Radial Maximal Function Characterizations for Hardy Spaces on RD-spaces

Transcription

1 Radial Maximal Function Characterizations for Hardy Spaces on RD-spaces Loukas Grafakos Liguang Liu and Dachun Yang Abstract An RD-space X is a space of homogeneous type in the sense of Coifman and Weiss with the additional property that a reverse doubling property holds. The authors prove that for a space of homogeneous type X having dimension n there exists a p n/n + such that for certain classes of distributions the L p X quasi-norms of their radial maximal functions and grand maximal functions are equivalent when p p ]. This result yields a radial maximal function characterization for Hardy spaces on X. Introduction The theory of Hardy spaces on Euclidean spaces plays an important role in harmonic analysis and partial differential equations and has been systematically studied and developed; see for example [ ]. It is well known that spaces of homogeneous type in the sense of Coifman and Weiss [3] are a natural setting of the Calderón-Zygmund theory of singular integrals; see also [4]. A space of homogeneous type is a set X equipped with a metric d and a regular Borel measure µ having the doubling property. Coifman and Weiss [4] introduced the atomic Hardy space H p at X for p ] and further established a molecular characterization for H at X. Moreover under the assumption that the measure of any ball in X is equivalent to its radius i. e. X is an Ahlfors -regular metric measure space when p /2 ] Macías and Segovia [5] used distributions acting on certain spaces of Lipschitz functions to obtain a grand maximal function characterization for H p at X ; Han [] further established a Lusin-area characterization for H p at X and Duong and Yan [5] characterized these atomic Hardy spaces in terms of Lusin-area functions associated with certain Poisson semigroups. Also in this setting a deep result of Uchiyama [23] states that if p p ] for some p near for functions in L X the L p X quasi-norms of the grand maximal functions as in [5] are equivalent to the L p X quasi-norms of the radial maximal functions defined via some kernels in [4]. An important special class of spaces of homogeneous type is called RD-spaces which is introduced in [] see also [2 6] and modeled on Euclidean spaces with A -weights 2 Mathematics Subject Classification. Primary 42B25; Secondary 42B3 47B38 47A3. Key words and phrases. Space of homogeneous type approximation of the identity space of test function grand maximal function radial maximal function Hardy space. The first author was supported by grant DMS 4387 of the National Science Foundation of the USA and the University of Missouri Research Council. The third author was supported by the National Science Foundation for Distinguished Young Scholars Grant No of China. Corresponding author.

2 2 L. Grafakos L. Liu and D. Yang Muckenhoupt s class Ahlfors n-regular metric measure spaces see for example [4] Lie groups of polynomial growth see for example [ 24 25] and Carnot-Carathéodory spaces with doubling measure see for example [ ]. A Littlewood-Paley theory of Hardy spaces on RD-spaces was established in [2] and these Hardy spaces are shown to coincide with some of Triebel-Lizorkin spaces in []. The grand nontangential and dyadic maximal function characterizations of these Hardy spaces have recently been established in [9]. The main purpose of this paper is twofold: first to generalize the results of Uchiyama [23] to the setting of RD-spaces and second to replace the space L X used by Uchiyama in [23] by certain spaces of distributions developed in [2 ]. In other words we build on the work of Uchiyama [23] to establish a radial maximal function characterization for the Hardy spaces in [2]. To state our main results we need to recall some definitions and notation. We begin with the classical notions of spaces of homogeneous type [3] [4] and RD-spaces []. Definition. Let X d be a metric space with a regular Borel measure µ such that all balls defined by d have finite and positive measures. For any x X and r > set Bx r {y X : dx y < r}. i The triple X d µ is called a space of homogeneous type if there exists a constant C such that for all x X and r > µbx 2r C µbx r doubling property.. ii Let < κ n. The triple X d µ is called a κ n-space if there exist constants < C and C 2 such that for all x X < r < diam X /2 and λ < diam X /2r C λ κ µbx r µbx λr C 2 λ n µbx r.2 where diam X sup x y X dx y. A space of homogeneous type is called an RD-space if it is a κ n-space for some < κ n i. e. some reverse doubling condition holds. Remark.2 i A regular Borel measure µ has the property that open sets are measurable and every set is contained in a Borel set with the same measure; see [4]. ii The number n in some sense measures the dimension of X. Obviously any κ n space is a space of homogeneous type with C = C 2 2 n. Conversely any space of homogeneous type satisfies the second inequality of.2 with C 2 = C and n = log 2 C. iii If µ is doubling then µ satisfies.2 if and only if there exist constants a > and C > such that for all x X and < r < diam X /a µbx a r C µbx r reverse doubling property If a = 2 this is the classical reverse doubling condition and equivalently for all x X and < r < diam X /a Bx a r \ Bx r ; see []. From this it follows that if X is an RD-space then µ{x} = for all x X.

3 Radial Maximal Function Characterizations 3 Throughout the whole paper we always assume that X is an RD-space and µx =. For any x y X and δ > set V δ x µbx δ and V x y µbx dx y. It follows from. that V x y V y x. The following notion of approximations of the identity on RD-spaces is a variant of that in [ Definitions ]; see also [2]. Definition.3 Let ɛ ] ɛ 2 > and ɛ 3 >. A sequence {S k } k Z of bounded linear integral operators on L 2 X is said to be a special approximation of the identity of order ɛ ɛ 2 ɛ 3 for short ɛ ɛ 2 ɛ 3 - SAOTI if there exists a constant C 3 > 2 such that for all k Z and all x x y and y X S k x y the integral kernel of S k is a function from X X into [ satisfying 2 i S k x y C kɛ 2 3 V 2 k x+v 2 k y+v xy 2 k +dxy ; ɛ 2 ii S k x y S k x dxx y C ɛ 3 2 k +dxy ɛ V 2 k x+v 2 k y+v xy 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 dxx ] C ɛ 3 V 2 k x+v 2 k y+v xy 2 kɛ 3 2 k + dx y/3; v C 3 V 2 kxs k x x > for all x X and k Z; vi X S kx y dµy = = X S kx y dµx. 2 k +dxy ɛ 2 kɛ 2 2 k +dxy ɛ 2 for dx x dyy ɛ 2 k +dxy ɛ 2 k +dxy ɛ 3 for dx x 2 k + dx y/3 and dy y We remark that i and v of Definition.3 imply that C 3 > 2. The existence of ɛ ɛ 2 ɛ 3 - SAOTI s was proved in [ Theorem 2.]. The following spaces of test functions play a key role in the theory of function spaces on RD-spaces; see [2 ]. Definition.4 Let x X r β ] and γ. A function ϕ on X is said to be a test function of type x r β γ if there exists a nonnegative constant C such that γ r i ϕx C V rx +V x x r+dx x for all x X ; β γ ii ϕx ϕy C dxy r r+dx x V rx +V x x r+dx x for all x y X 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. If ϕ 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. Throughout the whole paper we fix x X. Let Gβ γ Gx β γ. It is easy to see that for any x 2 X and r > we have Gx 2 r β γ = Gβ γ with equivalent norms but with constants depending on x x 2 and r. Moreover Gβ γ is a Banach space. For any given ɛ ] let G ɛ β γ be the completion of the space Gɛ ɛ in Gβ γ when β γ ɛ]. Obviously G ɛɛ ɛ = Gɛ ɛ. Moreover ϕ Gɛ β γ if and only if ϕ Gβ γ and there exists {φ i } i N Gɛ ɛ such that ϕ φ i Gβγ as i. If ϕ G ɛβ γ define ϕ G ɛβγ ϕ Gβγ. Obviously G ɛ β γ is a Banach space and

4 4 L. Grafakos L. Liu and D. Yang ϕ G ɛ βγ = lim i φ i Gβγ for the above chosen {φ i } i N. It is known that G ɛ β γ is dense in L p X for p [ ; see [ Corollary 2.]. Let G ɛβ γ be the set of all bounded linear functionals f from G ɛ β γ to C. Denote by f ϕ the natural pairing of elements f G ɛβ γ and ϕ G ɛ β γ. Let ɛ β γ ɛ and p ]. If f G ɛβ γ then for all x X we define the grand maximal function of f to be f x sup { f ϕ : ϕ Gɛ ɛ ϕ Gxrɛɛ for some r > }. Define the corresponding Hardy space by H p X { f G ɛ β γ : f L p X < }. For any f H p X set f H p X f L p X. Let {Q k α : k Z α I k } be the Christ dyadic cube collection of X where I k is some index set; see [2]. For any f G ɛβ γ we define the dyadic maximal function M d f of f by setting for all x X M d fx sup k Z α I k { µq k α Q k α S k fy dµy } χ Q k α x and define H p d X to be the corresponding Hardy space; see [9 Definition 2.9]. When p ] it was proved in [9 Corollary 3.2] that H p X = H p d X = Lp X with equivalent norms. Definition.5 Let ɛ ] ɛ 2 > ɛ 3 > ɛ ɛ ɛ 2 and {S k } k Z be an ɛ ɛ 2 ɛ 3 - SAOTI. Let p ] and f G ɛ β γ with β γ ɛ. Define the radial maximal function of f to be M fx sup k Z S k fx for all x X. The corresponding Hardy spaces are defined by H p X { f G ɛ β γ : M f L p X < } and moreover we define f H p X M f L p X. The properties i and ii in Definition.3 imply that M fx f x for all x X. In what follows for simplicity of presentation for any t > we use the notation S t x y k Z S k x yχ 2 k 2 k ]t..3 By.3 and Definition.5 it is easy to see that for all x X M fx = sup S t fx..4 t> Obviously S t satisfies i through vi in Definition.3 with 2 k replaced by t. From iv and v in Definition.3 it follows easily that there exist constants C 4 C 3 2/ɛ and C 5 > such that for all t > and all x y X satisfying dx y < C 4 t C 5 V t xs t x y >..5

5 Radial Maximal Function Characterizations 5 This observation is used in applications below. Denote by M the centered Hardy-Littlewood maximal operator. To be precise for any f L loc X and x X set Mfx sup fy dµy. r> µbx r Bx r Then M is weak-type and bounded on L p X for p ] in [3 4]. It is not so difficult to show that for all x X M fx Mfx and M d fx MM fx by their definitions and Lemma 2. iv below. Therefore we have H p X = Lp X with equivalent norms when p ]. The main result of this paper concerns the spaces H p X and H p X and is as follows. Theorem.6 Let ɛ ] ɛ 2 > ɛ 3 > and ɛ ɛ ɛ 2. Let {S k } k Z be an ɛ ɛ 2 ɛ 3 - SAOTI and M be as in.4. Then there exist σ /2 and η σ /ɛ /2 both depending only on X and ɛ such that for any given p n/n + log η σ ] and all f G ɛβ γ with β log η σ and γ ɛ f L p X C M f L p X where C is a positive constant independent of f. Theorem.6 will be a consequence of the following key proposition. Proposition.7 With the notation of Theorem.6 for any δ log η σ there exists a positive constant C depending on X ɛ and δ such that for all f G ɛ β γ with β log η σ and γ ɛ and all ϕ Gx r ɛ ɛ satisfying ϕ Gx r ɛɛ for some x X and r > f ϕ C [ M[M f] n/n+δ x ] n+δ /n. We remark that in Theorem.6 and Proposition.7 it is not necessary to assume that {S k } k Z has the property vi of Definition.3. Moreover Theorem.6 follows easily from Proposition.7; see Section 3 below. The main ingredient in the proof of Proposition.7 is to expand ϕ as in Proposition.7 into a sum of certain S t as in.3; see 3. below. When X is an Ahlfors -regular metric measure space for any Lipschitz function with bounded support Uchiyama in [23] established an expansion similar to 3. which holds pointwise. Unlike [23] we prove that for any ϕ as in Proposition.7 3. also holds in G ɛ β γ with β and γ as in Proposition.7. This allows us to relax the assumption f L X to f G ɛβ γ. From the fact M f f and Theorem.6 it follows that for p in a certain range of ] H p X coincides with H p X as a subspace of certain distribution spaces G ɛβ γ. Recall that when p n/n + ] [9 Remark 3.6] and [9 Corollary 4.9] tell us that the definition of H p X is independent of the choices of G ɛβ γ with β γ n/p ɛ. Therefore we deduce the following conclusion.

6 6 L. Grafakos L. Liu and D. Yang Corollary.8 Let ɛ ɛ 2 ɛ 3 σ and η be as in Theorem.6 and ɛ ɛ ɛ 2. Let p n/n + log η σ and p p ]. Then H p X = H p X with equivalent quasi-norms where H p X and H p X are defined via G ɛβ γ with some β n/p n/p and γ n/p ɛ. Consequently the definition of H p X is independent of the choices of ɛ ɛ 2 ɛ 3 - SAOTI and G ɛβ γ with β and γ as above. Remark.9 We point out that in Theorem.6 and Proposition.7 it is not necessary to assume that X satisfies the reverse doubling condition determined by the first inequality of.2. However the assertion H p X = Lp X when p ] and Corollary.8 do need this assumption since to obtain these conclusions we need to use the Calderón reproducing formulae in [] which depend on the reverse doubling condition. The organization of this paper is as follows. In Section 2 we give some technical lemmas which will be used in the proof of Proposition.7. Section 3 is the main part of this paper which contains a proof of Proposition.7 and also of Theorem.6. In this paper we use the following notation: N { 2 } Z + N {} and R + [. For any p [ ] we denote by p the conjugate index namely /p+/p =. We also denote by C positive constant independent of 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 we then write f g. For any a b R set a b min{a b} and a b max{a b}. For any set E we denote by E the cardinality of E. 2 Some technical lemmas In this section we establish several technical lemmas which will be used in the proof of Proposition.7. The following lemma includes some basic properties on RD-spaces which are used throughout the paper; see for example [2 9]. Lemma 2. Let δ > a > r > and θ. Then i For all x y X and r > V r x+v x y V r y+v y x µby r+dy x µbx r + dx y. ii If x x x X satisfy dx x θr + dx x then r + dx x r + dx x and µbx r + dx x µbx r + dx x. V rx+v xy iii X a > η. iv For all f L loc X and x X dx y>δ uniformly in δ > f L loc X and x X. r r+dxy a dx y η dµx < Cr η uniformly in x X and r > if V xy δ a dxy a fy dµy CMfx When δ = the following lemma provides a property of Carleson measures on RDspaces; see [ Proposition 5.4]. Lemma 2.2 Let p ] and δ. Let ν be a non-negative measure on X R + such that for all x X and r > νbx r r [µbx r] +δ. 2.

7 Radial Maximal Function Characterizations 7 Then there exists a positive constant C such that for all f L p X { } /p+δ F r y f p+δ dνy r C f L px X R + where and in what follows F r x f S r fx for all r > and x X. Proof. Fix λ > and let W λ {x r X R + : F r x f > λ}. For any l Z set { } W lλ x X : sup F r x f > λ 2 l <r 2 l. For each N N let E N {x X : sup r>2 N F r x f > λ}. It is easy to deduce that lim E N =. 2.2 N To prove 2.2 notice that lim N E N = N N E N since E N+ E N for any N N. Suppose that 2.2 fails that is there exists an x N N E N. Thus for any N N there exists r N > 2 N satisfying that F r N x f > λ. By this.3 Hölder s inequality and Lemma 2. iii we obtain λ < F r N x f = X { S rn x yfy dµy [V 2 N x] /p X fy p dµy} /p which implies that V 2 N x λ p f p L p X < for all N N and hence µx <. This contradicts the assumption µx =. Thus 2.2 holds. It is not so difficult to prove that for any given N N there exist L N < N with L N Z or L N = a set of indices I Nl with l {L N + N} and disjoint balls {Bylj N 2l } LN <l N j I Nl satisfying i ylj N W lλ; ii Bylj N 2l N m=l+ i I Nm Bymi N 2m = ; iii for any x W lλ Bx 2 l N m=l i I Nm Bymi N 2m. In fact we start with l = N and choose an arbitrary point in W Nλ as yn N. Then we find a point yn2 N W Nλ \ ByN N 2N such that ByN2 N 2N ByN N 2N =. Continuing in this way by Zorn s lemma and the doubling property of the measure µ we arrive at I NN = N or I NN will be a finite set. We then consider l = N. In this way one finds the desired balls. From i ii and iii it follows that for each N N N [ ] W λ Bylj N 2l+ 2 l EN 2 N. 2.3 l=l N + j I Nl To see 2.3 notice that for any x r W λ F r x f > λ. If r > 2 N then x r E N 2 N ; otherwise there exists l N such that 2 l < r 2 l which implies that

8 8 L. Grafakos L. Liu and D. Yang x W lλ. By Property iii above there exist integers l m N and j I Nm such that Bx 2 l By N mj 2m. Noticing that l m we have x By N mj 2m+ and thus x r By N mj 2m+ 2 m which yields 2.3 then. For any N N by 2.3 νw λ N l=l N + j I N.l νby N lj 2l+ 2 l + νe N 2 N. Letting N in the formula above then using 2. and 2.2 we obtain νw λ lim N N l=l N + j I N.l lim N N l=l N + [ ] +δ µbylj N 2l+ µbylj N 2l+ j I N.l +δ 2.4 where in the second step we use the fact j N a j κ j N a j κ for any κ ]. Choose p p. For any N l L N N] and any given j I Nl by Property i the size condition of S l.2 and Hölder s inequality we have λ < sup F r ylj N f 2 l <r 2 l V 2 lylj N fz dµz Bylj N 2l 2 l ɛ2 + k= 2 l+k dylj N V z<2l+k 2 lylj N + V yn lj z 2 l + dylj N z fz dµz { } ep/p 2 kɛ 2 V k= 2 l+kylj N fz p/ep dµz Bylj N 2l+k { [ inf M f p/ep ] } ep/p zlj N : z N lj Bylj N 2l which together with the pairwise disjointness of the balls {Bylj N 2l } LN <l N j I Nl L ep X -boundedness of M yields that for all N N N l=l N + µbylj N 2l+ λ p j I N.l X [ M f p/ep ] ep z dµz f p L p X. and Combining this with 2.4 shows that λ p+δ νw λ f p+δ L p X. Then the desired conclusion follows from the Marcinkiewicz interpolation theorem which completes the proof of Lemma 2.2.

9 Radial Maximal Function Characterizations 9 Lemma 2.3 Let x X r > and g be a non-negative function on X. Then for any t ] there exist {x j } j X with x j x j g t x r and positive constants C 6 and C 7 depending only on X such that for all x X j χ Bxj C 4 tr j x C and gx j /2 C 7 F tr j x j g /2 χ Bxj C 4 tr j 2.6 where r j r + dx j x and F is as in Lemma 2.2. In particular there exists a constant C 8 > such that for all j and all x Bx j C 4 tr j 2 a C a 8 r V V rj x j r x trj js x trj j xχ Bxj C4trjx j r a 2.7 V r x + V x x r + dx x where S trj is as in.3. Moreover C 6 through C 8 are independent of x r and g. Proof. For any x X set r x r + dx x. By Zorn s lemma there exists a set of points {y j } j X satisfying that y j / j i= By i C 4 tr yi /4 and X = j By j C 4 tr yj /4. From the selection of {y j } j it follows that for any i j and that for all x X dy i y j 4 C 4t min{r yi r yj } 2.8 χ Byj C 4 tr yj /4x. 2.9 j By.2 and the disjointness of {By i C 4 tr /8} i we know that {y j } j is at most countable. For every y j choose x j x j g t x r satisfying that and gx j /2 dx j y j < C 4 tr yj /4 2. gz /2 dµz. 2. µby j C 4 tr yj /4 By j C 4 tr yj /4 Let r j r xj. By 2. and the triangle inequality for d together with C 4 < /2 and t ] we have r yj /2 r j 2r yj 2.2 which together with 2.9 implies the left-hand side inequality of 2.5. For any x X set Jx {j : dx j x < C 4 tr j }. Notice that for any j Jx by 2.2 r j /2 r x 2r j. This together with 2. and 2.2 yields that Jx Jx where Jx {j : dy j x < 3C 4 tr x }. It follows from.2 and the pairwise disjointness of {By i C 4 tr /8} i that Jx <. Thus we may assume that r y = min{r yj : j Jx}.

10 L. Grafakos L. Liu and D. Yang Therefore by 2.8 {By j C 4 tr y /8} j Jx are mutually disjoint. Furthermore for any j Jx by 2.2 and r j /2 r x 2r j we have By j C 4 tr x /32 By j C 4 tr y /8 Bx 4C 4 tr x By j 7C 4 tr x. From this and.2 it follows that Jx is bounded by a positive constant which depends only on X. This implies the validity of 2.5. The fact that By j C 4 tr yj /4 Bx j C 4 tr j together with 2. and.5 implies 2.6. Since for any x Bx j C 4 tr j we have r j /2 r x 2r j. Using this fact and Lemma 2. i we obtain 2.7 which completes the proof of Lemma 2.3. Lemma 2.4 Let t ] a [ b a M [ and {x j } j X satisfying χ Bxj C 4 tr j x C j where r j r + dx j x with r > and x X and C 6 is as in 2.5. For any j and x X set a V C4 tr u j x j x j V rj x j r j V trj x j + V trj x + V x j x tr b j dxj x χ tr j + dx j x [M tr j where χ [M is the characteristic function of the interval [M. Then there exists C 9 > independent of x r and M such that for all x X u j x C 9 max{t b + M b a }. V r x + V x x r + dx x j Proof. For any k Z set Jk {j : 2 k r j < 2 k } and v k j Jk u j. For any fixed x X let and We then write u j x = j W {k Z : r + dx x/2 2 k < 4r + dx x} k W j Jk W 2 {k Z : 2 k < r + dx x/2} W 3 {k Z : 2 k 4r + dx x}. u j x + k W 2 j Jk u j x + k W 3 j Jk To estimate Z notice that for any k W and j Jk we have u j x Z + Z 2 + Z 3. µbx j r + dx j x µbx r j µbx 2 k V r x + V x x. 2.4

11 Radial Maximal Function Characterizations For any j Jk and z Bx j C 4 tr j we have dz x j < C 4 tr j < C 4 t2 k which together with Lemma 2. further implies that for all z Bx j C 4 tr j t2 k + dx j x t2 k + dz x V t2 kx j + V x j x V t2 kz + V z x 2.5 and dxj x dxj x χ [M χ tr [M j t2 k χ [M 2C4 dz x t2 k. 2.6 From 2.3 through 2.6 it follows that v k x 2 ak V r x + V x x V C4 tr j x j V t2 kx j + V t2 kx + V x j x j Jk t2 k b t2 k χ + dx j x [M 2 ak V r x + V x x t2 k b V t2 kz + V t2 kx + V z x t2 k χ + dz x [M 2C4 X dxj x t2 k dz x t2 k dµz. Denote by J the integral in the last formula. When M 4C 4 + by Lemma 2. J X t2 k b V t2 kz + V t2 kx + V z x t2 k dµz + M b. + dz x When M > + 4C 4 we have M 2C 4 > + M/2. For any i N and x X set We then obtain R i {z X : 2 i+k 2 M 2C 4 t dz x < 2 i+k M 2C 4 t}. J = i= R i V t2 kz + V t2 kx + V z x t2 k b t2 k dµz + dz x [ + 2 i M 2C 4 ] b M 2C 4 b + M b. i= Therefore for all k W and x X we have v k x 2 ak + M b V r x + V x x which together with the fact W 5 yields that Z k W 2 ak + M b V r x + V x x r + dx x a + M b V r x + V x x.

12 2 L. Grafakos L. Liu and D. Yang To estimate Z 2 notice that for any k W 2 and j Jk we have r + dx x r + dx x j + dx j x < 2 k + dx x j r + dx x/2 + dx x j. From this it follows that for any j Jk and thus j Jk dx x j > r + dx x/2 > 2 k 2.7 V x j x µbx r + dx x V r x + V x x. 2.8 For k Z and j Jk we have V 2 kx j V 2 kx and Bx j C 4 tr j Bx +C 4 t2 k which together with.2 and 2.3 yields that V C4 t2 kx j χ V 2 kx j V 2 kx Bxj C 4 tr j x dµx. 2.9 Bx +C 4 t2 k j Jk For any k Z and j Jk we have V rj x j V 2 kx j which together with and 2.9 implies that Z 2 k W 2 j Jk 2 ak t b V r x + V x x V C4 t2 kx j V 2 kx j V r x + V x x a. r + dx x To estimate Z 3 notice that for any k W 3 and j Jk t2 k t2 k + r + dx x V rj x j µbx j r + dx j x µbx 2 k V r x + V x x. 2.2 Moreover since 2 k r + dx j x r + dx j x + dx x 2 k 2 + dx j x we have dx j x 2 k 2 r j /4. Consequently for any k W 3 and j Jk V trj x j + V trj x + V x j x V x j x V rj /2x j. 2.2 b An argument similar to 2.9 yields that V C4 tr j x j j Jk V rj /2x j Applying 2.2 through 2.22 and the fact that dx j x r j /4 we obtain Z 3 V C4 tr j x j t b + t k W 3 j Jk 2 ka t b V r x + V x x which completes the proof of Lemma 2.4. V r x + V x x V rj /2x j a r + dx x

13 Radial Maximal Function Characterizations 3 3 Proofs of Theorem.6 and Proposition.7 Assuming Proposition.7 for the moment we now prove Theorem.6. Proof of Theorem.6 Choose δ log η σ satisfying p > n/n + δ. By Proposition.7 and the L pn+δ /n X -boundedness of M we obtain [ f L p X M[M f] ] n/n+δ n+δ /n L M f L p X p X which completes the proof of Theorem.6. The rest of this section is devoted to the proof of Proposition.7. The key for the proof of Proposition.7 is to obtain a desired expansion 3. for any ϕ Gx r ɛ ɛ in terms of the given ɛ ɛ 2 ɛ 3 - SAOTI and to show this expansion converges in G ɛ β γ. To this end for a given ϕ Gx r ɛ ɛ we construct a sequence of functions {ϕ s } s Z+ and obtain some desired estimates for these functions. Proof of Proposition.7 Let ϕ Gx r ɛ ɛ satisfy ϕ Gx r ɛɛ for some x X and r >. Write ϕ as ϕ = Rϕ Rϕ + i[iϕ Iϕ ] where Rϕ and Iϕ represent the real part and the imaginary part of the function ϕ respectively. Since ϕ Gx r ɛɛ it follows easily that each term of the decomposition above has a norm in Gx r ɛ ɛ at most. Thus we may assume that ϕ is non-negative. Fix A 2 ɛ+ C 8 and some σ / + AC 3 C 9. Choose positive numbers H and η such that H min { 2 σ AC3 C 9 σ 2 /ɛ 2 } σ AC3 C 9 σ σ /ɛ 4AC 3 C 9 σ and η < min { σ /ɛ H C 4 AC 3 C 9 σ /ɛ } /ɛ2 H. 2 AC 3 C 9 Choose δ log η σ. For any s N applying Lemma 2.3 with t = η s and g = M f we obtain {x sj : s N j = js} X where js can be finite or. For each s N and j = js let r sj r + dx sj x. Lemma 2.3 further implies that A for any x X and s N js j= χ B sj x C 6 where B sj Bx sj C 4 η s r sj ; B [M fx sj ] /2 C 7 F η s r sj x sj [M f] /2 χ Bsj. Let us now inductively construct {ɛ sj functions {ϕ s } s Z+ satisfying that for all x X C for all s Z + ϕ s x σ s V r x + V x x : s N j = js} { } and r r + dx x ɛ ;

14 4 L. Grafakos L. Liu and D. Yang D ϕ x ϕx and for any s N ϕ s x ϕx Aσ ji s i= j= σ i ɛ ij V C4 η s x ij V rij x ij r ɛ S η irij x ij x. 3. In fact obviously ϕ satisfies C. Suppose that {ɛ ij : i = s ; j = ji} and {ϕ i } s i= satisfying C have been constructed. Then for each j = js let ɛ sj signϕ s x sj and ϕ s be as in D. Now it remains to verify that ϕ s satisfies C. To this end for all x X set js ω s x Aσ σ s V C4 η ɛ s r sj x sj sj V rsj x sj j= r r sj ɛ S η srsj x sj x. 3.2 By the size condition of S t ɛ 2 > ɛ and Lemma 2.4 with t = η s a = ɛ b = ɛ 2 and M = we obtain js r ω s x AC 3 σ σ s V C 4 η s r sj x sj V j= rsj x sj r sj η s r ɛ2 sj V η s r sj x sj + V η s r sj x + V x sj x η s r sj + dx sj x AC 3 C 9 σ σ s. 3.3 V r x + V x x r + dx x If dx y η s r + dx x/2 then for any i = s we have that dx y η i + dx x ij /2. This combined with the regularity of S η i and Lemma 2.4 with t = η i a = ɛ + ɛ b = ɛ + ɛ 2 and M = yields that ϕ s x ϕ s y ji s ϕx ϕy + AC 3 σ σ i i= j= V C 4 η i x ij ɛ r S V rij x ij r η irij x ij x S η irij x ij y ij s ji σ i ɛ r η i ɛ2 ϕx ϕy + AC 3 σ V i= j= rij x ij η i + dx ij x V C4 η i x ij dx y ɛ V η i x ij + V η i x + V x ij x η i + dx ij x [ ϕx ϕy + AC 3C 9 σ σ s σ η ɛ η ɛ ] dx y ɛ. 3.4 V r x + V x x r + dx x r + dx x ɛ

15 Radial Maximal Function Characterizations 5 When dx y < Hη s r + dy x 3.5 the assumption that H < /4 gives dx y < 2Hη s r +dx x < η s r +dx x/2 which together with 3.4 and the regularity of ϕ yields that [ η ϕ s x ϕ s y 2H ɛ ɛ s + 2Hɛ AC 3 C 9 σ η ɛ ] s σ σ η ɛ σ σ s V r x + V x x r + dx x σ s λ H 3.6 V r x + V x x r + dx x where λ H 2H ɛ + 2Hɛ AC 3 C 9 σ σ η ɛ. For any λ R and s N set { Ω sλ x X : ϕ s x > λ σ s V r x + V x x If ϕ s x sj < then by 3.6 and the definition of Ω sλ we obtain r r + dx x ɛ }. Bx sj η s r sj /3 Ω sλh =. 3.7 In fact for any x Bx sj η s r sj /3 by 3.6 σ s r ϕ s x ϕ s x sj + λ H V r x + V x x r + dx x σ s < λ H V r x + V x x r + dx x which implies that x / Ω sλh and thus 3.7 holds. For any x X by 3.2 we write ω s x = {j: ϕ s x sj >} {j: ϕ s x sj <} Aσ σ s V C 4 η s r sj x sj V rsj x sj I II. r r sj ɛ ɛ S η srsj x sj x We now turn to obtain a desired upper bound for ϕ s by considering two cases: x Ω sλh and x / Ω sλh. For any x Ω sλh Property A implies that x Bx sj C 4 η s r sj for some j. By this and the assumption η < H/C 4 we have dx x sj < C 4 η s r sj Hη s r sj which together with 3.7 implies that ϕ s x sj >. Thus by 2.7 I Aσ σ s V C 4 η s r sj x sj ɛ r S η V rsj x sj srsj x x sj r sj

16 6 L. Grafakos L. Liu and D. Yang Aσ σs C 8 2 ɛ. V r x + V x x r + dx x On the other hand by 3.7 and Lemma 2.4 II AC 3C 9 σ σ s max{η sɛ 2 + Hη ɛ 2 }. V r x + V x x r + dx x From the assumption on η it follows easily that A C 8 2 ɛ AC 3C 9 max{η sɛ 2 + Hη ɛ 2 } which together with the estimates for I and II yields that when x Ω sλh σ σ s ω s x. 3.8 V r x + V x x r + dx x Thus by 2. and the fact that C holds for s we obtain that when x Ω sλh σ s ϕ s x = ϕ s x ω s x. V r x + V x x r + dx x Let now x / Ω sλh. Notice that C holds for s. From this and 3.3 we deduce that ϕ s x = ϕ s x ω s x σ s λ H V r x + V x x r + dx x +AC 3 C 9 σ σ s r V r x + V x x r + dx x σ s V r x + V x x r + dx x where in the last inequality we used the fact λ H + AC 3 C 9 σ σ. Thus we obtain a desired upper bound of ϕ s. Let us now show that ϕ s has the desired lower bound also by considering two cases: x / Ω s λh and x Ω s λh. For every x sj satisfying ϕ s x sj > by 3.6 and an argument similar to the proof of 3.7 we have Bx sj Hη s r sj X \ Ω s λh =. From this Lemmas 2.3 and 2.4 together with an argument similar to the estimates for I and II it follows that when x / Ω s λh σ σs ω s x. 3.9 V r x + V x x r + dx x Therefore 3.9 and the lower bound of ϕ s yields that when x / Ω s λh σ s ϕ s x = ϕ s x ω s x ; 3. V r x + V x x r + dx x ɛ

17 Radial Maximal Function Characterizations 7 and the validity of C for s together with 3.3 and the assumption λ H +AC 3 C 9 σ σ implies that when x Ω s λh ϕ s x ϕ s x ω s x σ s λ H V r x + V x x r + dx x AC 3 C 9 σ σ s r V r x + V x x r + dx x σ s V r x + V x x r + dx x which together with 3. further yields the desired lower bound of ϕ s. This finishes the proofs of C and D. It follows from C and D that for all x X ɛ Set ϕx = Aσ Φ L x Aσ ji i= j= L σ i ɛ ij V C4 η i x ij V rij x ij i= {j: η i L i } r σ i V C4 η ɛ i x ij ij V rij x ij ɛ S η irij x ij x. 3. r ɛ S η irij x ij x. By A and.2 it is easy to see that the series in Φ L has only finitely many terms. To verify that 3. holds in G ɛβ γ it suffices to show that Φ L converges to ϕ in G ɛ β γ as L. Notice that for any i and j S η i x ij Gx ij η i ɛ ɛ. Thus Φ L Gɛ ɛ since it has only finite terms. Now it remains to show To this end we write ϕx Φ L x Aσ + Aσ lim ϕ Φ L Gx r L βγ =. i= {j: η i >L i } i=l {j: η i L i } Φ Lx + Φ 2 Lx. σ i V C4 η ɛ i x ij ij V rij x ij σ i V C4 η ɛ i x ij ij V rij x ij r r ɛ S η irij x ij x ɛ S η irij x ij x Let us first prove that Φ L converges to zero in Gx r β γ as L. Notice that ɛ > γ and = r + dx ij x. By the size condition of S t and Lemma 2.4 we obtain that for all x X Φ Lx σ i r η i ɛ γ V C4 η i x ij γ r L i S V rij x ij η irij x ij x i= {j: η i >L i }

18 8 L. Grafakos L. Liu and D. Yang ση ɛ γ r γ L ɛ γ ση ɛ γ. 3.2 V r x + V x x r + dx x For any x y X satisfying dx y r + dx x/2 we write Φ Lx Φ Ly Aσ + i= j Wi i= j Wi 2 σ i V C 4 η i x ij V rij x ij + i= j Wi 3 r ɛ S η irij x ij x S η irij x ij y Z i + Zi 2 + Zi 3 i= where W i {j N : η i > L i dx y η i + dx x ij /2} W 2 i {j N : η i > L i dx y > η i + dx x ij /2 dy x ij dx x ij } and W 3 i {j N : η i > L i dx y > η i + dx x ij /2 dy x ij < dx x ij }. By the regularity of S η i Lemma 2.4 and η i r + dx ij x > L i we obtain Z i j W i σ i V C 4 η i x ij V rij x ij r η i ɛ γ r L i dx y ɛ η i V η i x ij + V η i x + V x ij x η i + dx ij x η i + dx ij x ση ɛ γ ɛ i dx y ɛ r γ L ɛ γ. V r x + V x x r + dx x r + dx x By the size condition of S η i dx y > η i + dx x ij /2 dy x ij dx x ij r and Lemma 2.4 Z 2 i j W 2 i σ i V C 4 η i x ij V rij x ij r γ ɛ [ S η irij x ij x + S η irij x ij y] σ i r η i ɛ γ γ r V C4 η i x ij dx y L i V rij x ij η i r j W 2 i η i ɛ2 +ɛ r ij V η i x ij + V η i x + V x ij x η i + dx ij x ση ɛ γ ɛ i dx y ɛ r γ L ɛ γ. V r x + V x x r + dx x r + dx x ɛ ɛ2

19 Radial Maximal Function Characterizations 9 From the assumption dx y r + dx x/2 it follows that r + dx x r + dy x and V r x + V x x V r x + V x y 3.3 which together with the definition of W 3 i and an argument similar to the estimation of Z2 i yields that ση Z 3 ɛ γ ɛ i L ɛ γ i dx y ɛ V r x + V x x r + dx x r r + dx x γ. By the estimates of Z i Z2 i L ɛ γ > ση ɛ γ ɛ and Z 3 i we obtain that when L is large enough and satisfies Φ Lx Φ ση ɛ γ ɛ dx y Ly L ɛ γ ση ɛ γ ɛ r + dx x r γ V r x + V x x r + dx x ɛ which together with 3.2 implies that lim L Φ L Gx r βγ =. Now we consider Φ 2 L Gx r βγ. By the size condition of S η i obtain that for all x X and Lemma 2.4 we Φ 2 Lx σ i V rij x ij i=l j γ r V C4 η i x ij η i ɛ2 V η i x ij + V η i x + V x ij x η i + dx ij x σ L r γ. 3.4 V r x + V x x r + dx x For any x y X satisfying dx y r + dx x/2 we write Φ 2 Lx Φ 2 Ly Aσ + i=l j Wi 4 i=l j Wi 5 σ i V C 4 η i x ij V rij x ij + i=l j Wi 6 i=l r ɛ S η irij x ij x S η irij x ij y Z 4 i + Z 5 i + Z 6 i where and W 4 i {j N : dx y η i + dx x ij /2} W 5 i {j N : dx y > η i + dx x ij /2 dy x ij dx x ij } W 6 i {j N : dx y > η i + dx x ij /2 dy x ij < dx x ij }.

20 2 L. Grafakos L. Liu and D. Yang For any i N and k = by dx y r + dx x/ and 3.3 Z k i ω i x + ω i y σ i. 3.5 V r x + V x x r + dx x The regularity of S η i Z 4 i j W 4 i and Lemma 2.4 show that σ i V C 4 η i x ij V rij x ij r ɛ dx y η i + dx ij x η i ɛ2 V η i x ij + V η i x + V x ij x η i + dx ij x σ i dx y ɛ η ɛ. 3.6 V r x + V x x r + dx x r + dx x For any j Wi 5 we have dy x ij dx x ij. From this the size condition of S η i and Lemma 2.4 we deduce that Z 5 i σ i V C 4 η i x ij ɛ r dx y ɛ V rij x ij η i + dx ij x j W 5 i η i ɛ2 V η i x ij + V η i x + V x ij x η i + dx ij x σ i dx y ɛ η ɛ 3.7 V r x + V x x r + dx x r + dx x By an argument similar to the estimate of Z 3 i we obtain σ Z 6 i η ɛ i dx y ɛ V r x + V x x r + dx x ɛ r r + dx x ɛ. 3.8 The geometric means of 3.6 and and and 3.5 respectively give that for all i N and k = σ Z k i η β i dx y β V r x + V x x r + dx x r r + dx x Using this and the assumption η β > σ we obtain for all x y X satisfying dx y r + dx x/2 the following estimate: σ Φ 2 Lx Φ 2 Ly η β L dx y β V r x + V x x r + dx x γ. r r + dx x This together with 3.4 yields that lim L Φ 2 L Gx r βγ =. Therefore we obtain lim L Φ L Gx r βγ = and thus lim L Φ L Gx βγ = which implies that 3. holds in G ɛ β γ. γ.

21 Radial Maximal Function Characterizations 2 Set ν ji i= j= σ i V C 4 η i x ij V rij x ij r ɛ δ xijη irij where δ xr is the Dirac measure of point x r X R +. Notice that for any given i and j by Property B above we have S η i x ij f M fx ij [F η i x ij [M f] /2 χ Bij ] 2. Then by this and 3. r ji f ϕ σ i V C 4 η i x ij ɛ [ ] 2 F η i x ij [M f] /2 χ Bij V i= j= rij x ij [ 2 = F r x [M f] /2 χ BxC4 r] dνx r. X R + Denote by log 2 r the largest integer no more than log 2 r. Since r we then have ν r ɛ r ɛ t= log 2 r + t= log 2 r + 2 tɛ V 2 tx {i j: 2 t <2 t } σ i V C4 η i x ij δ xij η i 2 tɛ ν t. 3.9 If 2 t < 2 t then for any z Bx ij C 4 η i we have dz x dz x ij + dx ij x 2 < 2 t+ which implies that Bx ij C 4 η i Bx 2 t+. x r supp ν t From this it follows that for any By 3.9 and 3.2 f ϕ r ɛ F r x [M f] /2 χ BxC4 r F r x [M f] /2 χ Bx 2 t t= log 2 r 2 tɛ X R + [ 2 F r x [M f] /2 χ Bx2 ] t+ dνt x r. 3.2 We claim that for any fixed δ log η σ there exists a positive constant C δ independent of t such that for all x X and r > Vr x n+δ /n ν t Bx r r C δ V 2 tx

22 22 L. Grafakos L. Liu and D. Yang Assume this claim for the moment. By this claim 3.2 and Lemma 2.2 we have f ϕ r ɛ t= log 2 r + { 2 tɛ V 2 tx Bx C 4 2 t+ [ M M f n/n+δ n+δ /n x ] [M fx] n/n+δ dµx } n+δ /n where in the last step we use r ɛ t= log 2 r + 2 tɛ. Thus the desired conclusion of Proposition.7 holds. To finish the proof of Proposition.7 we still need to verify the validity of For any x X and r > set W {i j : 2 t < 2 t η i < r dx x ij < r}. For any i j W since C 4 < we have Bx ij C 4 η i Bx 2r which further implies that V C4 η i x ij V r x {j: i j W} If i j W then 2 t < 2 < 2rη i. From this we deduce that for any z Bx 2 t dz x dz x + dx x ij + dx ij x 2 t + 2 t + r 4η i + r and thus Bx 2 t Bx 4η i + r. Moreover by.2 V 2 tx 4η i + n V r x η in V r x Combining and the assumption δ < log η σ yields that ν t Bx r r = [V 2 tx ] [V 2 tx ] Bxrr {i j: 2 t <2 t } i j W σ i V C4 η i x ij Vr x n+δ /n σ i V 2 tx i N η δ σ i V C4 η i x ij dδ xij η i η in δ /n V r x V 2 tx Vr x n+δ /n V 2 tx which implies the claim and hence completes the proof of Proposition.7. Remark 3.. By the proof of Theorem.6 we need the assumption p > n/n + δ while in the proof of Proposition.7 we also need δ < log η σ. Thus Theorem.6 holds for any p n/n + log η σ ]. In fact Theorem.6 and Proposition.7 hold for all σ > and η > if σ η A > in 3. and H > in 3.5 satisfy the following

23 Radial Maximal Function Characterizations 23 five inequalities: H /3; C 4 η < H; η ɛ < σ < η β ; A C 8 2 AC ɛ 3 C 9 max{η ɛ 2 + Hη ɛ 2 } ; and H ɛ H ɛ AC3 C 9 σ + H H σ η ɛ σ AC 3C 9 σ. Acknowledgments. The authors are greatly indebted to the referees for their very valuable remarks which made this article more readable. References [] G. Alexopoulos Spectral multipliers on Lie groups of polynomial growth Proc. Amer. Math. Soc [2] M. Christ A T b theorem with remarks on analytic capacity and the Cauchy integral Colloq. Math. 6/ [3] R. R. Coifman and G. Weiss Analyse Harmonique Non-commutative sur Certains Espaces Homogènes Lecture Notes in Math. 242 Springer Berlin 97. [4] R. R. Coifman and G. Weiss Extensions of Hardy spaces and their use in analysis Bull. Amer. Math. Soc [5] X. T. Duong and L. Yan Hardy spaces of spaces of homogeneous type Proc. Amer. Math. Soc [6] D. Danielli N. Garofalo and D. M. Nhieu Non-doubling Ahlfors Measures Perimeter Measures and the Characterization of the Trace Spaces of Sobolev Functions in Carnot-Carathéodory Spaces Mem. Amer. Math. Soc no [7] C. Fefferman and E. M. Stein H p spaces of several variables Acta Math [8] L. Grafakos Modern Fourier Analysis Second Edition Graduate Texts in Math. No. 25 Springer New York 28. [9] L. Grafakos L. Liu and D. Yang Maximal function characterizations of Hardy spaces on RD-spaces and their applications Sci. in China Ser. A to appear. [] Y. Han Triebel-Lizorkin spaces on spaces of homogeneous type Studia Math [] Y. Han D. Müller and D. Yang A theory of Besov and Triebel-Lizorkin spaces on metric measure spaces modeled on Carnot-Carathéodory spaces Abstr. Appl. Anal. 28 Art. ID to appear. [2] Y. Han D. Müller and D. Yang Littlewood-Paley characterizations for Hardy spaces on spaces of homogeneous type Math. Nachr [3] Y. Han and E. T. Sawyer Littlewood-Paley Theory on Spaces of Homogeneous Type and the Classical Function Spaces Mem. Amer. Math. Soc. 994 no [4] J. Heinonen Lectures on Analysis on Metric Spaces Springer-Verlag New York 2. [5] R. A. Macías and C. Segovia A decomposition into atoms of distributions on spaces of homogeneous type Adv. in Math

24 24 L. Grafakos L. Liu and D. Yang [6] D. Müller and D. Yang A difference characterization of Besov and Triebel-Lizorkin spaces on RD-spaces Forum Math. to appear. [7] A. Nagel and E. M. Stein Differentiable control metrics and scaled bump functions J. Differential Geom [8] A. Nagel and E. M. Stein On the product theory of singular integrals Rev. Mat. Ibero [9] A. Nagel E. M. Stein and S. Wainger Balls and metrics defined by vector fields I. Basic properties Acta Math [2] E. M. Stein Some geometrical concepts arising in harmonic analysis Geom. Funct. Anal. 2 Special Volume Part I [2] E. M. Stein Harmonic Analysis: Real-Variable Methods Orthogonality and Oscillatory Integrals Princeton University Press Princeton N. J [22] E. M. Stein and G. Weiss On the theory of harmonic functions of several variables. I. The theory of H p -spaces Acta Math [23] A. Uchiyama A maximal function characterization of H p on the space of homogeneous type Trans. Amer. Math. Soc [24] N. Th. Varopoulos Analysis on Lie groups J. Funct. Anal [25] N. Th. Varopoulos L. Saloff-Coste and T. Coulhon Analysis and Geometry on Groups Cambridge University Press Cambridge 992. Loukas Grafakos Department of Mathematics University of Missouri Columbia MO 652 USA loukas@math.missouri.edu Liguang Liu & Dachun Yang Corresponding author School of Mathematical Sciences Beijing Normal University Laboratory of Mathematics and Complex Systems Ministry of Education Beijing 875 People s Republic of China s: liuliguang@mail.bnu.edu.cn L. Liu; dcyang@bnu.edu.cn D. Yang