Atomic decompositions and duals of weak Hardy spaces of B-valued martingales

Acta Mathematica Scientia 2009,29B(5):1439-1452 http://actams.wipm.ac.cn

ATOMIC DECOMPOSITIONS AND DUALS OF WEAK HARDY SPACES OF B-VALUED MARTINGALES∗

)

Ma Tao (




Liu Peide (

)

School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China E-mail: [email protected]; [email protected]

Abstract In this article, several weak Hardy spaces of Banach-space-valued martingales are introduced, some atomic decomposition theorems for them are established and their duals are investigated. The results closely depend on the geometrical properties of the Banach space in which the martingales take values. Key words martingale; atomic decomposition; weak Hardy space; geometry of Banach space 2000 MR Subject Classification

1

60G42; 46E40

Introduction

The atomic decomposition of Hardy spaces of scalar-valued martingales were investigated by Herz [1], Bernard and Masisonneuve [2], and Weisz [3, 4]. Liu, Hou, and Yu [5, 6] investigated the atomic decompositions in Banach-space-valued martingale spaces with small index. Recently, Weisz [7, 8] introduced a kind of weak Hardy space of martingales and gave the corresponding atomic decomposition theorems in the real-valued case by using the ideas of Fefferman [9]. In this article, we establish the atomic decompositions for other types of weak Hardy spaces of B-valued martingales and characterize their duals. Liu and Yu [10, 11] obtained a series of duality results in Hardy spaces of B-valued martingales. It is necessary to point out that the results are connected closely with the p-uniform smoothness or q-uniform convexity of Banach spaces in which the martingales take values. Let (Ω, Σ, P ) be a complete probability space and (Σn )n∈N a non-decreasing sequence of sub-σ-algebras of Σ. (X,  · ) denotes a Banach space with norm  · . The expectation and the conditional expectation with respect to Σn is denoted by E and En , respectively. Let f = (fn )n∈N be an X-valued martingale and dfn = fn − fn−1 its martingale difference with f−1 = 0. We define the maximal function, p-mean-square and conditional p-mean-square functions of f for 1 ≤ p < ∞ as follows: fn∗ = sup fm , f ∗ = sup fn ; m≤n

∗ Received

(10671147)

n≥0

October 17, 2006; revised July 28, 2007. Supported by the National Natural Foundation of China

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Sn(p) (f ) =



dfm 

p

1/p

, S (p) (f ) = sup Sn(p) (f ); n≥0

m≤n

σn(p) (f ) =



p

Em−1 dfm 

m≤n

Vol.29 Ser.B

1/p

, σ (p) (f ) = sup σn(p) (f ). n≥0

As well known, the weak Lr space wLr (0 < r < ∞) is the collection of all measurable functions f for which 1/r f wLr = sup λP (f  > λ) < ∞, λ>0

wLr is a quasi-normed space with f + gwLr ≤ K (f wLr + gwLr ) for some constants K > 1, and f wLr ≤ f Lr , i.e., Lr ⊂ wLr obviously. For 0 < r < ∞, 1 ≤ p < ∞, we define weak Hardy spaces of X-valued martingales as follows:   wHr (X) = f = (fn )n≥0 ; f wHr = f ∗ wLr < ∞ ;   wp HrS (X) = f = (fn )n≥0 ; f wpHrS = S (p) (f )wLr < ∞ ;   wp Hrσ (X) = f = (fn )n≥0 ; f wpHrσ = σ (p) (f )wLr < ∞ . Let (λn )n≥0 be an adapted, non-decreasing and non-negative sequence with λ∞ = lim λn , n→∞ we define wDr = {f = (fn ) ; ∃(λn ), s.t. fn  ≤ λn−1 , λ∞ ∈ wLr } , f wDr = inf λ∞ wLr , wp Qr = {f = (fn ) ; ∃(λn ), s.t. Sn(p) (f ) ≤ λn−1 , λ∞ ∈ wLr }, f wpQr = inf λ∞ wLr . In this article, the notions of smoothness, convexity and Radon-Nikodym(RN) property for a Banach space and some theorems on them refer to [12]. In particular, we will apply the following two results: Lemma 1.1 Let X be a Banach space and 1 < p ≤ 2, then X is isomorphic to puniformly smooth space iff there is a Cp > 0 such that f p ≤ Cp  S (p) (f ) p , f = (fn )n≥0 . Lemma 1.2 Let X be a Banach space and 2 ≤ q < ∞, then X is isomorphic to quniformly convex space iff there is a Cq > 0 such that  S (q) (f ) q ≤ Cq f q , f = (fn )n≥0 . In this article, we denote by Cp the constant relative only to p, and it may be different at different occurrences.

2

Weak Lipschitz Space

In the scalar case, the Lipschitz space p λβ was introduced as the dual of Hardy space of martingales with small index 0 < r ≤ 1 ([4, 13]). Yu and Liu [11] extended the concepts of Lipschitz spaces p λβ and p Λβ to B-valued martingales, and then, using the Lipschitz space σ p λβ , they investigated the dual of Hardy space of B-valued martingales p Hr (X), p Qr and Dr with index 0 < r ≤ 1. To characterize the dual space of weak Hardy space wH1 , Fefferman [9] gave a kind of weak Lipschitz space w1 λβ in the classical analysis. Using this idea, Weisz [8]

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investigated the dual of weak Hardy space of real-valued martingales wp Hrσ (1 ≤ p < ∞, 0 < r < p). We will extend the concept of weak Lipschitz space wp λβ (X) to B-valued martingales, and then characterize the daul of wp Hrσ , p Qr and wDr . First, let X be a Banach space with RN property (1 ≤ q < ∞, 0 ≤ β). The Lipschitz space q λβ of B-valued martingales is defined by q λβ = {f = (fn ) : f q λβ < ∞} with its norm f q λβ = sup P (τ < ∞)−1/q−β f∞ − f τ q ,

(2.1)

τ

where τ is a stopping time, f∞ = lim fn and the supremum is taken over all stopping times. n→∞

It is easy to see BMOq = q λ0 ([4]). Now, we denote tqβ (f, x) = tqβ (x) = x−1/q−β

sup P (τ <∞)≤x

f − f τ q ,

where 1 ≤ q < ∞, β > −1/q and then define the weak Lipschitz space of X-valued martingales as following:  ∞ q   tβ (x) wq λβ (X) = f = (fn ); f wq λβ = dx < ∞ . x 0 Obviously, for x ≥ 1, tqβ (x) ∼ x−1/q−β f q . Thus,  f wq λβ (X) ∼

1 tq (x) β

0

x

dx +

f q , 1/q + β

which means f ∈ Lq (X), when f ∈ wq λβ (X).

3

Atomic Decompositions

Definition 3.1 Let 1 ≤ p < ∞, 0 < r < ∞. An X-valued measurable function a is said to be a (1, p, r, ∞), ( or (2, p, r, ∞), (3, r, ∞)) atom, if there exists a stopping time τ such that (i)

an = En a = 0, if τ ≥ n; 1

1

(ii) σ (p) (a)∞ ≤ P (τ < ∞)− r ; (or (ii )S (p) (a)∞ ≤ P (τ < ∞)− r , (ii ) a∗ ∞ ≤ 1 P (τ < ∞)− r ). The set of all (1, p, r, ∞) (or (2, p, r, ∞), (3, r, ∞), respectively) atoms will be denoted by A1 (or A2 , A3 , respectively). Lemma 3.1 Let X be a Banach space and 1 < p ≤ 2, 0 < r < ∞. If X is isomorphic to a p-uniformly smooth Banach space, then sup E(a∗ )r ≤ C and sup E(a∗ )r ≤ C,

a∈A1

a∈A2

where C is a constant. Proof Let a be a (1, p, r, ∞) atom. It is easy to see that a∗ = 0 and σ (p) (a) = 0 on {τ = ∞}. Then according to H¨ older’s inequality, for 0 < r ≤ p, we get  r/p E(a∗ )r = E (a∗ )r χ{τ <∞} ≤ (E(a∗ )p ) P (τ < ∞)1−r/p .

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Since X is isomorphic to p-uniformly smooth space, according to Lemma 1.1, we have E(a∗ )p ≤ CEσ (p) (a)p , where C = Cp is the p-smoothable coefficient constant. And then r/p  P (τ < ∞)1−r/p E(a∗ )r ≤ C E(σ (p) (a))p r/p  = C E(σ (p) (a))p χ{τ <∞} P (τ < ∞)1−r/p ≤ Cσ (p) (a)r∞ P (τ < ∞)r/p P (τ < ∞)1−r/p ≤ Cσ (p) (a)r∞ P (τ < ∞) ≤ C.  For p < r, noticing that E σ (p) (a)r χ{τ <∞} ≤ 1, we also get E(a∗ )r ≤ CEσ (p) (a)r ≤ C. So we get sup E(a∗ )r ≤ C. It gives sup E(a∗ )r ≤ C. a∈A1

a∈A2

Lemma 3.2 Let X be a Banach space and 2 ≤ q < ∞, 0 < r < ∞. If X is isomorphic to a q-uniformly convex space, then sup ES (q) (a)r ≤ C,

a∈A3

sup Eσ (q) (a)r ≤ C,

a∈A3

where C is a constant. Proof Let a be a (3, r, ∞) atom. From the q-uniform convexity of X and Lemma 1.2, we have ES (q) (a)q ≤ CE(a∗ )q , then for 0 < r ≤ q, r/q  ES (q) (a)r = ES (q) (a)r χ{τ <∞} ≤ E(S (q) (a)q P (τ < ∞)1−r/q r/p  ≤ C E(a∗ )q χ{τ <∞} P (τ < ∞)1−r/q ≤ Ca∗ r∞ P (τ < ∞) ≤ C. This means sup ES (q) (a)r ≤ C for 0 < r ≤ p. It is easy to show the inequality remains true a∈A3

for q < r. By using the same method, we get the second inequality. Theorem 3.1 Let X be a Banach space, 1 < p ≤ 2, 0 < r ≤ p, then, the following are equivalent: (i) X is isomorphic to a p-uniformly smooth Banach space; (ii) For every X-valued martingale f = (fn )n≥0 ∈ wp Hrσ , there exists a sequence of (1, p, r, ∞) atoms (ak )k∈Z with the corresponding stopping times τk such that fn =



μk En ak =

k∈Z



μk akn ,

(3.1)

k∈Z ∗

where 0 ≤ μk ≤ A2k P (τ < ∞)1/r < ∞ for some constant A, and sup E(ak )r < ∞. Moreover k∈Z

f wpHrσ ∼ inf sup 2k P (τk < ∞)1/r , k∈Z

where the infimum is taken over all of such decompositions of f .

(3.2)

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Proof (i) ⇒ (ii) Let f = (fn )n≥0 ∈ wp Hrσ . For every integer k ∈ Z, we define stopping time (p)

τk = inf{n ∈ N; σn+1 (f ) > 2k } (inf ∅ = ∞), and the corresponding stopping martingale f (τk ) = (fn∧τk ) . Obviously, τk ↑ ∞ (n → ∞). Since σ (p) (f )wLr < ∞, then σ (p) (f ) < ∞ a.e.. It is easy to see 

fn(τk+1 )

−

fn(τk )

 =

∞  

dfm χ{τk
k∈Z m=0

k∈Z

Set 1

μk = 3 · 2k P (τk < ∞) r , and akn =

(τk+1 )

fn

(τk )

− fn μk

(k ∈ Z, n ∈ N),

then, ak = (akn )n≥0 is an X-valued martingale for any fixed k ∈ Z. From σ (p) (f (τk ) ) ≤ 2k , we get (p) (τk+1 ) σ (p) (ak ) ≤ μ−1 (f ) + σ (p) (f (τk ) )) ≤ P (τk < ∞)−1/r . k (σ

(3.3)

Since X is p-uniformly smoothable, according to Lemma 1.1, ∗

ak p ≤ Cσ (p) (ak )p ≤ CP (τk < ∞)−1/r < ∞. So ak = (akn )n≥0 is a Lp -bounded martingale. Since the uniform smoothness implies RN property, akn converges a.e., we also denote its limit by ak . It is clear that akn = En ak and 1 akn = 0 if τk ≥ n, and σ (p) (ak )∞ < P (τk < ∞)− r by (3.3). Therefore, ak is a (1, p, r, ∞) atom for each k ∈ Z, then (ii) is proved. By the definition of τk , {τk < ∞} = {σ (p) (f ) > 2k }, then we have 2kr P (τk < ∞) = 2kr P (σ (p) (f ) > 2k ) ≤ f rwpH σ < ∞,

(3.4)

r

∗

which gives sup 2k P (τk < ∞)1/r < ∞. And by using Lemma 1.1, we get sup E(ak )r < ∞. k∈Z

k∈Z

On the other hand, suppose that fn has the decomposition as (3.2) and every ak is a (1, p, r, ∞) atom with stopping time τk and 0 ≤ μk ≤ A2k P (τk < ∞)1/r . For any λ > 0, choose j ∈ Z such that A2j ≤ λ < A2j+1 and let f = g + h, g = (gn ), h = (hn ), where gn =

j−1 

μk akn , hn =

k=−∞

∞ 

(3.5)

μk akn .

k=j

Since μk ≤ A · 2k P (τ < ∞)−1/r < ∞, we have σ (p) (g) ≤

j−1  k=−∞

≤

j−1  k=−∞

j−1 

|μk | σ (p) (ak ) ≤

A2k P (τk < ∞)1/r P (τk < ∞)−1/r

k=−∞

A2k = A2j ≤ λ.

(3.6)

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Thus P (σ (p) (g) > λ) = 0. From the simple facts σ (p) (ak ) = 0 on {τk = ∞} and σ (p) (h) ≤ ∞ ∞

 |μk | σ (p) (ak ), we have {σ (p) (h) > 0} ⊂ {τk < ∞}, and then k=j

k=j

P (σ (p) (h) > λ) ≤ P (σ (p) (h) > 0) ≤ ≤



∞ 

P (τk < ∞)

k=j

∞  sup 2 P (τk < ∞) 2−kr ≤ Cr λ−r sup 2kr P (τk < ∞). kr

k∈Z

k∈Z

k=j

(3.7)

Combining (3.5), (3.6) and (3.7), we get P (σ (p) (f ) > 2λ) ≤ P (σ (p) (g) > λ) + P (σ (p) (h) > λ) ≤ Cr λ−r sup 2kr P (τk < ∞). k∈Z

It means f wp Hrσ = sup 2λP (σ (p) (f ) > 2λ)−1/r ≤ C sup 2k P (τk < ∞). λ>0

k∈Z

Combining this and (3.4), we prove (3.2). (ii) ⇒ (i)

Let f = (fn )n≥0 be an X-valued martingale with E ∞

E(σ (p) (f )p ) = E(S (p) (f )p ) = E

n=0

∞

n=0

dfn p < ∞. From

dfn p , we have f ∈ p Hrσ . Since the fact p Hrσ ⊂ wp Hrσ , we

know f ∈ wp Hrσ and f has the atomic decomposition as (3.2). For this f , we have    μrk = 3r 2kr P (τk < ∞) = 3r 2kr P (σ (p) (f )kr > 2kr ) k∈Z

k∈Z

k∈Z

2r 3r 3r  kr r Eσ (p) (f )r 2 (2 − 1)P (σ (p) (f )kr > 2kr ) ≤ r ≤ r 2 −1 2 −1 k∈Z

2r 3r f rpHrσ , = r 2 −1 which means that (μk ) ∈ lr for every f ∈ p Hrσ . And noticing that sup E(a∗ )r < ∞, we have Efm − fn r ≤ E



μrk akm − akn r ≤ C

k∈Z



k∈Z



μrk + C

|k|>k0

Eakm − akn r .

(3.8)

|k|≤k0

Since akn → ak n → ∞ in L1 for k ∈ Z, (3.8) means that f = (fn )n≥0 is a Cauchy sequence in Lr and thus converges in probability, hence X is p-uniformly smoothable (see Lemma 1 in [12]). Theorem 3.2 Let X be a Banach space, 1 < p ≤ 2, 0 < r ≤ p, then, the following are equivalent: (i) X is isomorphic to a p-uniformly smooth space; (ii) For every X-valued martingale f = (fn )n≥0 ∈ wp Qr , there exists a sequence of (2, p, r, ∞) atoms (ak )k∈Z with the corresponding stopping times τk such that   μk En ak = μk akn , (3.9) fn = k∈Z

k∈Z ∗

where 0 ≤ μk ≤ A · 2k P (τ < ∞) < ∞ for some constant A, and sup E(ak )r < ∞. Moreover, k∈Z

k

1/r

f wpQr ∼ inf sup 2 P (τk < ∞) k∈Z

,

(3.10)

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where the infimum is taken over all of such as above decompositions of f . Proof (i) ⇒ (ii) Let f = (fn )n≥0 ∈ wp Qr and (λn )n≥0 be a non-decreasing, non-negative (p) and adapted sequence such that Sn (f ) ≤ λn−1 and λ∞ ∈ wLr . For all k ∈ Z, we define (τk+1 ) τk = inf{n ∈ N; λn > 2k }, μk = 3 · 2k P (τk < ∞)1/r and akn = μ−1 − fn(τk ) ). k (fn

It is similar to prove Theorem 3.1 that ak = (akn )n≥0 is a (2, p, r, ∞) atom for each k ∈ Z and (3.9) holds. Because {τk < ∞} = {λ∞ > 2k }, we can get 2kr P (τk < ∞) = 2kr P (λ∞ > 2k ) ≤ λ∞ rwLr ,

(3.11) ∗

which gives sup 2k P (τk < ∞)1/r < ∞. By Lemma 3.1, we also get sup E(ak )r < ∞. k∈Z

k∈Z

On the other hand, suppose that fn has a decomposition of the form (3.9). We set  μk χ{τk ≤n} S (p) (ak )∞ , γn = k∈Z (p)

then (γn )n≥0 is a non-decreasing, non-negative and adapted sequence. Since Sn+1 (ak ) ≤ (p) (p) Sn+1 (ak )χ{τk ≤n} , we get Sn+1 (f ) ≤ γn . For any λ > 0, choose j ∈ Z such that A2j ≤ λ < A2j+1 , and then set j−1 

(1) (2) γ∞ = γ∞ + γ∞ =

μk χ{τk <∞} S (p) (ak )∞ +

k=−∞

+∞ 

μk χ{τk <∞} S (p) (ak )∞ .

k=j

Replacing σ (p) (g) and σ (p) (h) in (3.5) by γ (1) and γ (2) , respectively, we can get γ∞ rwLr ≤ Cr 2rk P (τk < ∞) analogously. It follows that f wpQr ≤ Cr sup 2k P (τk < ∞)1/r k∈Z

and (3.10) holds. (ii) ⇒ (i) Let f = (fn )n≥0 be an X-valued martingale with S (p) (f ) ∈ L∞ . Then f ∈ p Qr , ∗ so f ∈ wp Qr and fn has a decomposition as (3.9). By sup E(ak )r < ∞, similar to the proof of k∈Z

(ii) ⇒ (i) of Theorem 3.1, (fn )n≥0 converges in probability. This shows that X is p-uniformly smoothable. Theorem 3.3 Let Let X be a Banach space, 1 < p ≤ 2, 0 < r ≤ p, then the following are equivalent: (i) X has the RN property; (ii) For every X-valued martingale f = (fn )n≥0 ∈ wDr , there exists a sequence of (3, r, ∞) atoms (ak )k∈Z with the corresponding stopping times τk such that   μk En ak = μk akn , (3.12) fn = k∈Z

k∈Z

where 0 ≤ μk ≤ A · 2k P (τ < ∞) < ∞ for some constant A. Moreover f wDr ∼ inf sup 2k P (τk < ∞)1/r , k∈Z

(3.13)

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where the infimum is taken over all of such as above decompositions of f . Proof (i) ⇒ (ii) Let f = (fn )n≥0 ∈ wDr . For a fixed k ∈ Z, defining stopping time τk ∗ and akn is similar as Theorem 3.1, we can also get akn = 0 for τk ≥ n and ak ∞ ≤ P (τk < ∞)−1/r . Since X has the RN property, ak is an integrable function such that akn = En ak . Thus, ak is a weak (3, r, ∞) atom, and (3.12) is established. The equivalence of (3.13) can be verified in the same way as in Theorem 3.2. (ii) ⇒ (i) Suppose f = (fn )n≥0 that is an X-valued martingale with sup fn ∞ < ∞. n≥0

Then, f ∈ Dr ⊂ wDr , and f has a (3, r, ∞) atomic decomposition. By the definition of atom, ∗ ∗ we have E(ak )r = E(ak )r χ{τk <∞} ≤ 1. Similar to the poof of (ii) ⇒ (i) of Theorem 3.2, fn converges in probability. So X has the RN property.

4

The Duality Results of Weak Hardy Spaces

In order to characterize the dual of the weak Hardy space of martingales, we introduce the space Lr (lp (X)) (1 < p < ∞, 1 ≤ r ≤ p) firstly, Lr (lp (X)) denotes all sequences θ = (θ1 , θ2 , · · ·) of X-valued random variables satisfying ∞   1/p θLr (lp (X)) = θi p < ∞, r

i=1

with θLr (lp (X)) as its norm. We denote by X ∗ the dual of a normed (or quasinormed) space X. Lemma 4.1 Let X be a reflexive Banach space 1 < p < ∞, 1 ≤ r ≤ p, and 1p + 1q = 1, 1 1 r + s = 1, then for any bounded lineal functional l(θ) on Lr (lp (X)), there exists some ξ ∈ Ls (lq (X ∗ )), such that ∞  Eσi ξi , ∀θ ∈ Lr (lp (X)) , l(θ) = i=1

ξLs (lq (X ∗ )) ≤ l. Consider p HrS (X) as a subspace of Lr (lp (X)), which consists of those θ = df = (df1 , df2 , · · ·) for all f = (fn )n≥0 ∈ p HrS (X). We give the following corollary. Corollary 4.1 Let X be a reflexive Banach space and 1 < p < ∞, 1 ≤ r ≤ p, then (p HrS (X))∗ ∼ q HsS (X ∗ ), where

1 p

+

1 q

= 1 and

1 r

+

1 s

= 1.

Proof For ϕ ∈ q HsS (X ∗ ), define the linear functional lϕ (f ) = E Then |lϕ (f )| ≤

∞ 

E|dfn dϕn | ≤ E

∞  

n=1

dfn p

∞ 1/p  

n=1

n=1

dϕn q

∞

n=1

1/q 

dfn dϕn , f ∈ p HrS (X).

≤ f p HrS ϕq HsS .

Conversely, if l ∈ (p HrS (X))∗ , then from Lemma 4.1, there exists sequence ξ = (ξn )n∈Z ∈ Ls (lq (X ∗ )) of random variables such that l(f ) = E

∞  n=1

dfn ξn = E

∞  n=1

dfn (En ξn − En−1 ξn ),

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and

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∞   1/q ξn q ξLr (lq (X ∗ )) = ≤ l. r

n=1

Set ϕ = (ϕn ), dϕn = En ξn − En−1 ξn , then it is easy to see that ϕ is a martingale and ∞   1/q En ξn − En−1 ξn q ϕq HsS = ≤ CξLs (lq (X)) ≤ 2l. s

n=1

We shall consider the duals of closed subspaces of several weak Hardy spaces of B-valued martingales. Since the space Lp or L∞ is not dense in wLp (0 < p < ∞), hence in the next σ theorem, we denote by p H r the closure of p Hpσ in wp Hrσ in its norm. It is shown in [5] that ∞

μk ak converges to f in p Hrσ , and every ak (k ∈ Z) is also in p HpS , where ak is a (1, p, r, ∞)

k=−∞

σ is dense in p Hrσ and p HpS is dense in p Hrσ . Therefore, atom as in Theorem 3.1. Hence p H∞ σ both p HpS and p Hrσ are dense in p H r . Theorem 4.1 Suppose that X is isomorphic to a p-uniformly smooth Banach space, σ 1 < p ≤ 2, 0 < r ≤ 1, then wq λβ (X ∗ ) can be embedded continuously into the dual of p H r (X), where p1 + 1q = 1, β = 1r − 1. Proof Let ϕ ∈ wq λβ (X ∗ ), then ϕ ∈ Lq (X ∗ ). Since X is isomorphic to a p-uniformly smooth space, then X ∗ is isomorphic to a q-uniformly convex space, and ϕ ∈ q HqS (X ∗ ). From Corollary 4.1, we define the linear functional lϕ as

lϕ (f ) = E

∞ 

dfn dϕn and f ∈ p HpS (X).

n=1

From Theorem 3.1, there is a (1, p, r, ∞) atomic decomposition of f as (3.2) with μk = 3 · 2 P (τk < ∞)1/r , and ∞  μk dakn . dfn = k

k=−∞

Hence lϕ (f ) = E

∞ ∞  

μk dakn dϕn .

n=1 k=−∞

Applying H¨ older inequality and the definition of atoms, we get that ∞ ∞    E μk dakn χ{τk
k=−∞ ∞  k=−∞ ∞ 

n=1

∞ ∞   1/p   1/q E μk E dakn p dϕn q χ{τk
n=1

∞   1/p μk E dakn p S (q) (ϕ∞ − ϕτk )q .

k=−∞

(4.1)

n=1

∞  

dfn p , then Since E σ (p) (f )p = E S (p) (f )p = E n=1

 E

∞ 

dakn p

1/p

≤ (Eσ (p) (ak )p )1/p ≤ (Eσ (p) (ak )p χ{τk
n=1

≤ σ (p) (ak )∞ P (τk < ∞)1/p ≤ P (τ < ∞)1/p−1/r .

(4.2)

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Applying f wpHrσ ∼ inf sup 2k P (τk < ∞)1/r , we have k∈Z

P (τk < ∞) ≤ 2−kr f rwpHrσ (X) . From (4.1), (4.2), and the q-uniform convexity of X ∗ , we get |lϕ (f )| ≤ ≤

∞  k=−∞ ∞  k=−∞

μk P (τk < ∞)1/p−1/r ϕ − ϕτk q 1/p−1/r  3 2−kr f rwpHrσ (X) f wpHrσ (X) ϕ − ϕτk q

= 3f wpHrσ (X)

∞  1/q−(1/r−1)  2−kr f rwpHrσ (X) ϕ − ϕτk q .

k=−∞

We denote D = f rwpH σ (X) and tqβ (x) as in Section 2, then r

|lϕ (f )| ≤ 3f wpHrσ (X)

∞  k=−∞

tqβ (2−kr D), β =

1 − 1. r

(4.3)

It is simple to show that ∞  k=−∞

tqβ (D2−kr ) =

In fact, we also can get

 ∞ q ∞  tqβ (D2−kr )(D2−(k−1)r − D2−kr ) tβ (x) 1 dx. ≤ C r r −kr 2 −1 D2 x 0 k=−∞

∞ 0

tqβ (x) x dx

 0

≤ Cr

∞ tq (x) β

x

∞

k=−∞

tqβ (D2−kr ), so

dx ∼ Cr

∞  k=−∞

tqβ (D2−kr ).

(4.4)

(4.3) and (4.4) imply that |lϕ (f )| ≤ Cf wpHrσ (X) ϕwq λβ (X ∗ ) , σ

which means l ≤ Cϕwq λβ (X ∗ ) , and wq λβ (X ∗ ) can be embedded continuously into (pH r (X))∗ . Theorem 4.2 Suppose that X is isomorphic to a q-uniform convex Banach space, 2 ≤ σ q < ∞, 0 < r ≤ 1, then the dual of q H r (X) can be embedded continuously into wp λβ (X ∗ ), where p1 + 1q = 1, β = 1r − 1. σ σ Proof Let l ∈ (q H r (X))∗ . By noticing that q HqS (X) ⊂ q H r (X), l is in the dual of S S ∗ q Hq (X). So, there exists ϕ ∈ p Hp (X ) such that lϕ (f ) = E

∞ 

dfn dϕn , f ∈ q HqS (X).

n=1

Let τk be the stopping time satisfying P (τk < ∞) ≤ 2−kr (k ∈ Z). Notice that dϕn ∈ Lp (X ∗ ), then σn = d ϕn χ{τk
No.5

Ma & Liu: ATOMIC DECOMPOSITIONS AND DUALS OF WEAK HARDY SPACES

where

1449

∞ 1/q   . dϕn p χ{τk
akn

k

Since ϕ ∈ M < ∞, and = En a = 0 for τk ≥ n, namely, the definition (i) of the j atom holds. For any λ > 0, let 2 ≤ λ < 2j+1 and define the martingales f N , g N and hN as in Theorem 3.1, respectively, by fnN

N 

=

akn ,

gnN

=

k=−N

j−1 

akn , and hN n =

k=−N

N 

akn .

k=j

We have σ (q) (g N )p = S (q) (g N )q ≤

j−1 

S (q) (ak )q ,

k=−N

and S (q) (ak )q =

∞ 

Edakn q

1/q

n=1

∞ 1/q   q M  σn p−1 ≤ E(u − E u ) χ  n n−1 n {τ
≤

∞ 

2q Eσn pq−q un q χ{τk
n=1

≤2

∞ 

σn qq Eun p

n=1 ∞ 



≤C E

1/q M

1/p M

dϕn p χ{τk
1/q M

n=1

= C2kr(1/r−1/q) , so σ (q) (g N )q ≤

j−1 

C2kr(1/r−1/p) ≤ Cr λ1−r/q .

k=−N

And then P (σ (q) (g N ) > λ) ≤ λ−q σ (q) (g N )qq ≤ Cr λ−r .

(4.5)

Noticing that akn = 0 on {τk = ∞} and P (τk < ∞) ≤ 2−kr (k ∈ Z), we also get P (σ (q) (hN ) > λ) ≤

N 

P (τk < ∞) ≤ Cr λ−r .

(4.6)

k=j

Thus from (4.5) and (4.6), we obtain f N wq Hrσ ≤ Cr . Hence ∞       N   Cr l ≥ lϕ (f ) = E dfnN dϕn  n=0

=

∞ 

N 

n=0 k=−N

σn p−1 E (un dϕn − En−1 un dϕn ) χ{τk
 M

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ACTA MATHEMATICA SCIENTIA

=

N ∞   

Eun dϕn χ{τk
Vol.29 Ser.B

 M

k=−N n=0 N

=

∞

k=−N n=0

Edϕn p χ{τk
∞ 

1/q 2−kr(1/r−1/q) E dϕn p χ{τk
N ∞   1/p  E = dϕn p χ{τk
n=0

Since X is q-uniformly convexfiable, X ∗ is p-uniformly smoothable, then we have Cr l ≥

N N    ϕ − ϕτk 2kr(1/r−1/q) = ϕ − ϕτk 2−kr −1/p−(1/r−1) . p p

k=−N

k=−N

Taking over all N ∈ N and the supremum over all of such stopping times τk satisfying P (τk < ∞) ≤ 2−kr , k ∈ Z, and using (4.4), we obtain ϕwp λβ (X ∗ ) ≤ Cr

∞  l=−∞

tpβ (2−kr ) ≤ Cr l.

σ

This means that (q H r (X))∗ can be embedded continuously into wp λβ (X ∗ ). Corollary 4.2 If X is isomorphic to a Hilbert space, 0 < r ≤ 1, β = 1r − 1, then σ ∗ 2 H r (X) ∼ w2 λβ (X ). Similarly, p Qr denotes the closure of p Qr in the wp Qr norm and D r denotes the closure of Dr in the wDr with the norm below. Theorem 4.3 Let X be a p-uniformly smoothable Banach space, 1 < p ≤ 2, 0 < r ≤ 1, then wq λβ (X ∗ ) can be embedded continuously into the dual of p Qr (X), where 1p + 1q = 1, β = 1 r − 1. Proof By using the same method as in the proof of Theorem 4.1, we get wq λβ (X ∗ ) ⊆ S ∗ p Qr . Noticing that p Qp (X) ⊂ p Hp (X), for ϕ ∈ w q λβ (X ), we define a linear functional lϕ (f ) = E

∞ 

dfn dϕn , f ∈ p HpS (X).

n=1

Then by using the atom decomposition of f with (2, p, r, ∞) atoms ak (k ∈ Z), similar to the proof in Theorem 4.1, we get |lϕ (f )| ≤ Cf wp Qr (X) ϕwq λβ (X ∗ ) , which means l ≤ ϕwq λβ (X ∗ ) . Thus wq λβ (X ∗ ) can be embedded continuously into p Qr (X). Theorem 4.4 If X is a reflexive Banach space, then w1 λβ (X ∗ ) can be embedded continuously into Dr , where 0 < r ≤ 1 and β = 1r − 1. Proof Let ϕ ∈ w1 λβ (X ∗ ), then ϕ ∈ L1 (X ∗ ). Notice that X is reflexive, and define lϕ (f ) = E(f ϕ),

∀ f ∈ L∞ (X ∗∗ ) = L∞ (X),

No.5

Ma & Liu: ATOMIC DECOMPOSITIONS AND DUALS OF WEAK HARDY SPACES

then according to Theorem 3.3, we can take (3, r, ∞) atomic decomposition fn = from the definition of (3, r, ∞) atom, we have

k∈Z

1451 μk akn ,

E(ak ϕ) = E(ak (ϕ − ϕτk )). Then by H¨older inequality, we get ∞      lϕ (f ) = |E(f ϕ)| =  E(μk ak ϕ) k=−∞

∞ ∞       μk Eak (ϕ − ϕτk ) ≤ |μk |ak ∞ ϕ − ϕτk 1 =

≤

k=−∞ ∞ 

k=−∞

μk P (τk < ∞)−1/r ϕ − ϕτk 1 .

k=−∞

From (3.13), P (τk < ∞) ≤ 2−kr f rwDr , it is easy to check lϕ (f ) ≤ 3f wDr ≤ 3f wDr

∞ 

(2−kr f rwDr )−1/r ϕ − ϕτk 1

k=−∞ ∞ 

t1β (2−kr D) ≤ Cr f wDr ϕw1 λβ (X ∗ ) .

k=−∞

where D = f rwDr . The proof is completed. It is well known that L1 (R) ⊆ L∞ (R)∗ , so we consider the subspaces of (Dr (X))∗ . Define (Dr (X))∗1 = {l ∈ (Dr (X))∗ ; ∃ϕ ∈ L1 (X ∗ ), such that l(f ) = E(f ϕ), ∀f ∈ L∞ (X)}. Theorem 4.5 If X is a reflexive Banach space, then (Dr (X))∗1 can be embedded continuously into w1 λβ (X ∗ ), where 0 < r ≤ 1 and β = 1r − 1. Proof If l ∈ (Dr (X))∗1 , then ∃ϕ ∈ L1 (X ∗ ), such that lϕ (f ) = E(f ϕ), f ∈ L∞ (X). Since X is reflexive, by noticing that ϕ − ϕτk ∈ L1 (X ∗ ), then, there exists hk ∈ L∞ (X) with hk ∞ ≤ 1 such that E(ϕ − ϕτk )hk = ϕ − ϕτk 1 . We define the function in the following way ak = 2k (hk − hτkk ), where τk satisfies P (τk < ∞) ≤ 2−kr . Let f N =

N

k ∈ Z, ak . Then, we define the non-decreasing,

k=−N

non-negative and adapted sequence (γn )n ≥ 0 by γn−1 =

N 

2k χ{τk
k=−N

Hence, we have fnN  ≤ γn−1 . For a fixed λ > 0, let 2j ≤ λ < 2j+1 and (1) (2) γ∞ = γ∞ + γ∞ =

j−1  k=−N

2k χ{τk <∞} hk ∞ +

N  k=j

2k χ{τk <∞} hk ∞ .

1452

ACTA MATHEMATICA SCIENTIA

Since (1) γ∞

≤

j−1 

k

2 χ{τk <∞} ≤

k=−N

we have

(1) P (γ∞

j−1 

Vol.29 Ser.B

2k ≤ λ,

k=−N

> λ) = 0. On the other hand,

(2) (2) > λ) ≤ P (γ∞ > 0) ≤ P (γ∞

N 

P (τk < ∞) ≤

k=j

N 

2−kr ≤ Cr λ−r .

k=j

And we obtain (1) (2) P (γ∞ > 2λ) ≤ P (γ∞ > λ) + P (γ∞ > λ) ≤ Cr λ−r ,

which means f N wDr ≤ Cr . Hence N      2k E(hk − hτkk )ϕ Cr l ≥ |lϕ (f N )| =  k=−N

N N       ≥ 2k Ehk (ϕ − ϕτk ) = (2−kr )−1/r ϕ − ϕτk 1 . k=−N

k=−N

Similar to Theorem 4.2, we get ϕw1 λβ (X ∗ ) ≤ Cr

∞ 

t1β (2−kr ) ≤ Cr l.

k=−∞

The proof is completed. References [1] Herz C. Bounded mean oscillation and regulated martingales. Trans Amer Math Soc, 1974, 193: 199–215 [2] Bernard A, Maisonneuve B. D´ecomposition Atomique de Martingales de la Classe H 1 , S´ eminaire de Probabilit´es XI. Berlin, Heidelberg, New York: Springer, 1977 [3] Weisz F. Martingale Hardy spaces for 0 < p ≤ 1. Probab Th Rel Fields, 1990, 84: 361–376 [4] Weisz F. Martingale Hardy Spaces and Their Applications in Fourier-analysis. Berlin, Heidelberg, New York: Springer, 1994 [5] Liu P D, Hou Y L. Atomic decompositions for B-valued martingales. Sci China Ser A, 1998, 28(10): 884–892 [6] Liu P D, Yu L. B-valued martingale spaces with small index and atomic decompositions. Sci China Ser A, 2001, 31(7): 615–625 [7] Weisz F. Martingale operators and Hardy spaces generated by them. Studia Math, 1995, 114: 39–70 [8] Weisz F. Weak martingale Hardy spaces. Prob Math Stab, 1998, 18: 133–148 [9] Fefferman R, Soria F. The space weak H1 . Studia Math, 1987, 85: 1–16 [10] Liu P D. Fefferman’s inequality and the dual of a Hp . Chin Ann Math, 1991, 12A: 356–364 (in Chinese) [11] Yu L, Liu P D. On vector-valued martingale lipschitz spaces p λβ (X) and p Λβ (X). Acta Math Sinica, 2001, 44(1): 59–68 (in Chinese) [12] Liu P D. Martingales and Geometry of Banach Spaces. Wuhan: Wuhan University Press, 1993 [13] Long R L. Martingale Spaces and Inequalities. Beijing: Peiking University Press, 1993