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

(1)

ATOMIC DECOMPOSITIONS FOR WEAK HARDY SPACES WQ_p AND WD_p

YANBO REN AND DEWU YANG DEPARTMENT OFMATHEMATICS ANDPHYSICS

HENANUNIVERSITY OFSCIENCE ANDTECHNOLOGY

LUOYANG471003, P.R. CHINA

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

Received 18 March, 2008; accepted 11 October, 2008 Communicated by S.S. Dragomir

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

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

2000 Mathematics Subject Classification. 60G42, 46E45.

1. INTRODUCTION ANDPRELIMINARIES

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 spaceswQpandwDp, 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 ofΣsuch thatΣ =σ S

n≥0Σ_n

. The expectation operator and the conditional expectation operators relative toΣ_n are denoted by E and En, respectively. For a martingale f = (f_n)n≥0 relative to (Ω,Σ,P,(Σ_n)n≥0), define df_i = f_i − fi−1 (i ≥ 0, with convention df₀ = 0) and

f_n^∗ = sup

0≤i≤n

|f_i|, f^∗ =f_∞^∗ = sup

n≥0

|f_n|,

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

086-08

(2)

Sn(f) =

n

X

i=0

|dfi|²

!¹₂

, S(f) =

∞

X

i=0

|dfi|²

!¹₂ .

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

y>0

yP(|f|> y)¹^p <∞

is called a weak L_p-space and denoted by wL_p. We set wL∞ = L∞. It is well-known that k·k_wL

p is a quasi-norm on wL_p and L_p ⊂ wL_p since kfk_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→∞λ_n. 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}, kfkwD_p = inf

(λn)∈Λkλ∞k_wL_p.

Remark 1. Similar to martingale Hardy spacesQ_p andD_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 thatkfkwQp =kλ⁽¹⁾∞kwLpandkfkwDp =kλ⁽²⁾∞kwLp, which are called the optimal control ofS(f)andf, respectively.

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

(i) a_n =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 by ZandN, respectively. We useC_p to denote constants which depend only onpand may denote different constants at different occurrences.

2. MAINRESULTS AND PROOFS

Atomic decompositions for weak martingale Hardy spaces wQ_p and wD_p have been estab- lished 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_pkfkwQp;

(ii) If f = (fn)n≥0 ∈ wQp, then there exist a sequence (a^k)k∈Z of (3, p,∞)atoms and a sequence(µ_k)_k∈_Zof nonnegative real numbers such that for alln∈N:

(2.1) fn =X

k∈Z

µkEⁿa^k

and

(2.2) sup

k∈Z

2^kP(ν_k<∞)¹^p ≤C_p kf k_wQ_p,

(3)

where 0 ≤ µ_k ≤ A·2^kP(ν_k 6= ∞)¹^p for some constant A andν_k is the stopping time associated witha^k.

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

(2.3) |fn| ≤f_n−1^∗ +λn−1.

Define stopping times for allk∈Z:

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

The sequence of stopping times is obviously non-decreasing. Let f^ν^k = (fn∧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≤νk+1)dfm−

n

X

m=0

χ(m≤νk)dfm

!

=

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 = 0 ifµ_k = 0).It is clear that for a fixed k ∈ 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 ∈L₂ such thatEna^k =a^k_n, n≥0.

It is clear thata^k_n = 0ifn ≤ν_kand by (2.6) we get ka^k∗k∞ ≤P(ν_k 6=∞)⁻¹^p. Therefore each a^k is a(3, p,∞)atom, (2.4) and (2.5) shows thatf has a decomposition of the form (2.1) and 0≤µ_k ≤A·2^kP(ν_k 6=∞)¹^p withA= 3, respectively. By (i), we have

2^kpP(ν_k<∞) = 2^kpP(f^∗+λ∞>2^k)

≤2^kp(P(f^∗ >2^k−1) +P(λ∞>2^k−1))

≤Cpkfk^p_wQ_p, which proves (2.2).

(ii)⇒(i). Letf = (f_n)≥0 ∈ wQ_p. Thenf can be decomposed as in (ii)f_n = P

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

f =X

k∈Z

µka^k =

j−1

X

k=−∞

µka^k+

∞

X

k=j

µka^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).

(4)

For0 < p <∞, chooseqso thatmax(1, p) < q <∞. By (ii) and the fact thata^k∗ = 0on the set(ν_k =∞), we have

kg^∗k_q ≤

j−1

X

k=−∞

µ_kka^k∗k_q=

j−1

X

k=−∞

µ_kka^k∗χ(ν_k 6=∞)k_q

≤

j−1

X

k=−∞

A·2^k(1−^p^q⁾2^kp^q P(ν_k6=∞)¹^q

≤C_p

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]≤C_py^−pkfk^p_wQ

p. 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(νk 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_pkfkwQp.

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)k_wL_p ≤C_pkfkwDp;

(ii) If f = (f_n)n≥0 ∈ wD_p, then there exist a sequence (a^k)k∈Z of (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

and

(2.10) sup

k∈Z

2^kP(ν_k<∞)^p¹ ≤C_pkfk_wD

p,

where 0 ≤ µk ≤ A·2^kP(νk 6= ∞)¹^p for some constant A andνk is the stopping time associated witha^k.

(5)

Proof. (i)⇒(ii). Letf = (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:

νk = inf{n≥0 :Sn(f) + 2λn>2^k}, (inf∅=∞), anda^k_nandµ_k are as in the proof of Theorem 2.1. 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 ana^k such thatEna^k=a^k_n, n≥0. It is clear thata^kis a(2, p,∞)atom. Similar to the proof of Theorem 2.1, we can prove (2.9) and (2.10).

The proof of the implication (ii)⇒(i) is similar to that of Theorem 2.1.

The proof is completed.

Remark 2. The two inequalities in (i) of Theorems 2.1 and 2.2 were obtained in [3]. Here we establish the relation between atomic decompositions of weak martingale Hardy spaces and martingale inequalities.

REFERENCES

[1] A. BERNARD ANDB. MUSISONNEUVE, Decomposition Atomique de Martingales de la Class H₁LNM, 581, Springer-Verlag, Berlin, 1977, 303–323.

[2] C. HERZ,Hp-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 decompositions, Science in China, Series A, 49(7) (2006), 912–921.

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

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

[6] F. WEISZ, Martingale Hardy Spaces and their Applications in Fourier Analysis. Lecture Notes in Math, Vol. 1568, Springer-Verlag, 1994.