ATOMIC DECOMPOSITIONS FOR WEAK HARDY
SPACES wQp AND wDp
Weak Hardy Spaces
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
Yanbo Ren and Dewu Yang
vol. 9, iss. 4, art. 101, 2008
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dewuyang0930@163.com
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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 wQp and wDp are obtained.
Acknowledgements:
The author would like to thank the referees for their detailed comments.
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Contents
1
Introduction and Preliminaries
3
2
Main Results and Proofs
5
Weak Hardy Spaces
Yanbo Ren and Dewu Yang
vol. 9, iss. 4, art. 101, 2008
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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 spaces wQp and wDp , and give some necessary and sufficient conditions.
Let (Ω, Σ, P) be a complete probability space, and
a non-decreasing
S (Σn )n≥0
sequence of sub-σ-algebras of Σ such that Σ = σ n≥0 Σn . The expectation
operator and the conditional expectation operators relative to Σn are denoted by E
and En , respectively. For a martingale f = (fn )n≥0 relative to (Ω, Σ, P, (Σn )n≥0 ),
define dfi = fi − fi−1 (i ≥ 0, with convention df0 = 0) and
fn∗ = sup |fi |,
Sn (f ) =
n
X
n≥0
! 12
2
|dfi |
,
S(f ) =
i=0
∞
X
|dfi |2
y>0
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.
i=0
1
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! 12
Let 0 < p < ∞. The space consisting of all measurable functions f for which
kf kwLp =: sup yP(|f | > y) p < ∞
Yanbo Ren and Dewu Yang
vol. 9, iss. 4, art. 101, 2008
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∗
f ∗ = f∞
= sup |fn |,
0≤i≤n
Weak Hardy Spaces
is called a weak Lp -space and denoted by wLp . We set wL∞ = L∞ . It is well-known
that k·kwLp is a quasi-norm on wLp and Lp ⊂ wLp since kf kwLp ≤ kf kLp . Denote
by Λ the collection of all sequences (λn )n≥0 of non-decreasing, non-negative and
adapted functions and set λ∞ = limn→∞ λn . If 0 < p < ∞, we define the weak
Hardy spaces as follows:
wQp = {f = (fn )n≥0 : ∃(λn )n≥0 ∈ Λ, s.t. Sn (f ) ≤ λn−1 , λ∞ ∈ wLp },
Weak Hardy Spaces
kf kwQp = inf kλ∞ kwLp ;
Yanbo Ren and Dewu Yang
(λn )∈Λ
vol. 9, iss. 4, art. 101, 2008
wDp = {f = (fn )n≥0 : ∃(λn )n≥0 ∈ Λ, s.t. |fn | ≤ λn−1 , λ∞ ∈ wLp },
kf kwDp = inf kλ∞ kwLp .
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(λn )∈Λ
Remark 1. Similar to martingale Hardy spaces Qp and Dp (see F.Weisz [6]), we
can prove that “ inf ” in the definitions of k·kwQp and k·kwDp is attainable. That is,
(1)
(2)
(1)
there exist (λn )n≥0 and (λn )n≥0 such that kf kwQp = kλ∞ kwLp and kf kwDp =
(2)
kλ∞ kwLp , which are called the optimal control of S(f ) and f , respectively.
Definition 1.1 ([6]). Let 0 < p < ∞. A measurable function a is called a (2, p, ∞)
atom (or (3, p, ∞) atom) if there exists a stopping time ν (ν is called the stopping
time associated with a) such that
(i) an = En a = 0 if ν ≥ n,
− p1
(ii) kS(a)k∞ ≤ P(ν 6= ∞)
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(or (ii)0
− p1
ka∗ k∞ ≤ P(ν 6= ∞)
).
Throughout this paper, we denote the set of integers and the set of non-negative
integers by Z and N, respectively. We use Cp to denote constants which depend only
on p and may denote different constants at different occurrences.
2.
Main Results and Proofs
Atomic decompositions for weak martingale Hardy spaces wQp and wDp have 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. Let 0 < p < ∞. Then the following statements are equivalent:
(i) There exists a constant Cp > 0 such that for each martingale f = (fn )≥0 :
kf ∗ kwLp ≤ Cp kf kwQp ;
(ii) If f = (fn )n≥0 ∈ wQp , then there exist a sequence (ak )k∈Z of (3, p, ∞) atoms
and a sequence (µk )k∈Z of nonnegative real numbers such that for all n ∈ N:
X
(2.1)
fn =
µk En ak
k∈Z
and
(2.2)
1
p
k
sup 2 P(νk < ∞) ≤ Cp k f kwQp ,
k∈Z
1
where 0 ≤ µk ≤ A · 2k P(νk 6= ∞) p for some constant A and νk is the stopping
time associated with ak .
Proof. (i)⇒(ii). Let f = (fn )≥0 ∈ wQp . Then there exists an optimal control
(λn )n≥0 such that Sn (f ) ≤ λn−1 . Consequently,
(2.3)
∗
|fn | ≤ fn−1
+ λn−1 .
Define stopping times for all k ∈ Z:
νk = inf{n ≥ 0 : fn∗ + λn > 2k },
(inf ∅ = ∞).
Weak Hardy Spaces
Yanbo Ren and Dewu Yang
vol. 9, iss. 4, art. 101, 2008
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The sequence of stopping times is obviously non-decreasing. Let f νk = (fn∧νk )n≥0
be the stopped martingale. Then
!
n
n
X X
X
X
χ(m ≤ νk+1 )dfm −
χ(m ≤ νk )dfm
(fnνk+1 − fnνk ) =
k∈Z
n
X
m=0
m=0
k∈Z
Yanbo Ren and Dewu Yang
where χ(A) denotes the characteristic function of the set A. Now let
vol. 9, iss. 4, art. 101, 2008
k∈Z
=
(2.4)
(2.5)
m=0
!
X
1
χ(νk < m ≤ νk+1 )dfm
νk+1
− fnνk ),
akn = µ−1
k (fn
µk = 2k · 3P(νk 6= ∞) p ,
= fn ,
Weak Hardy Spaces
(k ∈ Z, n ∈ N)
(akn = 0 if µk = 0). It is clear that for a fixed k ∈ Z, (akn )n≥0 is a martingale, and by
(2.3) we have
k
an ≤ µ−1 (|fnνk+1 | + |fnνk |) ≤ P(νk 6= ∞)− p1 .
(2.6)
k
(akn )n≥0
k
kp
∗
k
2 P (νk < ∞) = 2 P(f + λ∞ > 2 )
≤ 2kp (P(f ∗ > 2k−1 ) + P(λ∞ > 2k−1 ))
≤ Cp kf kpwQp ,
which proves (2.2).
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k
Consequently,
is L2 -bounded and so there exists a ∈ L2 such that En a =
akn , n ≥ 0. It is clear that akn = 0 if n ≤ νk and by (2.6) we get kak∗ k∞ ≤ P(νk 6=
1
∞)− p . Therefore each ak is a (3, p, ∞) atom, (2.4) and (2.5) shows that f has a
1
decomposition of the form (2.1) and 0 ≤ µk ≤ A · 2k P(νk 6= ∞) p with A = 3,
respectively. By (i), we have
kp
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Let f = (fn )≥0 ∈ wQp . Then f can be decomposed as in (ii) fn =
P (ii)⇒(i).
k
µ
of
(3, p, ∞) atoms such that (2.2) holds. For any fixed y > 0 choose
a
k∈Z k n
j ∈ Z such that 2j ≤ y < 2j+1 and let
f=
X
j−1
X
µ k ak =
k∈Z
µ k ak +
∞
X
k=−∞
µk ak =: g + h.
k=j
It follows from the sublinearity of maximal operators that we have f ∗ ≤ g ∗ + h∗ , so
∗
∗
∗
P(f > 2y) ≤ P(g > y) + P(h > y).
For 0 < p < ∞, choose q so that max(1, p) < q < ∞. By (ii) and the fact that
ak∗ = 0 on the set (νk = ∞), we have
kg ∗ kq ≤
j−1
X
µk kak∗ kq =
k=−∞
≤
j−1
X
j−1
X
A·2
Yanbo Ren and Dewu Yang
vol. 9, iss. 4, art. 101, 2008
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Contents
µk kak∗ χ(νk 6= ∞)kq
k=−∞
k(1− pq )
Weak Hardy Spaces
kp
q
2 P(νk 6= ∞)
1
q
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k=−∞
≤ Cp
j−1
X
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k(1− pq )
A·2
p
q
kf kwQp
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k=−∞
p
p
q
≤ Cp y 1− q kf kwQ
.
p
It follows that
(2.7)
P(g ∗ > y) ≤ y −q E[g ∗q ] ≤ Cp y −p kf kpwQp .
Close
On the other hand, we have
∗
∗
P (h > y) ≤ P(h > 0) ≤
∞
X
P(ak∗ > 0)
k=j
≤
(2.8)
≤
∞
X
k=j
∞
X
P(νk 6= ∞)
Weak Hardy Spaces
Yanbo Ren and Dewu Yang
2−kp · 2kp P(νk 6= ∞)
vol. 9, iss. 4, art. 101, 2008
k=j
≤ Cp y −p kf kpwQp .
∗
Combining (2.7) with (2.8), we get P(f > y) ≤ Cp y
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−p
kf kpwQp .
Hence
∗
kf kwLp ≤ Cp kf kwQp .
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The proof is completed.
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Theorem 2.2. Let 0 < p < ∞. Then the following statements are equivalent:
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(i) There exists a constant Cp > 0 such that for each martingale f = (fn )≥0 :
kS(f )kwLp ≤ Cp kf kwDp ;
(ii) If f = (fn )n≥0 ∈ wDp , then there exist a sequence (ak )k∈Z of (2, p, ∞) atoms
and a sequence (µk )k∈Z of nonnegative real numbers such that for all n ∈ N:
X
(2.9)
fn =
µk En ak
k∈Z
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and
(2.10)
1
sup 2k P(νk < ∞) p ≤ Cp kf kwDp ,
k∈Z
1
where 0 ≤ µk ≤ A · 2k P(νk 6= ∞) p for some constant A and νk is the stopping
time associated with ak .
Proof. (i)⇒(ii). Let f = (fn )≥0 ∈ wDp . Then there exists an optimal control
(λn )n≥0 such that |fn | ≤ λn−1 . Consequently,
! 12
n−1
X
(2.11)
Sn (f ) =
|dfi |2 + |dfn |2
≤ Sn−1 (f ) + 2λn−1 .
Weak Hardy Spaces
Yanbo Ren and Dewu Yang
vol. 9, iss. 4, art. 101, 2008
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i=0
Define stopping times for all k ∈ Z:
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k
νk = inf{n ≥ 0 : Sn (f ) + 2λn > 2 },
(inf ∅ = ∞),
and akn and µk are as in the proof of Theorem 2.1. Then by (2.11) we have
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1
νk+1
S(ak ) ≤ µ−1
) + S(f νk )) ≤ P(νk 6= ∞)− p .
k (S(f
1
Thus S(ak )∞ ≤ P(νk 6= ∞)− p and there exists an ak such that En ak = akn ,
n ≥ 0. It is clear that ak is 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.
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References
[1] A. BERNARD AND B. MUSISONNEUVE, Decomposition Atomique de Martingales de la Class H1 LNM, 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. HOU AND Y. 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.
Weak Hardy Spaces
Yanbo Ren and Dewu Yang
vol. 9, iss. 4, art. 101, 2008
Title Page
[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.
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