ATOMIC DECOMPOSITIONS FOR WEAK HARDY SPACES wQp

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Weak Hardy Spaces Yanbo Ren and Dewu Yang vol. 9, iss. 4, art. 101, 2008

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ATOMIC DECOMPOSITIONS FOR WEAK HARDY SPACES wQ

_p

AND wD

_p

YANBO REN AND DEWU YANG

Department of Mathematics and Physics Henan University of Science and Technology Luoyang 471003, P.R. China

EMail:ryb7945@sina.com.cn dewuyang0930@163.com

Received: 18 March, 2008

Accepted: 11 October, 2008

Communicated by: S.S. Dragomir 2000 AMS Sub. Class.: 60G42, 46E45.

Key words: Martingale, Weak Hardy space, Atomic decomposition.

Abstract: In this paper some necessary and sufficient conditions for new forms of atomic decompositions of weak martingale Hardy spaces wQpand wDpare obtained.

Acknowledgements: The author would like to thank the referees for their detailed comments.

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1. Introduction and Preliminaries

It is well known that the method of atomic decompositions plays an important role in martingale theory, such as in the study of martingale inequalities and of the duality theorems for martingale Hardy spaces. Many theorems can be proved more easily through its use. The technique of stopping times used in the case of one-parameter is usually unsuitable for the case of multi-parameters, but the method of atomic decompositions can deal with them in the same manner. F.Weisz [6] gave some atomic decomposition theorems on martingale spaces and proved many important martingale inequalities and the duality theorems for martingale Hardy spaces with the help of atomic decompositions. Hou and Ren [3] obtained some weak types of martingale inequalities through the use of atomic decompositions.

In this paper we will establish some new atomic decompositions for weak martingale Hardy spaceswQ_pandwD_p, and give some necessary and sufficient conditions.

Let (Ω,Σ,P) be a complete probability space, and (Σ_n)n≥0 a non-decreasing sequence of sub-σ-algebras of Σ such that Σ = σ S

n≥0Σ_n

. The expectation operator and the conditional expectation operators relative to Σ_n are denoted by E andEn, respectively. For a martingalef = (f_n)n≥0 relative to(Ω,Σ,P,(Σ_n)n≥0), definedfi =fi−fi−1 (i≥0, with conventiondf0 = 0) and

f_n^∗ = sup

0≤i≤n

|f_i|, f^∗ =f_∞^∗ = sup

n≥0

|f_n|,

S_n(f) =

n

X

i=0

|df_i|²

!¹₂

, S(f) =

∞

X

i=0

|df_i|²

!¹₂ .

Let0< p <∞. The space consisting of all measurable functionsf for which kfk_wL_p =: sup

y>0

yP(|f|> y)¹^p <∞

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is called a weakL_p-space and denoted by wL_p. We set wL_∞=L_∞. It is well-known thatk·k_wL

p is a quasi-norm on wL_p andL_p ⊂ wL_p sincekfk_wL_p ≤ kfk_L_p. Denote by Λ the collection of all sequences (λ_n)n≥0 of non-decreasing, non-negative and adapted functions and setλ∞ = limn→∞λn. If0 < p < ∞, we define the weak Hardy spaces as follows:

wQ_p ={f = (f_n)n≥0 :∃(λ_n)n≥0 ∈Λ, s.t. S_n(f)≤λn−1, λ∞ ∈wL_p}, kfk_wQ_p = inf

(λn)∈Λkλ_∞k_wL_p;

wD_p ={f = (f_n)n≥0 :∃(λ_n)n≥0 ∈Λ, s.t.|f_n| ≤λn−1, λ∞∈wL_p}, kfk_wD_p = inf

(λn)∈Λkλ_∞k_wL_p.

Remark 1. Similar to martingale Hardy spaces Q_p and D_p (see F.Weisz [6]), we can prove that“ inf ”in the definitions of k·k_wQ

p andk·k_wD

p is attainable. That is, there exist (λ⁽¹⁾n )n≥0 and (λ⁽²⁾n )n≥0 such that kfkwQ_p = kλ⁽¹⁾∞k_wL_p and kfkwDp = kλ⁽²⁾∞k_wL_p, which are called the optimal control ofS(f)andf, respectively.

Definition 1.1 ([6]). Let0< p < ∞. A measurable functionais called a(2, p,∞) atom (or (3, p,∞)atom) if there exists a stopping time ν (ν is called the stopping time associated witha) such that

(i) an =Eⁿa= 0ifν ≥n,

(ii) kS(a)k_∞≤P(ν 6=∞)⁻¹^p (or (ii)⁰ ka^∗k_∞≤P(ν 6=∞)⁻¹^p).

Throughout this paper, we denote the set of integers and the set of non-negative integers byZandN, respectively. We useC_pto denote constants which depend only onpand may denote different constants at different occurrences.

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2. Main Results and Proofs

Atomic decompositions for weak martingale Hardy spaces wQ_pand wD_phave been established in [3]. In this section, we give them new forms of atomic decompositions, which are closely connected with weak type martingale inequalities.

Theorem 2.1. Let0< p <∞. Then the following statements are equivalent:

(i) There exists a constantC_p >0such that for each martingalef = (f_n)_≥0: kf^∗k_wL_p ≤C_pkfk_wQ_p;

(ii) Iff = (f_n)n≥0 ∈wQ_p, then there exist a sequence(a^k)k∈Zof(3, p,∞)atoms and a sequence(µk)k∈Zof nonnegative real numbers such that for alln∈N:

(2.1) f_n=X

k∈Z

µ_kEⁿa^k

and

(2.2) sup

k∈Z

2^kP(ν_k<∞)^p¹ ≤C_p kf kwQp,

where0≤µ_k≤A·2^kP(ν_k 6=∞)¹^p for some constantAandν_kis the stopping time associated witha^k.

Proof. (i)⇒(ii). Let f = (f_n)≥0 ∈ wQ_p. Then there exists an optimal control (λ_n)n≥0such thatS_n(f)≤λn−1. Consequently,

(2.3) |fn| ≤f_n−1^∗ +λn−1.

Define stopping times for allk∈Z:

νk= inf{n ≥0 :f_n^∗+λn>2^k}, (inf∅=∞).

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The sequence of stopping times is obviously non-decreasing. Letf^ν^k = (f_n∧ν_k)_n≥0 be the stopped martingale. Then

X

k∈Z

(f_n^ν^k+1−f_n^ν^k) = X

k∈Z n

X

m=0

χ(m ≤ν_k+1)df_m−

n

X

m=0

χ(m≤ν_k)df_m

!

=

n

X

m=0

X

k∈Z

χ(ν_k < m≤ν_k+1)df_m

!

=f_n, (2.4)

whereχ(A)denotes the characteristic function of the setA. Now let

(2.5) µ_k = 2^k·3P(ν_k 6=∞)¹^p, a^k_n=µ⁻¹_k (f_n^ν^k+1 −f_n^ν^k), (k ∈Z, n∈N) (a^k_n= 0ifµ_k = 0).It is clear that for a fixedk∈Z,(a^k_n)n≥0 is a martingale, and by (2.3) we have

(2.6)

a^k_n

≤µ⁻¹_k (|f_n^ν^k+1|+|f_n^ν^k|)≤P(νk 6=∞)⁻¹^p.

Consequently,(a^k_n)n≥0 isL₂-bounded and so there existsa^k ∈ L₂such thatEna^k = a^k_n, n ≥ 0. It is clear thata^k_n = 0ifn ≤ ν_k and by (2.6) we getka^k∗k∞ ≤ P(ν_k 6=

∞)⁻¹^p. Therefore each a^k is a (3, p,∞) atom, (2.4) and (2.5) shows that f has a decomposition of the form (2.1) and 0 ≤ µ_k ≤ A·2^kP(ν_k 6= ∞)¹^p with A = 3, respectively. By (i), we have

2^kpP(νk<∞) = 2^kpP(f^∗+λ∞>2^k)

≤2^kp(P(f^∗ >2^k−1) +P(λ_∞>2^k−1))

≤C_pkfk^p_wQ_p, which proves (2.2).

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(ii)⇒(i). Let f = (f_n)_≥0 ∈ wQ_p. Then f can be decomposed as in (ii)f_n = P

k∈Zµ_ka^k_n of (3, p,∞) atoms such that (2.2) holds. For any fixed y > 0 choose j ∈Zsuch that2^j ≤y <2^j+1and let

f =X

k∈Z

µ_ka^k =

j−1

X

k=−∞

µ_ka^k+

∞

X

k=j

µ_ka^k =:g+h.

It follows from the sublinearity of maximal operators that we havef^∗ ≤g^∗+h^∗, so P(f^∗ >2y)≤P(g^∗ > y) +P(h^∗ > y).

For0 < p < ∞, choose q so that max(1, p) < q < ∞. By (ii) and the fact that a^k∗ = 0on the set(ν_k=∞), we have

kg^∗kq ≤

j−1

X

k=−∞

µkka^k∗kq =

j−1

X

k=−∞

µkka^k∗χ(νk6=∞)kq

≤

j−1

X

k=−∞

A·2^k(1−^p^q⁾2^kp^q P(νk6=∞)¹^q

≤Cp j−1

X

k=−∞

A·2^k(1−^p^q⁾kfk

p q

wQp

≤C_py¹⁻^p^qkfk

p q

wQp. It follows that

(2.7) P(g^∗ > y)≤y^−qE[g^∗q]≤Cpy^−pkfk^p_wQ_p.

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On the other hand, we have

P(h^∗ > y)≤P(h^∗ >0)≤

∞

X

k=j

P(a^k∗ >0)

≤

∞

X

k=j

P(ν_k6=∞)

≤

∞

X

k=j

2^−kp·2^kpP(ν_k 6=∞) (2.8)

≤C_py^−pkfk^p_wQ_p.

Combining (2.7) with (2.8), we getP(f^∗ > y)≤C_py^−pkfk^p_wQ

p. Hence kf^∗k_wL_p ≤C_pkfkwQ_p.

The proof is completed.

Theorem 2.2. Let0< p <∞. Then the following statements are equivalent:

(i) There exists a constantC_p >0such that for each martingalef = (f_n)≥0: kS(f)kwLp ≤CpkfkwDp;

(ii) Iff = (f_n)_n≥0 ∈ wD_p, then there exist a sequence (a^k)_k∈_Zof(2, p,∞)atoms and a sequence(µ_k)_k∈_Zof nonnegative real numbers such that for alln∈N:

(2.9) f_n=X

k∈Z

µ_kEna^k

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and

(2.10) sup

k∈Z

2^kP(ν_k <∞)¹^p ≤C_pkfk_wD

p,

where0≤µk≤A·2^kP(νk 6=∞)¹^p for some constantAandνkis the stopping time associated witha^k.

Proof. (i)⇒(ii). Let f = (f_n)≥0 ∈ wD_p. Then there exists an optimal control (λ_n)n≥0such that|f_n| ≤λn−1. Consequently,

(2.11) S_n(f) =

n−1

X

i=0

|df_i|²+|df_n|²

!¹₂

≤Sn−1(f) + 2λn−1. Define stopping times for allk∈Z:

ν_k = inf{n≥0 :S_n(f) + 2λ_n >2^k}, (inf∅=∞), anda^k_nandµ_kare as in the proof of Theorem2.1. Then by (2.11) we have

S(a^k)≤µ⁻¹_k (S(f^ν^k+1) +S(f^ν^k))≤P(ν_k 6=∞)⁻¹^p. Thus

S(a^k)

_∞ ≤ P(ν_k 6= ∞)⁻¹^p and there exists an a^k such that Ena^k = a^k_n, n ≥ 0. It is clear thata^k is a (2, p,∞)atom. Similar to the proof of Theorem2.1, we can prove (2.9) and (2.10).

The proof of the implication (ii)⇒(i) is similar to that of Theorem2.1.

The proof is completed.

Remark 2. The two inequalities in (i) of Theorems2.1and2.2were obtained in [3].

Here we establish the relation between atomic decompositions of weak martingale Hardy spaces and martingale inequalities.

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References

[1] A. BERNARD AND B. MUSISONNEUVE, Decomposition Atomique de Mar- tingales de la ClassH₁ LNM, 581, Springer-Verlag, Berlin, 1977, 303–323.

[2] C. HERZ,H_p-space of martingales,0< p ≤1, Z. Wahrs. verw Geb., 28 (1974), 189–205.

[3] Y. HOUANDY. REN, Weak martingale Hardy spaces and weak atomic decom- positions, Science in China, Series A, 49(7) (2006), 912–921.

[4] R.L. LONG, Martingale Spaces and Inequalities, Peking University Press, Bei- jing, 1993.

[5] P. LIU AND Y. HOU, Atomic decompositions of Banach-space-valued martin- gales, Science in China, Series A, 42(1) (1999), 38–47.

[6] F. WEISZ, Martingale Hardy Spaces and their Applications in Fourier Analysis.

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