)(cid:16) Q ∗ k Q k (cid:17) (cid:16) Q ∗ k Q k (cid:17)(cid:13)(cid:13)(cid:13)(cid:13) L ( X ) → L ( X ) . (cid:13)(cid:13)(cid:13) Q k Q ∗ k (cid:13)(cid:13)(cid:13) L ( X ) → L ( X ) . δ | k − k | β ′ . Therefore, by the fact that Q ∗ k Q k is self-adjoint and the Cotlar-Stein lemma (see [47, pp. 279–280] and [29, Lemma 4.5]), we obtain the boundedness of P ∞ k = −∞ Q ∗ k Q k on L ( X ) and hence theboundedness of g on L ( X ).Suppose that a is a ( p , B : = B ( x , r ) with x ∈ X and r ∈ (0 , ∞ ).By the Fubini theorem and the boundedness of g on L ( X ), we find that kS θ ( a ) k L ( X ) . (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X k ∈ Z | Q k a | / (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ( X ) ∼ k g ( a ) k L ( X ) . k a k L ( X ) . [ µ ( B )] − p , Z iyi H e , Y ongsheng H an , J i L i , L iguang L iu , D achun Y ang and W en Y uan which further implies that(5.8) Z B ( x , A θ r ) [ S θ ( a )( x )] p d µ ( x ) ≤ kS θ ( a ) k pL ( X ) h µ (cid:16) B (cid:16) x , A θ r (cid:17)(cid:17)i − p . θ ω (1 − p ) . Let x < B ( x , A θ r ) and y ∈ B ( x , θδ k ). Since now θ ∈ [1 , ∞ ), for any u ∈ B = B ( x , r ), wehave d ( u , x ) < (4 A θ ) − d ( x , x ) < (2 A ) − [ δ k + d ( x , y )] and hence | Q k a ( y ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z X Q k ( y , u ) a ( u ) d µ ( u ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Z B | Q k ( y , u ) − Q k ( y , x ) || a ( u ) | d µ ( u ) . Z B " d ( x , u ) δ k + d ( x , y ) η V δ k ( x ) + V ( x , y ) " δ k δ k + d ( x , y ) γ | a ( u ) | d µ ( u ) . [ µ ( B )] − p " r δ k + d ( x , y ) η V δ k ( x ) + V ( x , y ) " δ k δ k + d ( x , y ) γ . On the one hand, if δ k < (4 A θ ) − d ( x , x ), then d ( x , y ) ≥ (4 A ) − d ( x , x ) and hence | Q k a ( y ) | . [ µ ( B )] − p " r d ( x , x ) η V ( x , x ) " δ k d ( x , x ) γ , which further implies that X δ k < (4 A θ ) − d ( x , x ) Z d ( x , y ) <θδ k | Q k a ( y ) | d µ ( y ) V θδ k ( x ) . [ µ ( B )] − p " r d ( x , x ) η " V ( x , x ) X δ k < (4 A θ ) − d ( x , x ) " δ k d ( x , x ) γ . [ µ ( B )] − p " r d ( x , x ) η " V ( x , x ) . On the other hand, if δ k ≥ (4 A θ ) − d ( x , x ), then V ( x , x ) . µ ( B ( x , θδ k )) . θ ω V δ k ( x ) and | Q k a ( y ) | . θ ω [ µ ( B )] − p (cid:18) r δ k (cid:19) η V ( x , x ) , which further implies that X δ k ≥ (4 A θ ) − d ( x , x ) Z d ( x , y ) <θδ k | Q k a ( y ) | d µ ( y ) V θδ k ( x ) . θ ω [ µ ( B )] − p " V ( x , x ) X δ k ≥ (4 A θ ) − d ( x , x ) (cid:18) r δ k (cid:19) η ∼ θ ω + η [ µ ( B )] − p " r d ( x , x ) η " V ( x , x ) . Therefore, when x < B ( x , A θ r ), we have S θ ( a )( x ) . θ ω + η [ µ ( B )] − p " r d ( x , x ) η V ( x , x ) . ardy S paces on S paces of H omogenous T ype p ∈ ( η/ ( ω + η ) , B = B ( x , r ) and (1.1), we obtain Z [ B ( x , A θ r )] ∁ [ S θ ( a )( x )] p d µ ( x )(5.9) . θ ( ω + η ) p [ µ ( B )] p − Z [ B ( x , A θ r )] ∁ " r d ( x , x ) p η " V ( x , x ) p d µ ( x ) . θ p ω [ µ ( B )] p − ∞ X j = − jp η Z (2 A ) j θ r ≤ d ( x , x ) < (2 A ) j + θ r " µ ( B ( x , (2 A ) j θ r )) p d µ ( x ) . θ ω ∞ X j = − j [ p η − (1 − p ) ω ] . θ ω . Combining (5.8) and (5.9) implies that, when θ ∈ [1 , ∞ ),(5.10) kS θ ( a ) k L p ( X ) . θ ω/ p . Let f ∈ ˚ H p , ( X ). By the definition of ˚ H p , ( X ), we know that, for any ǫ ∈ (0 , ∞ ), there exist( p , { a j } ∞ j = and { λ j } ∞ j = ⊂ C such that f = P ∞ j = λ j a j in ( ˚ G η ( β, γ )) ′ and P ∞ j = | λ j | p ≤k f k p ˚ H p , ( X ) + ǫ . By (5.10) and the fact S θ ( f ) ≤ P ∞ j = | λ j |S θ ( a j ), we conclude that kS θ ( f ) k pL p ( X ) ≤ ∞ X j = | λ j | p kS θ ( a j ) k pL p ( X ) . θ ω ∞ X j = | λ j | p . θ ω [ k f k p ˚ H p , ( X ) + ǫ ] → θ ω k f k p ˚ H p , ( X ) as ǫ → + . This finishes the proof of (5.6) and hence of Proposition 5.6. (cid:3) Next, we use Proposition 5.5 to show the following converse of Proposition 5.6.
Proposition 5.7.
Let p ∈ ( ω/ ( ω + η ) , , β, γ ∈ ( ω (1 / p − , η ) and f ∈ ( ˚ G η ( β, γ )) ′ belongto H p ( X ) . Then there exist a sequence { a j } ∞ j = of ( p , -atoms and { λ j } ∞ j = ⊂ C such that f = P ∞ j = λ j a j in ( ˚ G η ( β, γ )) ′ and P ∞ j = | λ j | p ≤ C k f k pH p ( X ) , where C is a positive constant independent off . Consequently, H p ( X ) ⊂ ˚ H p , ( X ) .Proof. Assume that f ∈ ( ˚ G η ( β, γ )) ′ belongs to H p ( X ). To avoid the confusion of notation, we use { E k } k ∈ Z to denote an exp-ATI and then define S ( f ) as in (5.1) but with Q k therein replaced by E k .Denote by D the set of all dyadic cubes. For any k ∈ Z , we define Ω k : = { x ∈ X : S ( f )( x ) > k } and D k : = ( Q ∈ D : µ ( Q ∩ Ω k ) > µ ( Q ) and µ ( Q ∩ Ω k + ) ≤ µ ( Q ) ) . It is easy to see that, for any Q ∈ D , there exists a unique k ∈ Z such that Q ∈ D k . A dyadic cube Q ∈ D k is called a maximal cube in D k if Q ′ ∈ D and Q ′ ⊃ Q , then Q ′ < D k . Denote the set of allmaximal cubes in D k at level j ∈ Z by { Q j τ, k } τ ∈ I j , k , where I j , k ⊂ A j may be empty. The center of Q j τ, k is denoted by z j τ, k . Then D = S j , k ∈ Z S τ ∈ I j , k { Q ∈ D k : Q ⊂ Q j τ, k } .2 Z iyi H e , Y ongsheng H an , J i L i , L iguang L iu , D achun Y ang and W en Y uan From now on, we adopt the notation E Q : = E l and e E Q : = e E l whenever Q = Q l + α for some l ∈ Z and α ∈ A l + . Then, by Theorem 2.6, we find that f ( · ) = ∞ X l = −∞ e E l E l f ( · ) = ∞ X l = −∞ X α ∈A l + Z Q l + α e E l ( · , y ) E l f ( y ) d µ ( y )(5.11) = X Q ∈D Z Q e E Q ( · , y ) E Q f ( y ) d µ ( y ) = ∞ X k = −∞ ∞ X j = −∞ X τ ∈ I j , k X Q ∈D k , Q ⊂ Q j τ, k Z Q e E Q ( · , y ) E Q f ( y ) d µ ( y ) = : ∞ X k = −∞ ∞ X j = −∞ X τ ∈ I j , k λ j τ, k b j τ, k ( · ) , where all the equalities converge in ( ˚ G η ( β, γ )) ′ , λ j τ, k : = h µ (cid:16) Q j τ, k (cid:17)i p − X Q ∈D k , Q ⊂ Q j τ, k Z Q | E Q f ( y ) | d µ ( y ) and(5.12) b j τ, k ( · ) : = λ j τ, k X Q ∈D k , Q ⊂ Q j τ, k Z Q e E Q ( · , y ) E Q f ( y ) d µ ( y ) . For any Q ∈ D k and Q ⊂ Q j τ, k , assume that Q = Q l + α for some l ∈ Z and α ∈ A l + . Since δ isassumed to satisfy δ < (2 A ) − , it then follows that 2 A C ♮ δ < Q = Q l + α ⊂ B ( y , δ l ) forany y ∈ Q . By this and the fact that µ ( Q ∩ Ω k + ) ≤ µ ( Q ), we obtain µ ( B ( y , δ l ) ∩ [ Q j τ, k \ Ω k + ]) ≥ µ ( B ( y , δ l ) ∩ [ Q \ Ω k + ]) = µ ( Q \ Ω k + ) ≥ µ ( Q ) ∼ V δ l ( y ) . Thus, we have X Q ∈D k , Q ⊂ Q j τ, k Z Q | E Q f ( y ) | d µ ( y ) . ∞ X l = j − X α ∈A l + , D k ∋ Q l + α ⊂ Q j τ, k Z Q l + α µ ( B ( y , δ l ) ∩ ( Q j τ, k \ Ω k + )) V δ l ( y ) | E l f ( y ) | d µ ( y ) . ∞ X l = j − Z Q j τ, k µ ( B ( y , δ l ) ∩ ( Q j τ, k \ Ω k + )) V δ l ( y ) | E l f ( y ) | d µ ( y ) ∼ Z X ∞ X l = j − Z B ( y ,δ l ) ∩ ( Q j τ, k \ Ω k + ) | E l f ( y ) | d µ ( x ) V δ l ( y ) d µ ( y ) ardy S paces on S paces of H omogenous T ype . Z Q j τ, k \ Ω k + [ S ( f )( x )] d µ ( x ) . k µ (cid:16) Q j τ, k (cid:17) . From this and the fact µ ( Q j τ, k ) < µ ( Q j τ, k ∩ Ω k ), it follows that ∞ X k = −∞ ∞ X j = −∞ X τ ∈ I j , k (cid:16) λ j τ, k (cid:17) p . ∞ X k = −∞ kp ∞ X j = −∞ X τ ∈ I j , k µ (cid:16) Q j τ, k (cid:17) (5.13) . ∞ X k = −∞ kp ∞ X j = −∞ X τ ∈ I j , k µ (cid:16) Q j τ, k ∩ Ω k (cid:17) . ∞ X k = −∞ kp µ ( Ω k ) ∼ kS ( f ) k pL p ( X ) . Choose γ ′ ∈ ( ω (1 / p − , γ ) and let ~ǫ : = { δ m [ γ ′ − ω (1 / p − } m ∈ N . Assume for the moment that every b j τ, k as in (5.12) is a ( p , , ~ǫ )-molecule centered at a ball B j τ, k : = B ( z j τ, k , A δ j − ), whose proof isgiven in Lemma 5.8 below. Further, applying Proposition 5.5, we conclude that k b j τ, k k H p , ( X ) . b j τ, k can be written as a linear combination of ( p , L / p − ( X )) ′ when p ∈ ( ω/ ( ω + η ) ,
1) or in L ( X ) when p =
1, and hence converges in ( ˚ G η ( β, γ )) ′ because˚ G η ( β, γ ) ⊂ L / p − ( X ) (see Lemma 4.15). Invoking this, (5.11) and (5.13), we find that f ∈ ˚ H p , ( X )and k f k ˚ H p , ( X ) . kS ( f ) k L p ( X ) . This finishes the proof of Proposition 5.7. (cid:3) Lemma 5.8.
Let all the notation be as in the proof of Proposition 5.7. Then every b j τ, k as in (5.12) is a harmlessly positive constant multiple of a ( p , , ~ǫ ) -molecule centered at the ball B j τ, k : = B ( z j τ, k , A δ j − ) , where ~ǫ : = { δ m [ γ ′ − ω (1 / p − } m ∈ N and γ ′ ∈ ( ω (1 / p − , γ ) .Proof. Let b j τ, k be as in (5.12). For any h ∈ L ( X ) with k h k L ( X ) ≤
1, by the Fubini theorem and theH ¨older inequality, we conclude that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z X b j τ, k ( x ) h ( x ) d µ ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ λ j τ, k X Q ∈D k , Q ⊂ Q j τ, k Z Q | E Q f ( y ) | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z X e E Q ( x , y ) h ( x ) d µ ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d µ ( y ) ≤ λ j τ, k X Q ∈D k , Q ⊂ Q j τ, k Z Q | E Q f ( y ) | d µ ( y ) X Q ∈D k , Q ⊂ Q j τ, k Z X (cid:12)(cid:12)(cid:12) e E ∗ Q h ( y ) (cid:12)(cid:12)(cid:12) d µ ( y ) ≤ h µ (cid:16) Q j τ, k (cid:17)i − p k e g ( h ) k L ( X ) , where e g ( h ) : = [ P ∞ l = −∞ | e E ∗ l h | ] / . Noticing that the kernel of e E ∗ l has the regularity with respect tothe second variable, we follow the argument used in the beginning of the proof of Proposition 5.6to deduce that e g is bounded on L ( X ). Thus, we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z X b j τ, k ( x ) h ( x ) d µ ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . h µ (cid:16) Q j τ, k (cid:17)i − p k h k L ( X ) . h µ (cid:16) B j τ, k (cid:17)i − p . Z iyi H e , Y ongsheng H an , J i L i , L iguang L iu , D achun Y ang and W en Y uan Taking supremum over all h ∈ L ( X ) with k h k L ( X ) ≤
1, we further find that (cid:13)(cid:13)(cid:13)(cid:13) b j τ, k (cid:13)(cid:13)(cid:13)(cid:13) L ( X ) . h µ (cid:16) B j τ, k (cid:17)i − p . Let γ ′ ∈ ( ω (1 / p − , γ ). Fix m ∈ N and let R m : = ( δ − m B j τ, k ) \ ( δ − m + B j τ, k ). Then, for any x ∈ R m ,by the H ¨older inequality and the size condition of { e E l } l ∈ Z , we conclude that (cid:12)(cid:12)(cid:12)(cid:12) b j τ, k ( x ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ λ j τ, k X Q ∈D k , Q ⊂ Q j τ, k Z Q (cid:12)(cid:12)(cid:12) e E Q ( x , y ) E Q f ( y ) (cid:12)(cid:12)(cid:12) d µ ( y ) . λ j τ, k ∞ X l = j − X α ∈A l + , D k ∋ Q l + α ⊂ Q j τ, k Z Q l + α V δ l ( x ) + V ( x , y ) " δ l δ l + d ( x , y ) γ | E l f ( y ) | d µ ( y ) . λ j τ, k ∞ X l = j − X α ∈A l + , Q l + α ⊂ Q j τ, k Z Q l + α V δ l ( x ) + V ( x , y ) " δ l δ l + d ( x , y ) γ ′ d µ ( y ) × ∞ X l = j − X α ∈A l + D k ∋ Ql + α ⊂ Qj τ, k Z Q l + α V δ l ( x ) + V ( x , y ) " δ l δ l + d ( x , y ) γ − γ ′ ) | E l f ( y ) | d µ ( y ) = : 1 λ j τ, k Y( x )Z( x ) . Notice that, for any x ∈ R m , we have 4 A δ j − m − ≤ d ( x , z j τ, k ) < A δ j − m − and, for any y ∈ Q l + α ⊂ Q j τ, k , we have δ l + d ( x , y ) ∼ d ( x , y ) ∼ δ − m + j and henceY( x ) . ∞ X l = j − X α ∈A l + , Q l + α ⊂ Q j τ, k Z Q l + α µ ( B ( y , δ − m + j )) δ l δ − m + j ! γ ′ d µ ( y ) . ∞ X l = j − δ l δ − m + j ! γ ′ Z Q j τ, k µ ( B ( z j τ, k , δ − m + j )) d µ ( y ) . δ m γ ′ µ ( B j τ, k ) µ ( δ − m B j τ, k ) . Thus, for any x ∈ R m , we have (cid:12)(cid:12)(cid:12)(cid:12) b j τ, k ( x ) (cid:12)(cid:12)(cid:12)(cid:12) . λ j τ, k δ m γ ′ µ ( B j τ, k ) µ ( δ − m B j τ, k ) Z( x ) , which, together with the Fubini theorem and Lemma 2.2(ii), implies that (cid:13)(cid:13)(cid:13)(cid:13) b j τ, k χ R m (cid:13)(cid:13)(cid:13)(cid:13) L ( X ) . λ j τ, k δ m γ ′ µ ( B j τ, k ) µ ( δ − m B j τ, k ) (Z R m [Z( x )] d µ ( x ) ) ardy S paces on S paces of H omogenous T ype . λ j τ, k δ m γ ′ µ ( B j τ, k ) µ ( δ − m B j τ, k ) X Q ∈D k , Q ⊂ Q j τ, k Z Q (cid:12)(cid:12)(cid:12) E Q f ( y ) (cid:12)(cid:12)(cid:12) d µ ( y ) . δ m γ ′ µ ( B j τ, k ) µ ( δ − m B j τ, k ) h µ (cid:16) B j τ, k (cid:17)i − p . δ m [ γ ′ − ω ( p − h µ (cid:16) δ − m B j τ, k (cid:17)i − p . The cancelation of b j τ, k follows directly from that of e E l , the details being omitted.Letting ǫ m : = δ m [ γ ′ − ω ( p − for any m ∈ N , we find that { ǫ m } ∞ m = satisfies (5.4) and b j τ, k is aharmlessly positive constant multiple of a ( p , , ~ǫ )-molecule. This finishes the proof of Lemma5.8. (cid:3) Combining Propositions 5.6 and 5.7, we immediately obtain the following main result of thissection, the details being omitted.
Theorem 5.9.
Suppose that p ∈ ( ω/ ( ω + η ) , , β, γ ∈ ( ω (1 / p − , η ) and q ∈ ( p , ∞ ] ∩ [1 , ∞ ] . Assubspaces of ( ˚ G η ( β, γ )) ′ , it holds true that ˚ H p , q at ( X ) = H p ( X ) with equivalent (quasi-)norms. In this section, we characterize Hardy spaces H p ( X ) via the Lusin area functions with apertures,the Littlewood-Paley g -functions and the Littlewood-Paley g ∗ λ -functions, respectively. Theorem 5.10.
Let p ∈ ( ω/ ( ω + η ) , and β, γ ∈ ( ω (1 / p − , η ) . Assume that θ ∈ (0 , ∞ ) and λ ∈ ( ω [1 + / p ] , ∞ ) . Then, for any f ∈ ( ˚ G η ( β, γ )) ′ , it holds true that k f k H p ( X ) ∼ kS θ ( f ) k L p ( X ) ∼ k g ∗ λ ( f ) k L p ( X ) ∼ k g ( f ) k L p ( X ) , (5.14) provided that either one in (5.14) is finite. Here, the positive equivalent constants in (5.14) areindependent of f .Proof. Let f ∈ ( ˚ G η ( β, γ )) ′ with β, γ ∈ ( ω (1 / p − , η ). With { Q k } k ∈ Z being an exp-ATI, wedefine S θ ( f ), g ∗ λ ( f ) and g ( f ), respectively, as in (5.1), (5.2) and (5.3), where θ ∈ (0 , ∞ ) and λ ∈ ( ω [1 + / p ] , ∞ ).By Proposition 5.6 and Theorem 5.9, if f ∈ H p ( X ), then kS θ ( f ) k L p ( X ) . k f k ˚ H p , ( X ) ∼ k f k L p ( X ) .Conversely, if kS θ ( f ) k L p ( X ) < ∞ , then we proceed as the proof of Proposition 5.7 to deduce that f = P ∞ j = λ j a j in ( ˚ G η ( β, γ )) ′ , where { a j } ∞ j = are ( p , { λ j } ∞ j = ⊂ C satisfying P ∞ j = | λ j | p . kS θ ( f ) k pL p ( X ) . Combining this with Theorem 5.9 implies that k f k H p ( X ) = kS ( f ) k L p ( X ) ∼ k f k ˚ H p , ( X ) . kS θ ( f ) k L p ( X ) . Therefore, we have k f k H p ( X ) ∼ kS θ ( f ) k L p ( X ) whenever k f k H p ( X ) or kS θ ( f ) k L p ( X ) is finite.Noticing that S ( f ) . g ∗ λ ( f ) . P ∞ j = j ( ω − λ ) / S j ( f ) , we then apply (5.6) and λ ∈ ( ω [1 + / p ] , ∞ )to obtain kS ( f ) k pL p ( X ) . k g ∗ λ ( f ) k pL p ( X ) . ∞ X j = j ( ω − λ ) p / kS j ( f ) k pL p ( X ) Z iyi H e , Y ongsheng H an , J i L i , L iguang L iu , D achun Y ang and W en Y uan . ∞ X j = j ( ω − λ ) p / j ω k f k p ˚ H p , ( X ) . k f k p ˚ H p , ( X ) . Invoking Theorem 5.9, we then obtain k f k H p ( X ) ∼ k g ∗ λ ( f ) k L p ( X ) whenever k f k H p ( X ) or k g ∗ λ ( f ) k L p ( X ) is finite.If f ∈ H p ( X ) = ˚ H p , ( X ), then, by following the proof of (5.6), we also obtain k g ( f ) k L p ( X ) . k f k ˚ H p , ( X ) ∼ k f k H p ( X ) . To finish the proof of (5.14), it remains to prove k f k H p ( X ) . k g ( f ) k L p ( X ) . Indeed, for any x ∈ X , wehave S ( f )( x ) = X k ∈ Z X α ∈A k N ( k ,α ) X m = Z d ( x , y ) <δ k | Q k f ( y ) | χ Q k , m α ( x ) d µ ( y ) V δ k ( x ) (5.15) . X k ∈ Z X α ∈A k N ( k ,α ) X m = sup z ∈ B ( z k , m α ,δ k − ) | Q k f ( z ) | χ Q k , m α ( x ) , where Q k , m α is as in Section 2 and z k , m α the center of Q k , m α . With all the notation as in Theorem 2.7,we know that, for any z ∈ B ( z k , m α , δ k − ), Q k f ( z ) = X k ′ ∈ Z X α ′ ∈A k ′ N ( k ′ ,α ′ ) X m ′ = µ (cid:16) Q k ′ , m ′ α ′ (cid:17) Q k e Q k ′ (cid:16) z , y k ′ , m ′ α ′ (cid:17) Q k ′ f (cid:16) y k ′ , m ′ α ′ (cid:17) , where y k ′ , m ′ α ′ is an arbitrary point in Q k ′ , m ′ α ′ . Fix β ′ ∈ (0 , β ∧ γ ). Then, similarly to the proof of (3.7)(see also [27, (3.2)]), we conclude that, for any z ∈ B ( z k , m α , δ k − )(5.16) (cid:12)(cid:12)(cid:12)(cid:12) Q k e Q k ′ (cid:16) z , y k ′ , m ′ α ′ (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) . δ | k − k ′ | β ′ V δ k ∧ k ′ ( z ) + V ( z , y k ′ , m ′ α ′ ) δ k ∧ k ′ + d ( z , y k ′ , m ′ α ′ ) γ . The variable z in (5.16) can be replaced by any x ∈ Q k , m α , because max { d ( z , x ) , d ( z , z k , m α ) } . δ k . δ k ∧ k ′ . Further, from Lemma 3.7, we deduce that, for any fixed r ∈ ( ω/ ( ω + η ) , k ′ ∈ Z and z ∈ B ( z k , m α , δ k − ), (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X α ′ ∈A k ′ N ( k ′ ,α ′ ) X m ′ = µ (cid:16) Q k ′ , m ′ α ′ (cid:17) Q k e Q k ′ (cid:16) z , y k ′ , m ′ α ′ (cid:17) Q k ′ f (cid:16) y k ′ , m ′ α ′ (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . δ ( k ∧ k ′ − k ) ω ( r − M X α ′ ∈A k ′ (cid:12)(cid:12)(cid:12)(cid:12) Q k ′ f (cid:16) y k ′ , m ′ α ′ (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) r χ Q k ′ , m ′ α ′ ( x ) r and hence(5.17) | Q k f ( z ) | . X k ′ ∈ Z δ | k − k ′ | β ′ δ ( k ∧ k ′ − k ) ω ( r − M X α ′ ∈A k ′ N ( k ′ ,α ′ ) X m ′ = (cid:12)(cid:12)(cid:12)(cid:12) Q k ′ f (cid:16) y k ′ , m ′ α ′ (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) r χ Q k ′ , m ′ α ′ ( x ) r . ardy S paces on S paces of H omogenous T ype r and β ′ such that r ∈ ( ω/ ( ω + β ′ ) , p ) and applying theH ¨older inequality, we further conclude that, for any x ∈ X ,[ S ( f )( x )] . X k ∈ Z X α ∈A k N ( k ,α ) X m = X k ′ ∈ Z δ | k − k ′ | β ′ δ ( k ∧ k ′ − k ) ω ( r − × M X α ′ ∈A k ′ N ( k ′ ,α ′ ) X m ′ = (cid:12)(cid:12)(cid:12)(cid:12) Q k ′ f (cid:16) y k ′ , m ′ α ′ (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) r χ Q k ′ , m ′ α ′ ( x ) r χ Q k , m α ( x ) . X k ∈ Z X α ∈A k N ( k ,α ) X m = X k ′ ∈ Z δ | k − k ′ | β ′ δ ( k ∧ k ′ − k ) ω ( r − × M X α ′ ∈A k ′ N ( k ′ ,α ′ ) X m ′ = (cid:12)(cid:12)(cid:12)(cid:12) Q k ′ f (cid:16) y k ′ , m ′ α ′ (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) r χ Q k ′ , m ′ α ′ ( x ) r χ Q k , m α ( x ) . X k ∈ Z X k ′ ∈ Z δ | k − k ′ | [ β ′ − ω ( r − M X α ′ ∈A k ′ N ( k ′ ,α ′ ) X m ′ = (cid:12)(cid:12)(cid:12)(cid:12) Q k ′ f (cid:16) y k ′ , m ′ α ′ (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) r χ Q k ′ , m ′ α ′ ( x ) r . X k ′ ∈ Z M X α ′ ∈A k ′ N ( k ′ ,α ′ ) X m ′ = (cid:12)(cid:12)(cid:12)(cid:12) Q k ′ f (cid:16) y k ′ , m ′ α ′ (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) r χ Q k ′ , m ′ α ′ ( x ) r . From this and Lemma 5.2, we deduce that k f k H p ( X ) = k [ S ( f )] r k r L p / r ( X ) . (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X k ′ ∈ Z M X α ′ ∈A k ′ N ( k ′ ,α ′ ) X m ′ = (cid:12)(cid:12)(cid:12)(cid:12) Q k ′ f (cid:16) y k ′ , m ′ α ′ (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) r χ Q k ′ , m ′ α ′ r r (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) r L p / r ( X ) . (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X k ′ ∈ Z X α ′ ∈A k ′ N ( k ′ ,α ′ ) X m ′ = (cid:12)(cid:12)(cid:12)(cid:12) Q k ′ f (cid:16) y k ′ , m ′ α ′ (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) r χ Q k ′ , m ′ α ′ r r (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) r L p / r ( X ) ∼ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X k ′ ∈ Z X α ′ ∈A k ′ N ( k ′ ,α ′ ) X m ′ = (cid:12)(cid:12)(cid:12)(cid:12) Q k ′ f (cid:16) y k ′ , m ′ α ′ (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) χ Q k ′ , m ′ α ′ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ( X ) . By this and the arbitrariness of y k ′ , m ′ α ′ , we finally conclude that k f k H p ( X ) . (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X k ′ ∈ Z X α ′ ∈A k ′ N ( k ′ ,α ′ ) X m ′ = inf z ∈ Q k ′ , m ′ α ′ | Q k ′ f ( z ) | χ Q k ′ , m ′ α ′ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ( X ) . k g ( f ) k L p ( X ) . This finishes the proof of k f k H p ( X ) . k g ( f ) k L p ( X ) and hence of Theorem 5.10. (cid:3) Z iyi H e , Y ongsheng H an , J i L i , L iguang L iu , D achun Y ang and W en Y uan Remark 5.11. If X is a homogeneous group, Folland and Stein [12] showed that, for any given p ∈ (0 ,
2] and any f ∈ S ′ ( X ), k g ∗ λ ( f ) k L p ( X ) . kS ( f ) k L p ( X ) whenever λ ∈ (2 ω/ p , ∞ ), where S ′ ( X )denotes the space of tempered distributions on X (see [12, Corollary 7.4] by observing that λ in(5.3) be equal to 2 λ with λ as in the Littlewood-Paley g ∗ λ -function in [12]). Comparing with this,the range of λ in Theorem 5.10 is narrower, this is because it was proved in [12, Theorem 7.1]that, for any given p ∈ (0 , θ ∈ [1 , ∞ ) and f ∈ S ′ ( X ),(5.18) kS θ ( f ) k L p ( X ) . p θ ω (1 / p − / kS ( f ) k L p ( X ) while, in the proof of Theorem 5.10, we only show that (5.18) for an arbitrary space of homo-geneous type X holds true, with ω (1 / p − /
2) replaced by ω/ p , when p ∈ ( ω/ ( ω + η ) ,
1] and f ∈ ( ˚ G η ( β, γ )) ′ with β, γ ∈ ( ω (1 / p − , η ). However, it is still unclear whether or not (5.18) foran arbitrary space of homogeneous type X (and hence Theorem 5.10 with λ ∈ (2 ω/ p , ω (1 + / p )])holds true. In this section, we characterize the Hardy space via the wavelet orthogonal system { ψ k α : k ∈ Z , α ∈ G k } introduced in [1, Theorem 7.1]. The sequence { D k } k ∈ Z of operators on L ( X ) associatedwith integral kernels(6.1) D k ( x , y ) : = X α ∈G k ψ α ( x ) ψ α ( y ) , ∀ x , y ∈ X turns out to be an exp-ATI; see [25, 29]. Thus, all the conclusions in Section 5 hold true for { D k } k ∈ Z .For any f ∈ ( ˚ G η ( β, γ )) ′ with β, γ ∈ (0 , η ), define the wavelet Littlewood-Paley function S ( f ) bysetting, for any x ∈ X , S ( f )( x ) : = X k ∈ Z X α ∈G k h µ (cid:16) Q k + α (cid:17)i − (cid:12)(cid:12)(cid:12)(cid:12)D ψ k α , f E(cid:12)(cid:12)(cid:12)(cid:12) χ Q k + α ( x ) . For any p ∈ (0 , ∞ ), define the corresponding wavelet Hardy space H pw ( X ) by H pw ( X ) : = n f ∈ (cid:16) ˚ G η ( β, γ ) (cid:17) ′ : k f k H pw ( X ) : = k S ( f ) k L p ( X ) < ∞ o . For any p ∈ ( ω/ ( ω + η ) , ∞ ), the L p ( X )-norm equivalence between the wavelet Littlewood-Paleyfunction S ( f ) and the Littlewood-Paley g -function g ( f ) was proved in [25, Theorem 4.3] whenever f is a distribution. The proof of [25, Theorem 4.3] seems problematic because the authors thereinused an unknown fact that, when f ∈ ( ˚ G ( β, γ )) ′ and n ∈ N , X | k |≤ n X α ∈G k D f , ψ k α E ψ k α ∈ L ( X ) . (6.2)Although (6.2) may not be true for distributions, it is obviously true when f ∈ L ( X ). Indeed, theargument used in the proof of [25, Theorem 4.3] proves the following result. ardy S paces on S paces of H omogenous T ype Theorem 6.1.
Suppose p ∈ ( ω/ ( ω + η ) , ∞ ) and β, γ ∈ (0 , η ) . Then there exists a positive constantC such that, for any f ∈ ( ˚ G η ( β, γ )) ′ , (6.3) kG ( f ) k L p ( X ) ≤ C k S ( f ) k L p ( X ) and, if f ∈ L ( X ) , then (6.4) C − k S ( f ) k L p ( X ) ≤ kG ( f ) k L p ( X ) ≤ C k S ( f ) k L p ( X ) . Here and hereafter, G ( f ) is defined as in (5.2) , but with Q k therein replaced by D k in (6.1) . To show that (6.4) holds true for all distributions, we need the following basic property of H pw ( X ). Proposition 6.2.
Let p ∈ ( ω/ ( ω + η ) , and β, γ ∈ ( ω (1 / p − , η ) . Then H pw ( X ) is a (quasi-)Banach space that can be continuously embedded into ( ˚ G η ( β, γ )) ′ .Proof. Assume that f ∈ ( ˚ G η ( β, γ )) ′ belongs to H pw ( X ). By (6.3), Theorems 5.10 and 5.9, wehave k f k ˚ H p , ( X ) . k f k H pw ( X ) . Consequently, for any ǫ ∈ (0 , ∞ ), there exist ( p , { a j } ∞ j = and { λ j } ∞ j = ⊂ C satisfying ( P ∞ j = | λ j | p ) p ≤ k f k ˚ H p , ( X ) + ǫ such that f = P ∞ j = λ j a j in ( ˚ G η ( β, γ )) ′ .Combining this with Lemmas 4.14 and 4.15, we find that, for any ϕ ∈ ˚ G η ( β, γ ), |h f , ϕ i| ≤ ∞ X j = | λ j ||h a j , ϕ i| . ∞ X j = | λ j |k ϕ k L / p − ( X ) . k ϕ k ˚ G η ( β,γ ) ∞ X j = | λ j | p / p . k ϕ k ˚ G η ( β,γ ) [ k f k H pw ( X ) + ǫ ] . Letting ǫ → + , we obtain k f k ( ˚ G η ( β,γ )) ′ . k f k H pw ( X ) . Thus, H pw ( X ) can be continuously embeddedinto ( ˚ G η ( β, γ )) ′ .To prove that H pw ( X ) is a (quasi-)Banach space, we only prove its completeness. Let { f n } ∞ n = be aCauchy sequence in H pw ( X ). Then { f n } ∞ n = is also a Cauchy sequence in ( ˚ G η ( β, γ )) ′ , so it convergesto some element f in ( ˚ G η ( β, γ )) ′ . For any n ∈ N and x ∈ X , applying the Fatou lemma twice, weconclude that S ( f − f n )( x ) = S (cid:18) lim m →∞ [ f m − f n ] (cid:19) ( x ) = X k ∈ Z X α ∈G k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:28) ψ k α , lim m →∞ [ f m − f n ] (cid:29) e χ Q k + α ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = X k ∈ Z X α ∈G k lim m →∞ (cid:12)(cid:12)(cid:12)(cid:12)D ψ k α , f m − f n E e χ Q k + α ( x ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ lim inf m →∞ X k ∈ Z X α ∈G k (cid:12)(cid:12)(cid:12)(cid:12)D ψ k α , f m − f n E e χ Q k + α ( x ) (cid:12)(cid:12)(cid:12)(cid:12) = lim inf m →∞ S ( f m − f n )( x )and hence k f − f n k pH pw ( X ) = Z X (cid:2) S ( f − f n )( x ) (cid:3) p d µ ( x )0 Z iyi H e , Y ongsheng H an , J i L i , L iguang L iu , D achun Y ang and W en Y uan ≤ Z X lim inf m →∞ (cid:2) S ( f m − f n )( x ) (cid:3) p d µ ( x ) ≤ lim inf m →∞ Z X (cid:2) S ( f m − f n )( x ) (cid:3) p d µ ( x ) = lim inf m →∞ k f m − f n k pH pw ( X ) . Letting n → ∞ , we find that f ∈ H pw ( X ) and lim n →∞ k f − f n k H pw ( X ) =
0. Therefore, H pw ( X ) iscomplete. This finishes the proof of Proposition 6.2. (cid:3) Applying Theorem 6.1 and Proposition 6.2, we show the following wavelet characterizationsof Hardy spaces.
Theorem 6.3.
Suppose p ∈ ( ω/ ( ω + η ) , and β, γ ∈ ( ω (1 / p − , η ) . As subspaces of ( ˚ G η ( β, γ )) ′ ,H p ( X ) = H pw ( X ) with equivalent (quasi-)norms.Proof. Due to (6.3), Theorems 5.10 and 5.9, we obtain H pw ( X ) ⊂ H p ( X ) and k · k H p ( X ) . k · k H pw ( X ) .It remains to show H p ( X ) ⊂ H pw ( X ). To this end, by Theorem 5.9, we conclude that L ( X ) ∩ H p ( X ) is dense in H p ( X ). Thus, for any f ∈ H p ( X ), there exist { f n } ∞ n = ⊂ L ( X ) ∩ H p ( X ) suchthat lim n →∞ k f − f n k H p ( X ) =
0. Obviously, { f n } ∞ n = is a Cauchy sequence of H p ( X ). Noticing that { f n } ∞ n = ⊂ L ( X ), we use (6.4) and Theorem 5.10 to conclude that k f m − f n k H pw ( X ) = k S ( f m − f n ) k L p ( X ) ∼ kG ( f m − f n ) k L p ( X ) ∼ k f m − f n k H p ( X ) → m , n → ∞ , so that { f n } ∞ n = is also a Cauchy sequence of H pw ( X ). By Proposition 6.2, there exists e f ∈ H pw ( X ) such that f n → e f as n → ∞ in H pw ( X ), also in ( ˚ G η ( β, γ )) ′ . Meanwhile, f n → f as n → ∞ in H p ( X ), also in ( ˚ G η ( β, γ )) ′ . Therefore, e f = f in ( ˚ G η ( β, γ )) ′ and f ∈ H pw ( X ). Moreover, k f k pH pw ( X ) ≤ k f − f n k pH pw ( X ) + k f n k pH pw ( X ) ∼ k f − f n k pH pw ( X ) + k f n k pH p ( X ) . k f k pH p ( X ) when n is su ffi ciently large. Thus, we obtain H p ( X ) ⊂ H pw ( X ) and k · k H pw ( X ) . k · k H p ( X ) . Thisfinishes the proof of Theorem 6.3. (cid:3) Let p ∈ ( ω/ ( ω + η ) , H + , p ( X ), H p θ ( X ) with θ ∈ (0 , ∞ ), H ∗ , p ( X ), H p , q at ( X ), H p , q cw ( X ), ˚ H p , q at ( X ) with q ∈ ( p , ∞ ] ∩ [1 , ∞ ] and H pw ( X ) are essentially the same space in the sense of equivalent (quasi-)norms.From now on, we simply use H p ( X ) to denote either one of them if there is no confusion. Inthis section, we give criteria of the boundedness of sublinear operators on Hardy spaces via firstestablishing finite atomic characterizations of H p ( X ). For any p ∈ ( ω/ ( ω + η ) ,
1] and q ∈ ( p , ∞ ] ∩ [1 , ∞ ], we say f ∈ H p , q fin ( X ) if there exist N ∈ N , asequence { a j } Nj = of ( p , q )-atoms and { λ j } Nj = ⊂ C such that f = N X j = λ j a j . ardy S paces on S paces of H omogenous T ype k f k H p , q fin ( X ) : = inf N X j = (cid:12)(cid:12)(cid:12) λ j (cid:12)(cid:12)(cid:12) p p , where the infimum is taken over all the decompositions of f above. It is easy to see that H p , q fin ( X )is a dense subset of H p , q at ( X ) and k · k H p , q at ( X ) ≤ k · k H p , q fin ( X ) . Denote by the symbol UC( X ) the spaceof all uniformly continuous functions on X , that is, a function f ∈ UC( X ) if and only if, for anyfixed ǫ ∈ (0 , ∞ ), there exists σ ∈ (0 , ∞ ) such that | f ( x ) − f ( y ) | < ǫ whenever d ( x , y ) < σ . The nexttheorem characterizes H p , q at ( X ) via H p , q fin ( X ). Theorem 7.1.
Suppose p ∈ ( ω/ ( ω + η ) , . Then the following statements hold true: (i) if q ∈ ( p , ∞ ) ∩ [1 , ∞ ) , then k · k H p , q fin ( X ) and k · k H p , q at ( X ) are equivalent (quasi)-norms on H p , q fin ( X ) ; (ii) k · k H p , ∞ fin ( X ) and k · k H p , ∞ at ( X ) are equivalent (quasi)-norms on H p , q fin ( X ) ∩ UC( X ) ; (iii) H p , ∞ fin ( X ) ∩ UC( X ) is a dense subspace of H p , ∞ at ( X ) .Proof. First, we prove (i). It su ffi ces to show that k f k H p , q fin ( X ) . k f k H p , q at for any f ∈ H p , q fin ( X ) with q ∈ ( p , ∞ ) ∩ [1 , ∞ ). We may as well assume that k f k H ∗ , p ( X ) =
1. Let all the notation be as in theproof that H ∗ , p ( X ) ⊂ H p , q at ( X ) of Theorem 4.2. Then f = X j ∈ Z X k ∈ I j λ jk a jk = X j ∈ Z X k ∈ I j h jk = X j ∈ Z h j both in ( G η ( β, γ )) ′ and almost everywhere. Here and hereafter, for any j ∈ Z and k ∈ I j , thequantities h j , h jk , λ jk and a jk are as in (4.12) and (4.13). Since f ∈ H p , q fin ( X ), it follows that there exist x ∈ X and R ∈ (0 , ∞ ) such that supp f ⊂ B ( x , R ). We claim that there exists a positive constant ˜ c such that, for any x < B ( x , A R ),(7.1) f ⋆ ( x ) ≤ ˜ c [ µ ( B ( x , R ))] − p . We admit (7.1) temporarily and use it to prove (i) and (ii). Let j ′ be the maximal integer such that2 j ≤ ˜ c [ µ ( B ( x , R ))] − p and define(7.2) h : = X j ≤ j ′ X k ∈ I j λ jk a jk and ℓ : = X j > j ′ X k ∈ I j λ jk a jk In what follows, for the sake of convenience, we elide the fact whether I j or not is finite andsimply write the summation P k ∈ I j in (7.2) as P ∞ k = . If j > j ′ , then Ω j = { x ∈ X : f ⋆ ( x ) > j } ⊂ B ( x , A R ), which implies that supp ℓ ⊂ B ( x , A R ) because supp a jk ⊂ Ω j . From f = h + ℓ , itthen follows that supp h ⊂ B ( x , A ). Noticing that k h k L ∞ ( X ) ≤ X j ≤ j ′ (cid:13)(cid:13)(cid:13) h j (cid:13)(cid:13)(cid:13) L ∞ ( X ) . X j ≤ j ′ j ∼ [ µ ( B ( x , R ))] − p and R X h ( x ) d µ ( x ) =
0, we conclude that h is a harmlessly constant multiple of a ( p , ∞ )-atom.2 Z iyi H e , Y ongsheng H an , J i L i , L iguang L iu , D achun Y ang and W en Y uan Next we deal with ℓ . For any N : = ( N , N ) ∈ N , define ℓ N : = N X j = j ′ + N X k = λ jk a jk = N X j = j ′ + N X k = h jk . Then ℓ N is a finite linear combination of ( p , ∞ )-atoms and P N j = j ′ + P N k = | λ jk | p .
1. Notice thatsupp( ℓ − ℓ N ) ⊂ B ( x , A R ) and R X [ ℓ ( x ) − ℓ N ( x )] d µ ( x ) =
0. It su ffi ces to show that k ℓ − ℓ N k L q ( X ) → ffi ciently small when N and N are big enough. Noticing that ℓ = P ∞ j = N + h j + P N j = j ′ + P ∞ k = h jk , we have k ℓ − ℓ N k L q ( X ) ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X j = N + h j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L q ( X ) + N X j = j ′ + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X k = N + h jk (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L q ( X ) . For any j ∈ Z and k ∈ N , we recall that supp h jk ⊂ B jk ⊂ Ω j and k h j k L ∞ ( X ) . j . By f = P ∞ j = −∞ h j and supp( P ∞ j = N + h j ) ⊂ Ω N , we conclude that, for any z ∈ Ω N , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X j = N + h j ( z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f ( z ) − X j ≤ N h j ( z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ | f ( z ) | + X j ≤ N (cid:12)(cid:12)(cid:12) h j ( z ) (cid:12)(cid:12)(cid:12) . | f ( z ) | + N . Notice that, by [20, Proposition 3.9], there exists a constant e C > f ⋆ ≤ e C M ( f ). With f : = f χ { x ∈ X : | f ( x ) | > N − / e C } and f : = f − f , we have2 N q µ (cid:16) Ω N (cid:17) ≤ N q µ (cid:16)n x ∈ X : e C M ( f )( x ) > N o(cid:17) ≤ N q µ (cid:16)n x ∈ X : e C M ( f )( x ) > N − o(cid:17) . k f k qL q ( X ) → N → ∞ , because M is bounded from L q ( X ) to L q , ∞ ( X ) and f ∈ H p , q fin ( X ) ⊂ L q ( X ). Therefore, (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X j = N + h j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) qL q ( X ) . Z Ω N h | f ( z ) | q + N q i d µ ( z ) . (cid:13)(cid:13)(cid:13) f χ Ω N (cid:13)(cid:13)(cid:13) qL q ( X ) + N q µ (cid:16) Ω N (cid:17) → N → ∞ . Then, for any ǫ ∈ (0 , ∞ ), we choose N ∈ N such that k P ∞ j = N + h j k L q ( X ) < ǫ/ N ∈ N and N ≥ j > j ′ , then the fact P ∞ k = | h jk | . j χ Ω j ∈ L q ( X ) implies thatlim N → (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X k = N + h jk (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L q ( X ) = . So, we further choose N ∈ N such that P N j = j ′ + k P ∞ k = N + h jk k L q ( X ) < ǫ/
2. In this way, we have k ℓ − ℓ N k L q ( X ) < ǫ for large N . Then there exist a positive constant C ♭ , independent of N and ǫ , anda ( p , q )-atom a ( N ) such that ℓ − ℓ N = C ♭ ǫ a ( N ) . Therefore, we obtain k f k H p , q fin ( X ) . ∼ k f k H p , q at ( X ) andcomplete the proof of (i) under the assumption (7.1). ardy S paces on S paces of H omogenous T ype k f k H p , ∞ fin ( X ) . k f k H p , ∞ at whenever f ∈ H p , ∞ fin ( X ) ∩ UC( X ).We may also assume that k f k H ∗ , p ( X ) =
1. Notice that f ∈ L ∞ ( X ) and k f ⋆ k L ∞ ( X ) . kM ( f ) k L ∞ ( X ) ≤ c k f k L ∞ ( X ) , where c is a positive constant independent of f . Let j ′′ > j ′ be the largest integersuch that 2 j ≤ c k f k L ∞ ( X ) . We write f = h + ℓ with h as in (7.2) but now ℓ = P j ′ < j ≤ j ′′ P ∞ k = h jk . Asin the proof of (i), we know that h is a harmlessly positive constant multiple of some ( p , ∞ )-atom.Now we consider ℓ . Notice that f ∈ UC( X ). Then, for any ǫ ∈ (0 , ∞ ), there exists σ ∈ (0 , ∞ )such that | f ( x ) − f ( y ) | ≤ ǫ whenever d ( x , y ) ≤ σ . Split ℓ = ℓ σ + ℓ σ with ℓ σ : = X ( j , k ) ∈ G h jk = X ( j , k ) ∈ G λ jk a jk and ℓ σ : = X ( j , k ) ∈ G h jk , where G : = { ( j , k ) : 12 A r jk ≥ σ, j ′ < j ≤ j ′′ } and G : = { ( j , k ) : 12 A r jk < σ, j ′ < j ≤ j ′′ } . Notice that, for any j ′ < j ≤ j ′′ , Ω j is bounded. Thus, by Proposition 4.4(vi), we find that G is afinite set, which further implies that ℓ σ is a finite linear combination of ( p , ∞ )-atoms and X ( j , k ) ∈ G (cid:12)(cid:12)(cid:12)(cid:12) λ jk (cid:12)(cid:12)(cid:12)(cid:12) p . . To consider ℓ σ , it is obvious that supp ℓ σ ⊂ B ( x , A R ) and R X ℓ σ ( x ) d µ ( x ) =
0, so it remains toestimate k ℓ σ k L ∞ ( X ) . For any ( j , k ) ∈ G , applying the definition of h jk in (4.12) implies that (cid:12)(cid:12)(cid:12)(cid:12) h jk (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12) b jk (cid:12)(cid:12)(cid:12)(cid:12) + X l ∈ I j + (cid:12)(cid:12)(cid:12)(cid:12) b j + l φ jk (cid:12)(cid:12)(cid:12)(cid:12) + X l ∈ I j + (cid:12)(cid:12)(cid:12)(cid:12) L j + k , l φ j + l (cid:12)(cid:12)(cid:12)(cid:12) . By the definition of b jk , we have supp b jk ⊂ B ( x jk , A r jk ). Moreover, for any x ∈ B ( x jk , A r jk ), (cid:12)(cid:12)(cid:12)(cid:12) b jk ( x ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f ( x ) − k φ jk k L ( X ) Z B ( x jk , A r jk ) f ( ξ ) φ jk ( ξ ) d µ ( ξ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (7.3) ≤ (cid:12)(cid:12)(cid:12)(cid:12) f ( x ) − f (cid:16) x jk (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) + k φ jk k L ( X ) Z B ( x jk , A r jk ) (cid:12)(cid:12)(cid:12)(cid:12) f ( ξ ) − f (cid:16) x jk (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) φ jk ( ξ ) d µ ( ξ ) . ǫ. If b j + l φ jk ,
0, then B ( x jk , A r jk ) ∩ B ( x j + l , A r j + l ) , ∅ , which further implies that r j + l ≤ A r jk .Thus, for any x ∈ B ( x j + l , A r j + l ), we have d ( x , x j + l ) < A r jk and hence an argument similar tothe estimation of (7.3) gives (cid:12)(cid:12)(cid:12)(cid:12) b j + l ( x ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f ( x ) − k φ j + l k L ( X ) Z B ( x j + l , A r j + l ) f ( ξ ) φ j + l ( ξ ) d µ ( ξ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) φ j + l ( x ) . ǫφ j + l ( x ) , so that X l ∈ I j + (cid:12)(cid:12)(cid:12)(cid:12) b j + l ( x ) φ jk ( x ) (cid:12)(cid:12)(cid:12)(cid:12) . ǫφ jk ( x ) X l ∈ I j + φ j + l ( x ) ∼ ǫφ jk ( x ) . ǫ. Z iyi H e , Y ongsheng H an , J i L i , L iguang L iu , D achun Y ang and W en Y uan Using the definition of L j + k , l and arguing similarly as (7.3), we conclude that, for any x ∈ X , X l ∈ I j + (cid:12)(cid:12)(cid:12)(cid:12) L j + k , l φ jk ( x ) (cid:12)(cid:12)(cid:12)(cid:12) . ǫ, where L j + k , l is as in (4.10). Summarizing all gives k h jk k L ∞ ( X ) . ǫ . Recalling that supp h jk ⊂ B jk and P ∞ k = χ B jk ≤ L , we obtain k ℓ σ k L ∞ ( X ) . ǫ . Therefore, there exist a positive constant e C ♭ , independentof σ and ǫ , and a ( p , ∞ )-atom a ( σ ) such that ℓ σ = e C ♭ ǫ a ( σ ) . This proves that k f k H p , ∞ fin ( X ) . x < B ( x , A R ). Suppose that ϕ ∈ G η ( β, γ ) with k ϕ k G ( x , r ,β,γ ) . r ∈ (0 , ∞ ). First we consider the case r ≥ A d ( x , x ) /
3. For any y ∈ B ( x , d ( x , x )), we have k ϕ k G ( y , r ,β,γ ) .
1, which implies that |h f , ϕ i| . f ∗ ( y ) and hence(7.4) |h f , ϕ i| . ( µ ( B ( x , d ( x , x ))) Z B ( x , d ( x , x )) (cid:2) f ∗ ( y ) (cid:3) p d µ ( y ) ) p . [ µ ( B ( x , R ))] − p . Next we consider the case r < A d ( x , x ) /
3. Choose a function ξ satisfying χ B ( x , (2 A ) − d ( x , x )) ≤ ξ ≤ χ B ( x , (2 A ) − d ( x , x )) and k ξ k ˙ C η ( X ) . [ d ( x , x )] − η . Since supp f ⊂ B ( x , R ), it follows that f ξ = f .Let e ϕ : = ϕξ . For any y ∈ B ( x , d ( x , x )), assuming for the moment that(7.5) k e ϕ k G ( y , r ,β,γ ) . , we obtain |h f , ϕ i| = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z X f ( z ) ϕ ( z ) d µ ( z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z X f ( z ) ξ ( z ) ϕ ( z ) d µ ( z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = |h f , e ϕ i| . f ∗ ( y ) , which implies that (7.4) remains true in this case. Therefore, by the arbitrariness of ϕ and the factthat f ∗ ∼ f ⋆ , we obtain (7.1).Now we fix y ∈ B ( x , d ( x , x )) and prove (7.5). First we consider the size condition. Indeed, if e ϕ ( z ) ,
0, then d ( z , x ) < (2 A ) − d ( x , x ) and hence d ( z , y ) < (16 A / d ( x , z ), which implies that | e ϕ ( z ) | ≤ | ϕ ( z ) | ≤ V r ( x ) + V ( x , z ) " rr + d ( x , z ) γ ∼ V r ( y ) + V ( y , z ) " rr + d ( y , z ) γ . To consider the regularity condition of e ϕ , we may assume that d ( z , z ′ ) ≤ (2 A ) − [ r + d ( y , z )]due to the size condition. For the case d ( z , x ) > (2 A ) − d ( x , x ), we have e ϕ ( z ) = y ∈ B ( x , d ( x , x )) and r < A d ( x , x ) /
3, we further obtain d ( z , z ′ ) ≤ (2 A ) − [ r + d ( y , z )] ≤ (2 A ) − [ r + A d ( y , x ) + A d ( x , z )] ≤ (2 A ) − [4 A d ( x , x ) + A d ( x , z )] ≤ (2 A ) − d ( x , z ) , which further implies that d ( z ′ , x ) ≥ A d ( x , z ) − d ( z , z ′ ) ≥ (2 A ) − d ( x , x ) and hence e ϕ ( z ′ ) = d ( z , x ) ≤ (2 A ) − d ( x , x ). Then we have (2 A ) − d ( x , x ) ≤ d ( z , x ) ≤ A d ( x , x ) and d ( y , z ) ≤ A [ d ( y , x ) + d ( x , x ) + d ( x , z )] ≤ A d ( x , x ) + A d ( x , z ) ≤ (2 A ) d ( x , z ) , ardy S paces on S paces of H omogenous T ype d ( z , z ′ ) ≤ (2 A ) − [ r + d ( x , z )] and r + d ( y , z ) . min { r + d ( x , z ) , r + d ( x , z ′ ) , d ( x , x ) } .Therefore, by the regularity of ϕ and the definition of ξ , we conclude that (cid:12)(cid:12)(cid:12)e ϕ ( z ) − e ϕ ( z ′ ) (cid:12)(cid:12)(cid:12) ≤ ξ ( z ) | ϕ ( z ) − ϕ ( z ′ ) | + | ϕ ( z ′ ) || ξ ( z ) − ξ ( z ′ ) | . " d ( z , z ′ ) r + d ( x , z ) β V r ( x ) + V ( x , z ) " rr + d ( x , z ) γ + V r ( x ) + V ( x , z ′ ) " rr + d ( x , z ′ ) γ " d ( z , z ′ ) d ( x , x ) β . " d ( z , z ′ ) r + d ( y , z ) β V r ( y ) + V ( y , z ) " rr + d ( y , z ) γ . This proves (7.5) and hence finishes the proofs of (i) and (ii).Now we prove (iii). According to [23, pp. 3347–3348] (see also [27, Theorem 2.6]), thereexists a sequence { S k } k ∈ Z of bounded operators on L ( X ) with their kernels satisfying the followingconditions:(i) S k ( x , y ) = d ( x , y ) ≥ C ♯ δ k and, for any x , y ∈ X , | S k ( x , y ) | . V δ k ( x ) + V δ k ( y ) , where C ♯ is a fixed positive constant greater than 1;(ii) for any x , x ′ , y ∈ X with d ( x , x ′ ) ≤ C ♯ δ k , | S k ( x , y ) − S k ( x ′ , y ) | + | S k ( y , x ) − S k ( y , x ′ ) | . " d ( x , x ′ ) δ k θ V δ k ( x ) + V δ k ( y ) , where θ is as in [23, Theorem 2.4];(iii) for any x ∈ X , R X S k ( x , y ) d µ ( y ) = = R X S k ( y , x ) d µ ( y ) . For any g ∈ S p ∈ [1 , ∞ ] L p ( X ) and x ∈ X , define S k g ( x ) : = Z X S k ( x , y ) g ( y ) d µ ( y ) . Then, for any ( p , ∞ )-atom a supported on B ( z , r ) with z ∈ X and r ∈ (0 , ∞ ), we observe that S k a satisfies the following properties:(a) k S k a k L ∞ ( X ) . k a k L ∞ ( X ) and lim k →∞ k S k a − a k L ( X ) = k is su ffi ciently large, supp S k ( a ) ⊂ B ( z , A r );(c) R X S k a ( x ) d µ ( x ) = S k a ∈ UC( X ).6 Z iyi H e , Y ongsheng H an , J i L i , L iguang L iu , D achun Y ang and W en Y uan Consequently, S k a is a harmlessly constant multiple of a ( p , ∞ )-atom and hence of a ( p , k S k a − a k H p , ∞ at ( X ) ∼ k S k a − a k H p , ( X ) → k → ∞ . For any f ∈ H p , ∞ at ( X ), there exists asequence { f n } n ∈ N ⊂ H p , ∞ fin ( X ) such that lim n →∞ k f n − f k H p , q at ( X ) =
0. Then, for any n ∈ N , by theabove (a) through (d), we find that S k ( f n ) ∈ H p , ∞ fin ( X ) ∩ UC( X ) and lim k →∞ k S k f n − f n k H p , ∞ at ( X ) = k S k f n − f k H p , ∞ at ( X ) → n , k → ∞ , which completes the proof of (iii) and henceof Theorem 7.1. (cid:3) In this section, applying the finite atomic characterizations of Hardy spaces, we obtain twocriteria on the boundedness of sublinear operators on Hardy spaces.Recall that a complete vector space B is called a quasi-Banach space if its quasi-norm k · k B satisfies the following condition:(i) for any f ∈ B , k f k B = f is the zero element in B ;(ii) for any λ ∈ C and f ∈ B , k λ f k B = | λ |k f k B ;(iii) there exists C ∈ [1 , ∞ ) such that, for any f , g ∈ B , k f + g k B ≤ C ( k f k B + k g k B ).Next we recall the definition of r -quasi-Banach spaces (see, for example, [35, 51, 53, 52, 20]). Definition 7.2.
Suppose that r ∈ (0 ,
1] and B r is a quasi-Banach space with its quasi-norm k · k B r .The space B r is called an r-quasi-Banach space if there exists κ ∈ [1 , ∞ ) such that, for any m ∈ N and { f j } mj = ⊂ B r , (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) m X j = f j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) r B r ≤ κ m X j = k f j k r B r . Obviously, when p ∈ (0 , L p ( X ) and H ∗ , p ( X ) are p -quasi-Banach-spaces. Let Y be a linearspace and B r is an r -quasi-Banach space with r ∈ (0 , T : Y → B r is said to be B r -sublinear if there exists a positive constant κ ∈ [1 , ∞ ) such that(i) for any f , g ∈ Y , k T ( f ) − T ( g ) k B r ≤ κ k T ( f − g ) k B r ;(ii) for any m ∈ N , { f j } mj = ⊂ Y and { λ j } mj = ⊂ C , (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) T m X j = λ j f j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) r B r ≤ κ m X j = | λ j | r k T ( f j ) k r B r . (see, for example, [35, Definition 2.5], [51, Definition 1.6.7], [53, Remark 1.1(3)], [52, Definition1.6] and [20, Definition 5.8]).The next theorem gives us a criteria for B r -sublinear operators that can be extended to bounded B r -sublinear operators from Hardy spaces to B r . It can be proved by following the proof of [20,Theorem 5.9] with slight modifications, the details being omitted. ardy S paces on S paces of H omogenous T ype Theorem 7.3.
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E-mails: [email protected] (Z. He) [email protected] (D. Yang) [email protected] (W. Yuan)Yongsheng HanDepartment of Mathematics, Auburn University, Auburn, AL 36849-5310, USA
E-mail: [email protected]
Ji LiDepartment of Mathematics, Macquarie University, Sydney, NSW 2109, Australia
E-mail: [email protected]
Liguang LiuDepartment of Mathematics, School of Information, Renmin University of China, Beijing 100872,People’s Republic of China