Maximal function characterizations of Hardy spaces on RD-spaces and their applications

Transcription

1 Science in China Series A: Mathematics Dec., 2008, Vol. 5, No. 2, math.scichina.com Maximal function characterizations of Hardy spaces on RD-spaces and their applications Loukas GRAFAKOS,LIULiGuang 2 & YANG DaChun 2 Department of Mathematics, University of Missouri, Columbia, MO 652, USA 2 School of Mathematical Sciences, Beijing Normal University; Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 00875, China loukas@math.missouri.edu, liuliguang@mail.bnu.edu.cn, dcyang@bnu.edu.cn) Abstract Let be an RD-space, i.e., a space of homogeneous type in the sense of Coifman and Weiss, which has the reverse doubling property. Assume that has a dimension n. Forα 0, ) denotebyhα p ), H p d ), and H,p ) the corresponding Hardy spaces on defined by the nontangential maximal function, the dyadic maximal function and the grand maximal function, respectively. Using a new inhomogeneous Calderón reproducing formula, it is shown that all these Hardy spaces coincide with L p )whenp, ] and with each other when p n/n +), ]. An atomic characterization for H,p )withp n/n +), ] is also established; moreover, in the range p n/n +), ],itisprovedthatthespaceh,p ), the Hardy space H p ) defined via the Littlewood-Paley function, and the atomic Hardy space of Coifman and Weiss coincide. Furthermore, it is proved that a sublinear operator T uniquely extends to a bounded sublinear operator from H p )to some quasi-banach space B if and only if T maps all p, q)-atoms when q p, ) [, ) or continuous p, )-atoms into uniformly bounded elements of B. Keywords: space of homogeneous type, Calderón reproducing formula, space of test function, maximal function, Hardy space, atom, Littlewood-Paley function, sublinear operator, quasi- Banach space MSC2000): 42B25, 42B30, 47B38, 47A30 Introduction The theory of Hardy spaces on the Euclidean space R n certainly plays an important role in harmonic analysis and has been systematically studied and developed; see, for example, [ 3]. It is well known that spaces of homogeneous type in the sense of Coifman and Weiss [4] present a natural setting for the Calderón-Zygmund theory of singular integrals; see, for instance, [5]. Recall that a space of homogeneous type is a set equipped with a metric d and a Borel regular measure μ that satisfies the doubling property. Coifman and Weiss [5] introduced the atomic Hardy spaces H p at )forp 0, ]. Under the assumption that the measure of any ball in is equivalent to its radius i. e., is an Ahlfors -regular metric measure space), Coifman and Weiss [5] established a molecular characterization for Hat ). Also in this setting, Received November 5, 2007; accepted January 5, 2008; published online September 20, 2008 DOI: 0.007/s Corresponding author This work was supported by the National Science Foundation of USA Grant No. DMS ), the University of Missouri Research Council Grant No. URC ), the National Science Foundation for Distinguished Young Scholars of China Grant No ) and the Program for New Century Excellent Talents in University of the Ministry of Education of China Grant No )

2 2254 Loukas GRAFAKOS et al. if p /2, ], Macías and Segovia [6] obtained the grand maximal function characterization for H p at ) via distributions acting on certain spaces of Lipschitz functions; Han [7] established a Lusin-area characterization for H p at ); Duong and Yan[8] characterized these atomic Hardy spaces in terms of Lusin area functions associated with certain Poisson semigroups; Li [9] also characterized H p at ) by grand maximal functions defined via test functions introduced in [0]. Structures of spaces of homogeneous type encompass several important examples in harmonic analysis, such as Euclidean spaces with A -weights of the Muckenhoupt class), Ahlfors n- regular metric measure spaces see, for example, [, 2]), Lie groups of polynomial growth see, for instance, [3 5]), and Carnot-Carathéodory spaces with doubling measures see [6 22]). All these examples fall under the scope of the study of RD-spaces introduced in [23] see also [24]). An RD-space is a space of homogeneous type which has a dimension n and satisfies the following reverse doubling property: there exists a constant a 0 > such that for all x and 0 <r< diam )/a 0, Bx, a 0 r) \ Bx, r), where in this article diam ):=sup x, y dx, y). A Littlewood-Paley theory of Hardy spaces on RD-spaces was established in [24], and these Hardy spaces are known to coincide with some of Triebel-Lizorkin spaces in [23]. Let be an RD-space with dimension n. In this paper we achieve two goals. First, we introduce various Hardy spaces on via the nontangential maximal function, the dyadic maximal function, and the grand maximal function, and show that these Hardy spaces coincide with L p )whenp, ] and with each other when p n/n +), ]. When p n/n + ), ], we further identify these Hardy spaces with the Hardy space H p ) defined via the Littlewood-Paley function in [23, 24] and with the atomic Hardy space of Coifman and Weiss in [5]. Secondly, we prove that a sublinear operator T uniquely extends to a bounded sublinear operator from H p ) to some quasi-banach space B if and only if T maps all p, q)-atoms when q p, ) [, ) or continuous p, )-atoms into uniformly bounded elements of B. Thislast result answers a question posed by Meda, Sjögren and Vallarino in [25]. To be precise, in Section 2, we introduce various Hardy spaces Hα p )forα 0, ), H p d ), and H,p ), defined in terms of the nontangential maximal function, the dyadic maximal function, and the grand maximal function, respectively. We also introduce a slight variant of test functions in [23, 24], which is crucial in applications; see Remark 2.9 iii) below. One of the contributions of this paper is to establish a new inhomogeneous discrete Calderón reproducing formula; see Theorem 3.3 of Section 3. This formula plays a key role in the whole paper and may be useful in the study of other problems. Moreover, in Section 3, we prove that all Hardy spaces mentioned above coincide with L p )whenp, ], and all these Hardy spaces are equivalent to each other when p n/n +), ]. Section 4 is devoted to obtaining an atomic characterization for the Hardy space H,p ) when p n/n +), ] by using certain ideas from [6]. Some applications are given in Section 5. First, when p n/n +), ], we prove that H,p )andh p ) coincide by means of their atomic characterizations, where H p )isthe Hardy space defined via the Littlewood-Paley function in [23, 24]; see Theorem 5.4. From a key observation Lemma 5.3) which establishes the connection between the space of test functions and the Lipschitz space introduced by Coifman and Weiss cf. [5, 2.2)]), it follows directly

3 Maximal function characterizations of Hardy spaces 2255 that H p ) and the atomic Hardy space of Coifman and Weiss [5] coincide; see Remark 5.5 ii). This also answers a question in [24, Remark 2.30]. Our second application concerns the boundedness of sublinear operators from H p )tosome quasi-banach space B. It is well-known that atomic characterizations play an important role in obtaining boundedness of sublinear operators in Hardy spaces, namely, boundedness of these operators can be deduced from their behavior on atoms in principle. However, Meyer in [26, p. 53] see also [27, 28]) gave an example of f H R n ) whose norm cannot be achieved [28, Theorem2] by its finite atomic decompositions via, )-atoms. Based on this fact, Bownik constructed a surprising example of a linear functional defined on a dense subspace of H R n ), which maps all, )-atoms into bounded scalars, but cannot extend to a bounded linear functional on the whole H R n ). It follows that proving that a linear operator T maps all p, )-atoms into uniformly bounded elements of B does not guarantee the boundedness of T from the entire H p R n )withp 0, ]) to some quasi-banach space B; this phenomenon was essentially observed by Meyer in [29, p. 9]. Motivated by this, Yabuta [30] gave some sufficient conditions for the boundedness of T from H p R n )withp 0, ] to L q R n )withq or H q R n )withq [p, ]. It was proved in [3] that a linear operator T extends to a bounded linear operator from Hardy spaces H p R n )withp 0, ] to some quasi-banach space B if and only if T maps all p, 2,s)-atoms for some s n/p ) into uniformly bounded elements of B, where n/p ) denotes the maximal integer no more than n/p ). For q, ], denote by H,q fin Rn ) the vector space of all finite linear combinations of,q)-atoms endowed with the following norm: f H,q fin Rn ) { N N } := inf λ j : f = λ j a j,n N,{λ j } N j= C, and {a j } N j= are,q)-atoms. j= j= Recently, by means of the maximal characterization of H R n ), Meda, Sjögren and Vallarino [25] proved that H R n ) and H,q fin Rn ) are equivalent quasi-norms on H,q fin Rn )withq, ) or on H, fin Rn ) CR n ), where CR n ) denotes the set of continuous functions. It was also claimed in [25] that this equivalence of norms remains true for H p R n )andp, q)-atoms with p 0, ) and q [, ].) From this these authors deduced that a linear operator defined on H,q fin Rn ) which maps,q)-atoms or continuous, )-atoms into uniformly bounded elements of some Banach space B uniquely extends to a bounded linear operator from H R n )tob. In [32], the results in [25] were generalized to weighted Hardy spaces on R n with a general expansive matrix dilation. In [25], Meda, Sjögren and Vallarino also pointed out that it is not evident whether their results for H p R n ) can be extended to Hardy spaces on spaces of homogeneous type. In Section 5, using some ideas from [25], we give an affirmative answer to this question; see Theorems 5.6 and 5.9 below. We should mention that the result of [30] was generalized to Ahlfors -regular metric measure spaces in [33], and the result of [3] was extended to RD-spaces in [34]. Also, by using the dual theory, Meda, Sjögren and Vallarino [25] showed that T extends uniquely to a bounded linear operator from H )tol )ifandonly if T maps all,q)-atoms with q, ) to uniformly bounded elements of L ), where is a space of homogeneous type. However, this result is valid only for linear operators and the

4 2256 Loukas GRAFAKOS et al. Hardy space H ). Finally, we mention that a radial maximal function characterization of these Hardy spaces was also recently given in [35]. In this paper we use the following notation: N := {, 2,...}, Z + := N {0} and R + := [0, ). For any p [, ], we denote by p the conjugate index, namely, /p +/p =. Wealso denote by C a positive constant independent of the main parameters involved, which may vary at different occurrences. Constants with subscripts do not change through the whole paper. We use f g and f g to denote f Cg and f Cg, respectively. If f g f, wethen write f g. For any a, b R, seta b) :=min{a, b} and a b) :=max{a, b}. 2 Preliminaries We first recall the notions of spaces of homogeneous type in the sense of [4, 5] and RD-spaces in [23]. Definition 2.. Let,d) be metric space with Borel regular measure μ such that all balls defined by d have finite and positive measures. For any x and r>0, setbx, r) :={y : dx, y) <r}. i) The triple,d,μ) is called a space of homogeneous type if there exists a constant C such that for all x and r>0, μbx, 2r)) C μbx, r)) doubling property). 2.) ii) Let 0 <κ n. The triple,d,μ) is called a κ, n)-space if there exist constants 0 < C 2 and C 3 such that for all x, 0 <r< diam )/2 and λ< diam )/2r), C 2 λ κ μbx, r)) μbx, λr)) C 3 λ n μbx, r)), 2.2) where diam )=sup x, y dx, y). A space of homogeneous type is called an RD-space, if it is a κ, n)-space for some 0 <κ n, i. e., if some reverse doubling condition holds. Remark 2.2. i) In some sense, n measures the dimension of. Obviously a κ, n) space is a space of homogeneous type with C := C 3 2 n. Conversely, a space of homogeneous type satisfies the second inequality of 2.2) with C 3 := C and n := log 2 C. ii) If μ is doubling, then μ satisfies 2.2) if and only if there exist constants a 0 > and C 0 > such that for all x and 0 <r< diam )/a 0, μbx, a 0 r)) C 0 μbx, r)) reverse doubling property) If a 0 = 2, this is the classical reverse doubling condition), and equivalently, for all x and 0 <r< diam )/a 0, Bx, a 0 r) \ Bx, r) ; see [23, 36]. iii) From ii) of this remark, it follows that if is an RD-space, then μ{x}) = 0 for all x. iv) Let d be a quasi-metric, which means that there exists A 0 such that for all x, y, z, dx, y) A 0 dx, z)+dz,y)). Recall that Macías and Segovia [37, Theorem 2] proved that there exists an equivalent quasi-metric d such that all balls corresponding to d are open in the

5 Maximal function characterizations of Hardy spaces 2257 topology induced by d, and there exist constants A 0 > 0andθ 0, ) such that for all x, y, z, dx, z) dy, z) A 0 dx, y) θ [ dx, z)+ dy, z)] θ. It is known that the approximation of the identity as in Definition 2.3 below also exists for d; see [23]. Obviously, all results in this paper are invariant on equivalent quasi-metrics. From these facts, we deduce that all conclusions of this paper are still valid for quasi-metrics. In this paper, we always assume that is an RD-space and μ )=. For any x, y and δ>0, set V δ x) :=μbx, δ)) and V x, y) :=μbx, dx, y))). It follows from 2.) that V x, y) V y, x). The following notion of approximations of the identity on RD-spaces were first introduced in [23]; see also [24]. Definition 2.3. Let ɛ 0, ], ɛ 2 > 0 and ɛ 3 > 0. A sequence {S k } k Z of bounded linear integral operators on L 2 ) is said to be an approximation of the identity of order ɛ,ɛ 2,ɛ 3 ) in short, ɛ,ɛ 2,ɛ 3 )- AOTI ), if there exists a positive constant C 4 such that for all k Z and all x, x, y and y, S k x, y), the integral kernel of S k is a function from into C satisfying 2 i) S k x, y) C kɛ 2 4 V 2 k x)+v 2 k y)+v x,y) 2 k +dx,y)) ; ɛ 2 ii) S k x, y) S k x dx,x,y) C ) ɛ 4 2 k +dx,y)) ɛ V 2 k x)+v 2 k y)+v x,y) 2 k + dx, y))/2; iii) Property ii) holds with x and y interchanged ; iv) [S k x, y) S k x, y )] [S k x,y) S k x,y dx,x )] C ) ɛ 4 V 2 k x)+v 2 k y)+v x,y) 2 kɛ 3 2 k +dx,y)) ɛ 2 kɛ 2 2 k +dx,y)) ɛ 2 for dx, x ) dy,y ) ɛ 2 k +dx,y)) ɛ 2 k +dx,y)) for dx, ɛ 3 x ) 2 k + dx, y))/3 and dy, y ) 2 k + dx, y))/3; v) S kx, y) dμy) = S kx, y) dμx) =. Remark 2.4. It was proved in [23, Theorem 2.] that for any ɛ, ɛ 2, ɛ 3 as in Definition 2.3, there exists an ɛ,ɛ 2,ɛ 3 )- AOTI {S k } k Z with bounded support on, which means that there exists a positive constant C such that for all k Z and dx, y) C2 k, S k x, y) =0. The following space of test functions plays a key role in this paper, which is an equivalent variant of the space of test functions in [23, Definition 2.3]; see also [24]. Definition 2.5. Let x, r 0, ), β 0, ] and γ 0, ). A function ϕ on is said to be a test function of type x,r,β,γ) if r γ i) ϕx) C μbx,r+dx, x ))) r+dx,x)) for all x ; ii) ϕx) ϕy) C dx,y) ) β r γ r+dx,x) μbx,r+dx,x ))) r+dx,x)) for all x, y satisfying dx, y) r + dx,x))/2. We denote by Gx,r,β,γ) the set of all test functions of type x,r,β,γ). Ifϕ Gx,r,β,γ), we define its norm by ϕ Gx,r,β,γ) := inf{c :i)and ii) hold}. The space Gx,r,β,γ) is called the space of test functions. Remark 2.6. i) If μbx, r + dx, x ))) is replaced by V r x )+Vx,x) in Definition 2.5, then Gx,r,β,γ) was introduced in [23, Definition 2.3]; see also [24]. ii) Notice that μbx, r + dx, x ))) V r x )+Vx,x) by 2.2). This implies that both

6 2258 Loukas GRAFAKOS et al. spaces of test functions in Definition 2.5 and [23, Definition 2.3] are equivalent and with equivalent norms. Moreover, see Remark 2.9 iii) for the advantage of Definition 2.5. Throughout the paper, we fix x. Let Gβ,γ) :=Gx,,β,γ). It is easy to see that for any x 2 and r>0, we have Gx 2,r,β,γ)=Gβ,γ) with equivalent norms but with constants depending on x, x 2 and r). Furthermore, Gβ,γ) is a Banach space. For any given ɛ 0, ], let G0 ɛ β,γ) be the completion of the space Gɛ, ɛ) in Gβ,γ) when β, γ 0,ɛ]. Obviously G0ɛ, ɛ ɛ) = Gɛ, ɛ). Moreover, ϕ G0β,γ) ɛ if and only if ϕ Gβ,γ) and there exists {φ i } i N Gɛ, ɛ) such that ϕ φ i Gβ,γ) 0asi. If ϕ G0β,γ), ɛ define ϕ G ɛ 0 β,γ) := ϕ Gβ,γ). Obviously G0β,γ) ɛ is a Banach space and ϕ G ɛ 0 β,γ) = lim i φ i Gβ,γ) for the above chosen {φ i } i N. We define { } Gx,r,β,γ):= ϕ Gx,r,β,γ): ϕx) dμx) =0, which is called the space of test functions with mean zero. The space G 0β,γ) ɛ is defined to be the completion of Gɛ, ɛ) in Gβ,γ) whenβ, γ 0,ɛ]. Moreover, if ϕ G 0 ɛ β,γ), we then define ϕ Gɛ 0 β,γ) := ϕ Gβ,γ). The notation G0 ɛβ,γ)) denotes the dual space of G0 ɛ β,γ), that is, the set of all linear functionals f from G0β,γ) ɛ toc with the property that there exists a positive constant C such that for all ϕ G0 ɛβ,γ), f,ϕ C ϕ Gβ,γ). Wedenoteby f,ϕ the natural pairing of elements f G0β,γ)) ɛ and ϕ G0β,γ). ɛ Similarly, G 0β,γ)) ɛ denotes the set of all bounded linear functionals from G 0 ɛ β,γ) toc. The following cube constructions, which provide an analogue of the grid of Euclidean dyadic cubes on spaces of homogeneous type, were given by Christ [38]. Lemma 2.7. Let be a space of homogeneous type. Then there exists a collection {Q k α : k Z, α I k } of open subsets, where I k is some index set, and constants δ 0, ) and C 5, C 6 > 0 such that i) μ \ α Q k α )=0for each fixed k and Qk α Qk β = if α β; ii) for any α, β, k, l with l k, eitherq l β Qk α or Q l β Qk α = ; iii) for each k, α) and each l<k,thereisauniqueβ such that Q k α Ql β ; iv) diam Q k α) C 5 δ k ; v) each Q k α contains some ball Bz k α,c 6 δ k ),wherez k α. In fact, we can think of Q k α as being a dyadic cube with diameter rough δ k centered at z k α. In what follows, for simplicity, we may assume that δ =/2; see [23, p. 25] or [0, pp ] for how to remove this restriction. For any dyadic cube Q and any function g, set m Q g) := gx) dμx). μq) Let ɛ, ɛ 2 and {S k } k Z be as in Definition 2.3 and ɛ 0,ɛ ɛ 2 ). Observe that for any fixed x, S k x, ) G ɛ 0β,γ) withβ, γ 0,ɛ). In what follows, for any k Z, f G ɛ 0β,γ)) and x, we define S k f)x) := f,s k x, ). Definition 2.8. Let ɛ 0, ], ɛ 2 > 0, ɛ 3 > 0, ɛ 0,ɛ ɛ 2 ) and {S k } k Z be an ɛ,ɛ 2,ɛ 3 )- AOTI.Letp 0, ], α 0, ) and f G ɛ 0β,γ)) with some β, γ 0,ɛ). For any x, Q

7 Maximal function characterizations of Hardy spaces 2259 the grand maximal function of f is defined by f x) :=sup{ f,ϕ : ϕ G ɛ 0 β,γ), ϕ Gx,r,β,γ) for some r>0}; 2.3) the nontangential maximal function of f is defined by and the dyadic maximal function of f is defined by M α f)x) :=sup sup S k f)y) ; 2.4) k Z dx, y) α2 k M d f)x) := where {Q k α} k Z,α Ik is as in Lemma 2.7. The corresponding Hardy spaces are defined, respectively, by sup m Q k α S k f) )χ Q k α x), 2.5) k Z,α I k H,p ):={f G ɛ 0β,γ)) : f L p ) < }, H p α ):={f Gɛ 0 β,γ)) : M α f) Lp ) < } and H p d ):={f Gɛ 0 β,γ)) : M d f) Lp ) < }. Moreover, we define f H,p ) := f L p ), f H p α ) := M α f) L p ) and f H p d ) := M d f) L p ). Remark 2.9. i) Lemma 2.7 implies that M d f)x) can be equivalently defined by M d f)x) :=sup{m Q k α S k f) ) : k Z, α I k,q k α x}, 2.6) where {Q k α} k Z,α Ik is as in Lemma 2.7. ii) Let {S k } k Z and α be as in Definition 2.8. Observing that there exists a positive constant C α such that sup k Z sup dx, y) α2 k S k y, ) Gx,2 k,ɛ,ɛ 2) =: C α <, we then obtain that for all x, M α f)x) C α f x). iii) For any λ 0, ), the set Ω := {x : f x) >λ} is open, which is a key fact used in Section4. Infact,ifx 0 Ω, then there exists ϕ G ɛ 0 β,γ) such that ϕ Gx 0,r,β,γ) forsome r > 0 and f,ϕ > λ.let η be a sufficiently small positive quantity such that f,ϕ > λ+η). For any x Bx 0,r r/ + η) /γ ), choose s 0, ) satisfying r/ + η) /γ <s<r dx, x 0 ). It is easy to verify that ϕ Gx,s,β,γ) < +η. Thusf x) >λfor all x Bx 0,r r/ + η) /γ ), which implies that Ω is open. iv) It is proved in Corollary 3. below that if p, ], then H,p ), H p α )andhp d ) are all equivalent to L p ). v) Let ɛ 0, ) and p n/n+ɛ), ]. It is proved in Remark 3.5 below that if n/p ) < β, γ < ɛ, then the definitions of Hardy spaces H,p ), Hα p )andh p d ) are independent of the choices of AOTI s and G0 ɛβ,γ)). Moreover, they coincide with each other. 3 Maximal function characterizations The following basic properties concerned with RD-spaces are useful throughout the paper, which are proved in [23, Lemma 2., Propositions 2.2 and 3.]. In what follows, for any f L loc ), the centered Hardy-Littlewood maximal function Mf) is defined by that for all x, Mf)x) :=sup fy) dμy). r>0 μbx, r)) Bx, r)

8 2260 Loukas GRAFAKOS et al. Lemma 3.. Let δ>0, a>0, r>0 and θ 0, ). Then, i) For all x, y and r>0, μbx, r + dx, y))) μby, r + dx, y))). ii) If x, x, x satisfy dx, x ) θr + dx, x )), thenr + dx, x ) r + dx,x ) and μbx, r + dx, x ))) μbx,r+ dx,x ))). iii) a>η 0. μbx,r+dx,y))) r r+dx,y) )a dx, y) η dμx) Cr η uniformly in x and r > 0 if iv) For all f L loc ) and x, dx, y)>δ V x,y) dx,y) fy) dμy) CMf)x) uniformly in δ>0, f L loc a ) and x. v) Let {S k } k Z be an ɛ,ɛ 2,ɛ 3 )- AOTI with ɛ 0, ], ɛ 2 > 0 and ɛ 3 > 0. Then S k is bounded on L p ) for p [, ] uniformly in k Z. For any f L p ) with p, ), S k f) L p ) 0 as k. For any f L p ) with p [, ), S k f) f L p ) 0 as k. We introduce the following inhomogeneous approximation of the identity on,d,μ), which is a slight variant of the one in [23]. Definition 3.2. Let ɛ 0, ], ɛ 2 > 0, ɛ 3 > 0 and l 0 Z. A sequence {S k } k=l 0 of linear operators is said to be an inhomogeneous approximation of the identity of order ɛ,ɛ 2,ɛ 3 ) for short, ɛ,ɛ 2,ɛ 3 )-l 0 - AOTI ), if S k satisfies i) through v) of Definition 2.3 for any k {l 0,l 0 +,...}. In the following, for k Z and I k,wedenotebyq k,ν, ν =, 2,...,Nk, ), the set of all cubes Q k+j0 Qk,whereQk is the dyadic cube as in the Lemma 2.7 and j 0 is a positive integer satisfying δ a 2 j0 C 5 < /3. 3.) Denote by z k,ν the center of Q k,ν and y k,ν any point of Q k,ν. For any ɛ,ɛ 2,ɛ 3 )-l 0 -AOTI, following the procedure of the proof for [23, Theorem 4.6], we obtain a new inhomogeneous discrete Calderón reproducing formula, which starts from any l 0 Z. The details of the proof are omitted by similarity. This inhomogeneous discrete Calderón reproducing formula plays a key role in this paper, which has independent interest. Theorem 3.3. Let ɛ 0, ], ɛ 2 > 0, ɛ 3 > 0, ɛ 0,ɛ ɛ 2 ) and l 0 Z. Let{S k } k Z be an ɛ,ɛ 2,ɛ 3 )-l 0 - AOTI. Set D l0 := S l0 and D k := S k S k for k l 0 +. Then for any fixed j 0 satisfying 3.) large enough, there exists a family of functions { D k x, y)} k=l 0 such that for any fixed y k,ν Q k,ν with k l 0 +, I k and ν {, 2,...,Nk, )}, andallf G0β,γ)) ɛ with β, γ 0,ɛ), fx) = I l0 + = I l0 Nl 0,) ν= k=l 0+ I k Nl 0,) ν= Q l 0,ν Q l 0,ν Nk,) ν= D l0 x, y) dμy)d l0,ν, f) μq k,ν ) D k x, y k,ν )D k f)y k,ν ) D l0 x, y) dμy)d l0,ν, f)

9 Maximal function characterizations of Hardy spaces k=l 0+ I k Nk,) ν= μq k,ν ) D k x, y k,ν )D k,ν, f), where the series converges in G0 ɛβ,γ)),andfork l 0, I k and ν =, 2,...,Nk, ), D k,ν, z) is the corresponding integral operator with the kernel Dk,ν, z) :=m Q k,ν S k,z)). Moreover, Dk for k l 0 satisfies i) and ii) of Definition 2.3 with ɛ and ɛ 2 replaced by any ɛ [ɛ, ɛ ɛ 2 ),and when k = l 0 ;=0when k>l 0. D k x, y) dμx) = D k x, y) dμy) = Remark 3.4. Let l 0 Z. From the proof of Theorem 3.3, we obtain that the constant C, that appears in i) and ii) of Definition 2.3 for { D k } k=l 0, depends on j 0 and ɛ, but not on l 0. This observation plays a key role in applications. Moreover, a continuous version of Theorem 3.3 also holds by following the proof of [23, Theorem 3.4]. Now we show that all Hardy spaces defined in Definition 2.8 are equivalent to L p )when p, ]. We begin with some technical lemmas. Applying Theorem 2.3 of [23] yields the following result; we omit the details. Lemma 3.5. Let ɛ 0, ], ɛ 2 > 0, ɛ 3 > 0, ɛ 0,ɛ ɛ 2 ), β, γ 0,ɛ] and {S k } k Z be an ɛ,ɛ 2,ɛ 3 )-AOTI. Then there exists a positive constant C such that for all k 0 and ϕ Gβ,γ), S k ϕ) Gβ,γ) C ϕ Gβ,γ). Lemma 3.6. Let all the notation be as in Lemma 3.5. Then for all ϕ Gβ,γ) and β 0,β), lim k S k ϕ) =ϕ in Gβ,γ). Proof. Notice that Lemma 3.5 implies that S k ϕ) Gβ,γ) Gβ,γ) for any k 0. It remains to show that lim k S k ϕ) ϕ Gβ,γ) = 0. Set W := {z : dz,x) + dx, x ))/2} and W 2 := {z : dz,x) > + dx, x ))/2}. Using v) of Definition 2.3, we write S k ϕ)x) ϕx) S k x, z) ϕz) ϕx) dμz) W + ϕx) S k x, z) dμz)+ S k x, z) ϕz) dμz) W 2 W 2 =: Z +Z 2 +Z 3. The size condition of S k together with the regularity of ϕ and the assumption β<ɛ 2 together with Lemma 3. iii) yield that ) γ Z 2 kβ. μbx, +dx, x ))) +dx, x ) By the definition of W 2, the size conditions of S k and ϕ, and a procedure similar to the estimation of Z,weobtain ) γ Z 2 2 kγ. μbx, +dx, x ))) +dx, x )

10 2262 Loukas GRAFAKOS et al. Notice that ϕx) dμx). This together with the size condition of S k and the definition of W 2 yields that Therefore, for all x, ) γ Z 3 2 kγ. μbx, +dx, x ))) +dx, x ) S k ϕ)x) ϕx) 2 kβ γ) μbx, +dx, x ))) ) γ. 3.2) +dx, x ) For x, x satisfying dx, x ) + dx, x ))/2, by Lemma 3.5 and the regularity of ϕ, [S k ϕ)x) ϕx)] [S k ϕ)x ) ϕx )] dx, x ) β ) +dx, x ) μbx, +dx, x ))) ) γ ; +dx, x ) on the other hand, by 3.2) and the assumption dx, x ) + dx, x ))/2, [S k ϕ)x) ϕx)] [S k ϕ)x ) ϕx )] ) γ 2 kβ γ). μbx, +dx, x ))) +dx, x ) Then the geometric mean between the last two formulae above implies that [S k ϕ)x) ϕx)] [S k ϕ)x ) ϕx )] dx, x 2 kβ γ) σ) ) σβ ) +dx, x ) μbx, +dx, x ))) ) γ, 3.3) +dx, x ) where σ 0, ). The estimates 3.2) and 3.3) imply the desired conclusion, which completes the proof of Lemma 3.6. Proposition 3.7. Let ɛ 0, ], ɛ 2 > 0, ɛ 3 > 0, ɛ 0,ɛ ɛ 2 ), β 0,ɛ), γ 0,ɛ] and {S k } k Z be an ɛ,ɛ 2,ɛ 3 )- AOTI.Thenlim k S k ϕ) =ϕ in G0 ɛβ,γ) for all ϕ Gɛ 0 β,γ); and lim k S k f) =f in G0β,γ)) ɛ for all f G0β,γ)) ɛ. Proof. We may assume that k N. If ϕ G0 ɛ β,γ), then ϕ Gβ,γ) andthereexists {φ j } j N Gɛ, ɛ) such that ϕ φ j Gβ,γ) 0asj. By Lemma 3.5, we know S k φ j ) Gɛ, ɛ) and lim S kϕ) S k φ j ) Gβ,γ) lim ϕ φ j Gβ,γ) =0, j 0 j 0 which implies that S k ϕ) G0β,γ). ɛ For any j, k N, by Lemma 3.5 again, we have S k ϕ) ϕ Gβ,γ) S k φ j ) φ j Gβ,γ) + φ j ϕ Gβ,γ), which together with Lemma 3.6 here we require β<ɛ) implies that lim k S kϕ) ϕ Gβ,γ) =0. A standard duality argument yields the second conclusion of Proposition 3.7. Proposition 3.8. Let H, ) be as in Definition 2.8. Then H, ) L ).

11 Maximal function characterizations of Hardy spaces 2263 Proof. Let ɛ 0, ], ɛ 2 > 0, ɛ 3 > 0and{S k } k Z be an ɛ,ɛ 2,ɛ 3 )- AOTI. For any given f H, ), by the definition, we have f L ) <. Notice that 2.3) and Definition 2.3 imply that sup S k f) L ) f L ). k Z Thus, {S k f)} k Z is a bounded set in L ). Thus by [39, Theorem III.C.2], {S k f)} k Z is relatively weakly compact in L ). The Eberlein- Smulian theorem cf. [39, II.C]) implies that there exists a subsequence {S kj f)} j N of {S k f)} k Z which converges weakly in L ), and hence in G ɛ 0β,γ)). This together with Proposition 3.7 implies that f L ), which completes the proof of Proposition 3.8. Proposition 3.9. Let p 0, ] and all the notation be as in Definition 2.8. Then for any given α 0, ), there exists a positive constant C such that for all f L p ) and all x, f x) CMf)x) and M d f)x) CMM α f))x). Furthermore, when p, ], there exists a positive constant C such that for all f L p ), M d f) Lp ) C M α f) Lp ) C f Lp ) C f Lp ). Proof. For any x and ϕ G0β,γ) ɛ satisfying ϕ Gx,r,β,γ) forsomer>0, by the size condition of ϕ and Lemma 3. iv), we have f,ϕ fz) dμz) dz, x) r μbx, r)) r γ + fz) dμz) Mf)x), 3.4) μbz,dz,x))) dx, z) γ dz, x)>r which further implies that f x) Mf)x) for all x. For any x,ifxiscontained in some dyadic cube Q k α, then by Lemma 2.7, Bz k α,c 62 k ) Q k α Bx, C 52 k ) Bz k α, 2C 52 k ). Thus μq k α) μbz k α,c 6 2 k )) μbx, C 5 2 k )) by 2.2). From this and 2.6), it follows that for all x, M d f)x) sup k Z { μbx, C 5 2 k )) Bx, C 52 k ) } M α f)y) dμy) MM α f))x). This together with Remark 2.9 ii) and the L p )-boundedness of M with p, ] cf. [5]) then yields the desired norm inequalities, which completes the proof of Proposition 3.9. Theorem 3.0. Let p, ]. With the same notation as in Definition 2.8, we have H p d )=Lp ) in the following sense : there exists a positive constant C independent of f such that if f L p ), then f H p d ) and f H p d ) C f L p ); conversely, for any f H p d ), there exists a function f L p ) such that for all ϕ G0β,γ), ɛ f,ϕ = fx)ϕx) dμx). Moreover, there exists a positive constant C independent of f and x such that f Lp ) C f H p d ) and fx) Cf x) almost everywhere.

12 2264 Loukas GRAFAKOS et al. Proof. The fact L p ) G0 ɛβ,γ)) and Proposition 3.9 imply that L p ) H p d ). Conversely, if f H p d ), then we first show that for all ϕ Gɛ 0β,γ), f,ϕ f H p d ) ϕ L p ), 3.5) where /p +/p =. To this end, fix j 0 as in 3.) large enough. Then applying Theorem 3.3 with l 0 = j Z yields that in G ɛ 0 β,γ)). fx) = Nj, ) I j + ν= k=j+ I k Moreover, for any Q k,ν Q j,ν Nk,) ν= y k,ν D j x, y) dμy)d j,ν, f) μq k,ν ) D k x, y k,ν )D k,ν, f) 3.6) with k j, by the definition, we have that Q k,ν Q k for some I k. From this and 2.2), we deduce that μq k ) 2nj0 μq k,ν ). Thus, for any y k,ν Q k,ν, D k,ν, f) 2nj0 μq k ) Q k S k f)z) dμz) M d f)y k,ν ). 3.7) For any ϕ G0 ɛβ,γ), Proposition 3.7 implies that S jϕ) G0 ɛ β,γ) for any j Z. By this, 3.6) and 3.7), we obtain that for all ϕ G0 ɛ β,γ), j Z and yj,ν Q j,ν, f,s j ϕ Nj, ) I j + ν= k=j+ I k Q j,ν Nk,) ν= D j S jϕ)y) dμy) M df)y j,ν μq k,ν ) ) D k S jϕ)y k,ν ) M d f)y k,ν ), 3.8) where and in what follows, D k denotes the integral operator with the kernel D k x, y) = D k y, x) for all x, y. For any k, j Z, by similar arguments as in [23, Lemma 3.8] when k = j and [23, Lemma 3.] when k>j, respectively, we obtain that for any ɛ 0,ɛ ɛ 2 ), D k S 2 jɛ jy, z) 2 k j)ɛ. 3.9) V 2 j y)+v 2 j z)+vy, z) 2 j + dx, y)) ɛ An argument similar to the proof of 3.4) together with 3.9) yields that for any k j and x, D ks j ϕ)x) 2 k j)ɛ Mϕ)x). 3.0) Moreover, it follows directly from 3.9) and Lemma 3. iii) that for all k j, DkS j ϕ)y) dμy) 2 k j)ɛ ϕ L ). 3.) When p, ), by 3.8) together with the arbitrariness of y k,ν, 3.0), Hölder s inequality and the L p )-boundedness of M with p, ], we further have f,s j ϕ Dk S jϕ)y) M d f)y) dμy) k=j 2 k j)ɛ Mϕ) L p ) M df) L p ) ϕ L p ) f H p d ). 3.2) k=j

13 Maximal function characterizations of Hardy spaces 2265 When p =, this can be deduced directly from 3.8) and 3.). For any ϕ G0 ɛ β,γ), by Proposition 3.7, we have lim j S j ϕ) =ϕ in G0β,γ), ɛ which together with 3.2) implies 3.5). From 3.5) and the density of G ɛ 0 β,γ) inlp ) see [23, Corollary 2.]), it follows that f uniquely extends to a bounded linear functional on L p ), where /p +/p =. By the wellknown Riesz representation theorem, there exists a unique function f L p ) such that for all ϕ L p ), f,ϕ = fx)ϕx) dμx), and moreover, f L p ) f H p d ). Lemma 3. v) together with the Riesz lemma further implies that fx) = lim ki S ki fx) almosteverywhere for some {k i } i N. This combined with 2.3) yields that for almost every x, which completes the proof of Theorem 3.0. fx) lim k i S k i fx) f x) =f x), Combining Proposition 3.9 and Theorem 3.0 yields the following conclusion. Corollary 3.. Let all the notation be as in Definition 2.8 and p, ]. ThenH p d )= Hα p )=H,p )=L p ) with equivalent norms. To prove the equivalence of Hardy spaces defined as above when p, we need the following technical lemma see [23, Lemma 5.2]). Lemma 3.2. Let ɛ>0, k, k Z, andy k,ν be any point in Q k,ν for I k and ν =, 2,...,Nk, ). If r n/n + ɛ), ], then there exists a positive constant C depending on r such that for all a k,ν I k Nk,) ν= C and all x, μq k,ν ) C2 [k k) k]n /r) V 2 k k)x)+v x, y k,ν { M where C is also independent of k, k, and ν. I k ) Nk,) ν= 2 k k)ɛ 2 k k) + dx, y k,ν )) ɛ ak,ν ) /r a k,ν r χ Q k,ν x)}, Theorem 3.3. Let all the notation be as in Definition 2.8. Let p n/n + ɛ), ] and β, γ n/p ),ɛ). ThenH p d )=Hp α )=H,p ) with equivalent quasi-norms. Proof. Given any α 0, ), by Remark 2.9 ii), H,p ) H p α )andforallf H,p ), f H p α ) f H,p ). To prove H p α ) H,p ), we need only to show that for any κ n/[n +β γ)], ], there exists a positive constant C κ such that for all f H p α )andx, f x) C κ {M[M α f)] κ )x)} /κ. 3.3) Suppose now that 3.3) holds for some κ n/[n +β γ)],p). Then by this and the L p )- boundedness of M with p, ], we obtain that for all f H p α ), f H,p ) = f L p ) {M[M α f)] κ )} /κ L p ) M α f) L p ) = f H p α ), which implies that H p α ) H,p ).

14 2266 Loukas GRAFAKOS et al. To show 3.3), fix any x. Then for any ϕ G ɛ 0 β,γ) satisfying ϕ Gx,r,β,γ) with some r>0, we choose l 0 Z such that 2 l0 r<2 l0+. Then it is easy to verify that there exists a positive constant C which is independent of x and l 0 such that ϕ Gx,2 l 0,β,γ) C. Choose j 0 N satisfying 3.) large enough and C 5 2 j0 <α. Applying Theorem 3.3 with such a j 0, we obtain that for any y k,ν f,ϕ + I l0 Q k,ν, Nl 0,) ν= k=l 0+ I k Q l 0,ν Nk,) ν= D l 0 ϕ)y) dμy)d l0,ν, f) μq k,ν ) D k ϕ)yk,ν )D k f)y k,ν ). 3.4) Since diam Q k,ν ) C 5 2 k j0 <α2 k for all k l 0, I k and ν =, 2,...,N,k), then for any y l0,ν Q l0,ν, D l0,ν, f) = μq l0,ν ) and when k>l 0, for any y k,ν D k f)y k,ν Q k,ν, Q l 0,ν ) S k f)y k,ν S l0 f)w) dμw) M αf)y l0,ν ); 3.5) ) + S k f)y k,ν ) 2M α f)y k,ν ). 3.6) By an argument similar to [23, Lemma 3.], we obtain that for any k l 0, y and ɛ 0,β γ), D k ϕ)y) 2 l0γ 2 k l0)ɛ μby, 2 l0 + dy, x))) 2 l0 + dy, x)) γ. 3.7) Furthermore, the triangle inequality of d and 2.2) imply that for any y l0,ν, z Q l0,ν, and 2 l0 + dy l0,ν,x) 2 l0 + dz,x) 3.8) μby l0,ν, 2 l0 + dy l0,ν,x))) μbz,2 l0 + dz,x))). 3.9) From 3.4) together with the estimates 3.5) through 3.9), Lemma 3.2 and the arbitrariness of y k,ν in Q k,ν, we deduce that for any κ n/n +β γ)), ], f,ϕ k=l 0 2 k l0)[ɛ {M +n /κ)] I k Nk,) ν= [M α f)] κ χ Q k,ν ) x)} /κ. 3.20) Notice that M I k Nk,) ν= [M α f)] κ χ Q k,ν ) x) =M[M α f)] κ )x). 3.2) Therefore, for any κ n/n + β γ),ɛ), if we choose ɛ 0,β γ) withɛ >n/κ ), then combining 3.20) with 3.2) yields 3.3). Thus, for any given α 0, ), H,p )=H p α ) with equivalent quasi-norms.

15 Maximal function characterizations of Hardy spaces 2267 Instead of 3.4), 3.5) and 3.6), respectively, by 3.6) and 3.7), repeating the proof for H p α ) H,p ) then yields that H p d ) H,p ). By Lemma 2.7 iv) and 2.6), we have M d f)x) M C5 f)x), which implies that H p C 5 ) H p d ). This together with H p C 5 )=H,p ) further implies that H,p ) H p d ). Thus, H,p )=H p d )with equivalent quasi-norms, which completes the proof of Theorem 3.3. Using the Calderón reproducing formula in Theorem 3.3, we can verify that definitions of Hardy spaces are independent of the choices of underlying spaces of distributions; see [23, Proposition 5.3] for some details. Proposition 3.4. With the notation of Definition 2.8, let α 0, ) and p n/n + ɛ), ]. If f G0 ɛβ,γ )) with n/p ) <β, γ <ɛ, 3.22) and f H p α ) <, thenf G ɛ 0 β 2,γ 2 )) for every β 2 and γ 2 satisfying 3.22). Remark 3.5. Let ɛ 0, ) and p n/n + ɛ), ]. Then, i) For any given β and γ satisfying 3.22), Theorem 3.3 implies that the definitions of Hα p )andhp d ) are independent of the choices of any α 0, ) andanyɛ,ɛ 2,ɛ 3 )- AOTI {S k } k Z satisfying ɛ ɛ 2 ) >ɛ, ɛ andɛ 3 > 0. ii) Theorem 3.3 and Proposition 3.4 imply that the definitions of H,p ), Hα p )with α 0, ) andh p d ) are also independent of the choices of Gɛ 0 β,γ)) with β and γ satisfying 3.22). iii) In the sequel, given p n/n +), ], when we mention the Hardy spaces H,p ), Hα p )withα 0, ), and Hp d ), we will assume that ɛ n/p ), ) is chosen so that p n/n + ɛ), ] and these Hardy spaces are defined via some G0β,γ)) ɛ with β and γ satisfying 3.22), and some ɛ,ɛ 2,ɛ 3 )- AOTI {S k } k Z with ɛ ɛ 2 ) >ɛ, ɛ andɛ 3 > 0. By Theorem 3.3, these Hardy spaces coincide with each other. By an argument similar to the proof of [6, Lemma 2.8)] see also [9, Theorem 2.]), we can verify the following conclusion, we omit the details. Proposition 3.6. Let p n/n +), ]. Then the space H,p ) is a complete quasi- Banach space. 4 The atomic decomposition The procedure to obtain the atomic decomposition for the Hardy space H,p )whenp n/n +), ] is quite similar to that presented in [6] see also [9]). To shorten the presentation of this paper, we only give an outline by beginning with the following notions of atoms and atomic Hardy spaces. Definition 4.. Let p 0, ] and q [, ] p, ]. Afunctiona L q ) is said to be a p, q)-atom if A) supp a Bx 0,r) for some x 0 and r>0; A2) a Lq ) [μbx 0,r))] /q /p ; A3) ax) dμx) =0. Definition 4.2. Let p 0, ] and q [, ] p, ]. Let ɛ 0, ) and β, γ 0,ɛ). The distribution f G0β,γ)) ɛ is an element of H p, q at ), if there exist {λ j } j N C and

16 2268 Loukas GRAFAKOS et al. p, q)-atoms {a j } j N such that f = j N λ ja j in G ɛ 0 β,γ)) and j N λ j p <. Moreover, we define { ) /p f H p, q at Rn ) := inf λ j }, p j N where the infimum is taken over all the decompositions of f as above. Remark 4.3. ),ɛ). Since each p, )-atom is a p, q)-atom, H p, at i) Let ɛ 0, ), p n/n + ɛ), ], q [, ] p, ] andβ, γ n/p ) H p, q at ). We will further show in ) H,p )andh,p ) H p, at ). This implies Theorems 4.5 and 4.6 below that H p, q at that H p, at )=H p, q at ). ii) We simply write H p at ) instead of Hp, at ) without confusion. An argument similar to the proof of [6, Lemma 2.3)] yields the following lemma, we omit the details. Lemma 4.4. Let ɛ 0, ], p n/n + ɛ), ], q [, ] p, ], β n/p ),ɛ] and γ 0,ɛ]. There exists a positive constant C p, q such that for all h L q ) with support contained in Bx 0,r 0 ) with x 0 and r 0 > 0, and hx) dμx) =0, [h x)] p dμx) C p, q [μbx 0,r 0 ))] p/q h p L q ). Lemma 4.4 together with a standard argument yields the following proposition. Proposition 4.5. Let all the notation be as in Lemma 4.4. Then H p, q at ) H,p ); moreover, there exists a positive constant C such that for all f H p, q at ), f H,p ) C f H p, q at ). The rest of this section is devoted to showing that H,p ) H p at ) by following the procedure of [6]. The following lemma when Ω is a bounded set i. e., Ω is contained in some ball of ) was proved in [4, pp. 70 7] and in [5, Theorem 3.2], although the current version was also claimed in [4, p. 70] without a proof; see also [6, p. 277] for another variant. A detailed proof can be given by borrowing some ideas from [3, pp. 5 6], we omit the details. Lemma 4.6. Let Ω be an open proper subset of and let dx) :=inf{dx, y) : y/ Ω}. For any given C, letrx) :=dx)/2c). Then there exist a positive number M, which depends only on C and C 3 but independent of Ω, and a sequence {x k } k such that if we denote rx k ) by r k,then i) {Bx k,r k /4)} k are pairwise disjoint ; ii) k Bx k,r k )=Ω; iii) for every given k, Bx k,cr k ) Ω; iv) for every given k, x Bx k,cr k ) implies that Cr k <dx) < 3Cr k ; v) for every given k, thereexistsay k / Ω such that dx k,y k ) < 3Cr k ; vi) for every given k, the number of balls Bx i,cr i ) whose intersections with the ball Bx k,cr k ) are non-empty is at most M. Remark 4.7. If Ω is an open bounded subset of and {r k } k in Lemma 4.6 is an infinite sequence of positive numbers, then by the proofs in [5, pp ] and [4, p. 69], we further have that lim k r k =0.

17 Maximal function characterizations of Hardy spaces 2269 Using Lemma 4.6, similarly to the proof of [6, Lemma 2.6)], we obtain the following technical lemma. Lemma 4.8. Let Ω be an open subset of with finite measure. Consider the sequence {x k } k and {r k } k given in Lemma 4.6 for C =5. Then there exist non-negative functions {φ k } k satisfying : i) for any given k, 0 φ k, supp φ k Bx k, 2r k ) and k φ k = χ Ω ; ii) for any given k and x Bx k,r k ), φ k x) /M ; iii) there exists a positive constant C independent of Ω such that for all k and all ɛ 0, ], φ k Gxk,r k,ɛ,ɛ) CV rk x k ). From Lemma 4.8 and an argument similar to the proof of [6, Lemma 3.)], we deduce the following conclusion. Lemma 4.9. Let ɛ 0, ], β, γ 0,ɛ], and{φ k } k be the partition of unity as in Lemma 4.8 associated with some open set Ω. Then for any given k, the linear operator [ Φ k ϕ)x) :=φ k x) ] φ k z) dμz) [ϕx) ϕz)]φ k z) dμz) is bounded on G0 ɛ β,γ) with an operator norm depending on k. Lemma 4.0. Let β>0, q n/n + β), ) and M N. Then there exists a positive constant C depending only on q, β and M such that for any given sequences of points {x k } k and any positive numbers {r k } k satisfying that any point in belongs to no more than M balls of {Bx k,r k )} k,then [ ) β ] q μbx k,r k )) r k ) dμx) Cμ Bx k,r k ). 4.) μbx, r k + dx, x k ))) r k + dx k,x) k k Proof. By 2.2) and the definition of M, wehave ) /n r k r k + dx k,x) μbx k,r k )) [Mχ Bxk,rk))x)] /n. μbx, r k + dx, x k ))) By this, the assumption qn + β)/n >, the Fefferman-Stein inequality on RD-spaces see [40]) and the finite intersection property of balls {Bx k,r k )} k, we obtain that the left-hand side of 4.) is dominated by a multiple of [ ] q Mχ Bxk,r k ))x)) n+β)/n dμx) k [ ] q ) χ Bxk,r k )x)) n+β)/n dμx) μ Bx k,r k ), k which completes the proof of Lemma 4.0. Let ɛ 0, ), p n/n + ɛ), ], β, γ n/p ),ɛ)andh,p ) be as in 2.3). For f H,p )andt 0, ), set Ω := {x : f x) >t}. By Remark 2.9 iii), Ω is open. Obviously μω) <. Denote by {φ k } k the partition of unity in Lemma 4.8 associated to Ω. k

18 2270 Loukas GRAFAKOS et al. Let {Φ k } k be the corresponding linear operator in Lemma 4.9. For any ϕ G0 ɛ β,γ), define the distribution b k by setting b k,ϕ := f,φ k ϕ). 4.2) By following an approach in [6, Lemma 3.2)] or [9, Lemma 3.7], we obtain the following Calderón-Zygmund type decomposition. To limit the length of the paper, we omit the details. In what follows, for any set E,wedenote\E by E. Proposition 4.. for all k and x, and With the previous notation, there exists a positive constant C such that ) β b V rk x k ) r k kx) Ct χ μbx k,r k + dx, x k ))) r k + dx k,x) Bxk,0rk) x) + Cf x)χ Bxk,0r k )x) 4.3) [b k x)]p dμx) C [f x)] p dμx); 4.4) Bx k,0r k ) moreover, the series k b k converges in G ɛ 0 β,γ)) to a distribution b satisfying and b x) Ct k ) β V rk x k ) r k + Cf x)χ Ω x) 4.5) μbx k,r k + dx, x k ))) r k + dx k,x) the distribution g := f b satisfies that g G ɛ 0 β,γ)) and g x) Ct k [b x)] p dμx) C [f x)] p dμx); 4.6) Ω ) β V rk x k ) r k + Cf x)χ μbx k,r k + dx, x k ))) r k + dx k,x) Ω x). 4.7) Applying Proposition 4., Lemma 4.0 and Corollary 3., and following an argument similar to [6, Theorem 3.34)], we obtain the following density result on H,p ). Proposition 4.2. Let ɛ 0, ), p n/n + ɛ), ], β, γ n/p ),ɛ) and H,p ) be defined via the distribution space G ɛ 0 β,γ)) as in Definition 2.8. If q, ), thenl q ) H,p ) is dense in H,p ). Combining Lemmas 4.6, 4.8, 4.9 and Proposition 4.2, similarly to the proof of [6, Lemma 3.36)], we obtain the following proposition. Proposition 4.3. Let ɛ 0, ), p n/n + ɛ), ], β, γ n/p ),ɛ), q, ) and f L q ) H,p ). Assume that there exists a positive constant C such that for all x, fx) Cf x). With the same notation as in Proposition 4., then there exists a positive constant C independent of f, k and t such that i) if m k := [ φ kξ) dμξ)] fξ)φ kξ) dμξ), then m k Ct for all k; ii) if b k := f m k )φ k,then supp b k Bx k, 2r k ) and the distribution on G0 ɛ β,γ) induced by b k coincides with b k in Proposition 4.; iii) the series k b k converges in L q ). It induces a distribution on G0 ɛ β,γ) which coincides with b in Proposition 4. and is still denoted by b. Moreover, supp b Ω t ;

19 Maximal function characterizations of Hardy spaces 227 iv) set g := f b, theng = fχ Ω + k m kφ k and for all x, gx) Ct. Moreover, g induces a distribution on G ɛ 0β,γ) which coincides with g appearing in Proposition 4.. Remark 4.4. a) If f does not satisfy fx) Cf x) for all x, then by Theorem 3.0, there exists a function h such that f and h induce the same distribution and hx) Cf x) = Ch x) for all x. Therefore, we can replace f by h since we only need to consider the behavior of f as a distribution not as a function. b) Denote by C ) the space of continuous functions on.iff C ), by Proposition 4.3 ii), then b k C ) for all k. Applying Lemma 4.6 vi) to k b k, we obtain that for any given x, k b kx) has only finite terms when it is restricted to some small neighborhood of x, which further implies that b C ). Lemma 4.5. Let ɛ 0, ), q n/n + ɛ), ), p q, ] and β, γ n/p ),ɛ). If h L 2 ) satisfying hx) for all x and h L q ), then there exist {λ k } k N C and p, )-atoms {a k } k N such that h = k N λ ka k in G ɛ 0 β,γ)) and almost everywhere. Moreover, there exists a positive constant C independent of h such that k N λ k p C h q L q ), k N λ ka k L ) C and k N λ ka k r L r ) C h q L q ) if r [, ). Proof. For any θ 0, ), we define a sequence of functions {H m } m N as follows. Set H 0 := h. Proceeding by induction, assume that H m is defined. Then set Ω m := {x :H m ) x) > θ m } and define H m as the function g in Proposition 4.3 associated to f = H m and t = θ m. Notice that each Ω m has a decomposition of Ω m = i Bx m,i,r m,i ) satisfying i) through vi) of Lemma 4.6. Then define {φ m,i } i in the same way as in Lemma 4.8. If m, then we have H m = H m i b m,i, 4.8) where b m,i is as in Proposition 4.2 ii). Moreover, there exists a positive constant C 7 such that for all m N and all x, H m x) C 7 θ m 4.9) and H mx) h x)+c 7 m i= θ i j ) β μbx i,j,r i,j )) r i,j. 4.0) μbx i,j,r i,j + dx i,j,x))) r i,j + dx i,j,x) Notice that 4.9) follows immediately from the definition of H m and Proposition 4.3 iv). To prove 4.0), we first have that for any x/ Ω m, by 4.8) and 4.5), Hmx) Hm x)+cθ ) β m μbx m,j,r m,j )) r m,j, 4.) μbx m,j,r m,j + dx m,j,x))) r m,j + dx m,j,x) j N where C is the constant given in 4.5). Observe that for any x Ω m, Lemma 4.6 ii) implies that x Bx m,j,r m,j )forsomej N. From this and 4.7), we deduce that if x Ω m,then Hm x) t ) β μbx m,j,r m,j )) r m,j θm. 4.2) μbx m,j,r m,j + dx m,j,x))) r m,j + dx m,j,x) j N Combining 4.) and 4.2) and repeating this process m times yield 4.0).

20 2272 Loukas GRAFAKOS et al. For every k N, by 4.8), h = H k + k m= j b m,j almost everywhere. By Proposition 4.3 j b m,j almost everywhere. The assumption iii), Remark 4.4 and 4.9), we obtain h = m= h L 2 ) together with 4.0) and Lemma 4.0 further implies that H m L 2 ). Thus by 4.9) and the size condition of ϕ, we obtain that for any ϕ G0β,γ), ɛ k h, ϕ b m,j,ϕ = H k,ϕ = H k x)ϕx) dμx) θk ϕ Gβ,γ) 0, m= as k, which further implies that in G ɛ 0 β,γ)). j h = lim k m= k b m,j = b m,j 4.3) j m N j By the expression given in Proposition 4.3 ii) for b m,j, 4.9) and Proposition 4.3 i), we obtain that for all x, [ ] b m,j x) H m x) + φ m,j x) dμx) H m x)φ m,j x) dμx) 2C 7 θ m. 4.4) Let λ m,j := 2C 7 θ m μbx m,j, 2r m,j )) /p and e m,j := λ m,j ) b m,j. Itiseasytoverifythat each e m,j is a p, )-atom. Then from 4.3), we deduce that h = λ m,j e m,j 4.5) m N j holds in G0β,γ)) ɛ and almost everywhere. To estimate m N j λ m,j p, by Lemma 4.6, λ m,j p θ mp μω m ). m N m N j By the definition of Ω m, 4.0) and Lemma 4.0, we obtain that there exists a positive constant C 8 such that for all m N, θ mq μω m ) [Hm x)]q dμx) m [h x)] q dμx)+c 7 ) q C 8 [ [ j i= θ iq μbx i,j,r i,j )) μbx i,j,r i,j + dx i,j,x))) m ] [h x)] q dμx)+ θ iq μω i ). i= ) β ] q r i,j dμx) r i,j + dx i,j,x) Let b 0 := h x) q dμx) andb m := θ mq μω m )form N. Then the formula above can be written as b m C 8 m i=0 b i. By induction, we obtain b i b 0 C 8 +2) i for every i Z +.Thatis, θ iq μω i ) C 8 +2) i [h x)] q dμx). Choose θ>0 small enough such that θ p q C 8 +2)<. Then, λ m,j p θ p q)m C 8 +2) m m N j m N [h x)] q dμx) [h x)] q dμx).

21 Maximal function characterizations of Hardy spaces 2273 By the fact supp e m,j Bx m,j,r m,j ), 4.4) and Lemma 4.6, we obtain that for all x, λ m,j e m,j x) θ m χ Ωm x), m N m,j which implies that m,j λ m,je m,j L ). When r [, ), by Hölder s inequality, λ m,j e m,j [ /r θ m [μω m )] /r θ mp μω m )]. m N m,j L r ) Thus m,j λ m,je m,j r L r ) h q L q ), which completes the proof of Lemma 4.5. Theorem 4.6. Let ɛ 0, ), p n/n + ɛ), ] and β, γ n/p ),ɛ). Iff H,p ), then there exist {λ k } k N C and p, )-atoms {a k } k N such that f = k N λ ka k in G0 ɛβ,γ)). Moreover, there exists a constant C such that C f H,p ) m N { } /p λ k p C f H,p ). k N Proof. We only give an outline of the proof since it is similar to that of [6, Theorem 4.3)]. First assume that f L p ) L 2 ), then Theorem 3.0 tells us that f can be represented by a function satisfying fx) Cf x). For any given integer k, setω k := {x : f x) > 2 k }. By Proposition 4.3, we have f = B k + G k,whereb k and G k are the functions b and g in Proposition 4.3 corresponding to t =2 k.sinceb k + G k = B k+ + G k+, we then define h k := G k+ G k = B k B k+. 4.6) Thus for any m N, wehavef m k= m h k = B m+ + G m. Notice that G m x) 2 m for all x and supp B m Ω m by Proposition 4.3. Then following the arguments in [6, p. 300], we deduce that for any q n/n + ɛ), ], h k L q ) 2 k μω k ) by Lemma 4.0 and Proposition 4.; and moreover f = k Z h k holds in G ɛ 0 β,γ)) here, we need to use Proposition 3.6) and almost everywhere. By 4.6), we obtain that supp h k Ω k and there exists a positive constant C such that h k x) C2 k for all x and k Z. Applying Lemma 4.5 to C 2 k h k and q n/n + ɛ),p) yields that C 2 k h k = i N λ k,ia k,i in G0β,γ)) ɛ and almost everywhere, where {a k,j } j are p, )-atoms and {λ k,j } j C satisfying i λ k,i p 2 k h k q L q ) μω k). Let ρ k,i := C2 k λ k,i. Then h k = i ρ k,ia k,i, and thus f = k Z i ρ k,ia k,i in G0 ɛβ,γ)) and almost everywhere. Moreover, k Z i ρ k,i p k Z 2kp μω k ) f p L p ). This together with Proposition 4.5 shows the theorem with the additional assumption f L 2 ). The general case of the theorem then follows from the density of L 2 ) H,p )inh,p ) and standard arguments as in [6, pp ], which completes the proof of Theorem 4.6. For ɛ 0, ] and β, γ 0,ɛ), let G b ɛβ,γ) be the set of functions in G 0β,γ) ɛ with bounded support. Let C 0 ) be the set of continuous functions on which tends to zero at infinity. From Theorem 4.6 and the existence of an approximation of the identity with bounded support see [23, Theorem 2.]), we can deduce the following density result. We omit the details. Proposition 4.7. Let ɛ 0, ), p n/n+ɛ), ] and β, γ n/p ),ɛ). Then G b ɛβ,γ) is a dense subset of H,p ); for any p [, ), G b ɛβ,γ) is a dense subset of Lp ); and if p =, G b ɛβ,γ) is a dense subset of C 0 ).