Volume 9 (2008), Issue 4, Article 101, 5 pp.
ATOMIC DECOMPOSITIONS FOR WEAK HARDY SPACES WQp AND WDp
YANBO REN AND DEWU YANG
D EPARTMENT OF M ATHEMATICS AND P HYSICS
H ENAN U NIVERSITY OF S CIENCE AND T ECHNOLOGY
L UOYANG 471003, P.R. C HINA
ryb7945@sina.com.cn
dewuyang0930@163.com
Received 18 March, 2008; accepted 11 October, 2008
Communicated by S.S. Dragomir
A BSTRACT. In this paper some necessary and sufficient conditions for new forms of atomic
decompositions of weak martingale Hardy spaces wQp and wDp are obtained.
Key words and phrases: Martingale, Weak Hardy space, Atomic decomposition.
2000 Mathematics Subject Classification. 60G42, 46E45.
1. I NTRODUCTION AND P RELIMINARIES
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
and (Σn )n≥0 a non-decreasing sequence of
S space,
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 |,
f ∗ = f∞
= sup |fn |,
0≤i≤n
The author would like to thank the referees for their detailed comments.
086-08
n≥0
2
YANBO R EN AND D EWU YANG
Sn (f ) =
n
X
! 12
|dfi |2
,
S(f ) =
i=0
∞
X
! 12
|dfi |2
.
i=0
Let 0 < p < ∞. The space consisting of all measurable functions f for which
1
kf kwLp =: sup yP(|f | > y) p < ∞
y>0
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 },
kf kwQp = inf kλ∞ kwLp ;
(λn )∈Λ
wDp = {f = (fn )n≥0 : ∃(λn )n≥0 ∈ Λ, s.t. |fn | ≤ λn−1 , λ∞ ∈ wLp },
kf kwDp = inf kλ∞ kwLp .
(λn )∈Λ
Remark 1. Similar to martingale Hardy spaces Qp and Dp (see F.Weisz [6]), we can prove that
(1)
“ inf ” in the definitions of k·kwQp and k·kwDp is attainable. That is, there exist (λn )n≥0 and
(2)
(1)
(2)
(λn )n≥0 such that kf kwQp = kλ∞ kwLp and kf kwDp = 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,
1
1
(ii) kS(a)k∞ ≤ P(ν 6= ∞)− p (or (ii)0 ka∗ k∞ ≤ P(ν 6= ∞)− p ).
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. M AIN R ESULTS AND P ROOFS
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
µk En ak
(2.1)
fn =
k∈Z
and
(2.2)
1
sup 2k P(νk < ∞) p ≤ Cp k f kwQp ,
k∈Z
J. Inequal. Pure and Appl. Math., 9(4) (2008), Art. 101, 5 pp.
http://jipam.vu.edu.au/
W EAK H ARDY S PACES
3
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,
∗
|fn | ≤ fn−1
+ λn−1 .
(2.3)
Define stopping times for all k ∈ Z:
νk = inf{n ≥ 0 : fn∗ + λn > 2k },
(inf ∅ = ∞).
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
νk+1
νk
(fn − fn ) =
χ(m ≤ νk+1 )dfm −
χ(m ≤ νk )dfm
k∈Z
=
(2.4)
k∈Z
m=0
n
X
X
m=0
k∈Z
m=0
!
χ(νk < m ≤ νk+1 )dfm
= fn ,
where χ(A) denotes the characteristic function of the set A. Now let
(2.5)
1
µk = 2k · 3P(νk 6= ∞) p ,
νk+1
akn = µ−1
− fnνk ),
k (fn
(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
Consequently, (akn )n≥0 is L2 -bounded and so there exists ak ∈ L2 such that En ak = akn , n ≥ 0.
1
It is clear that akn = 0 if n ≤ νk and by (2.6) we get kak∗ k∞ ≤ P(νk 6= ∞)− p . Therefore each
ak is a (3, p, ∞) atom, (2.4) and (2.5) shows that f has a decomposition of the form (2.1) and
1
0 ≤ µk ≤ A · 2k P(νk 6= ∞) p with A = 3, respectively. By (i), we have
2kp P (νk < ∞) = 2kp P(f ∗ + λ∞ > 2k )
≤ 2kp (P(f ∗ > 2k−1 ) + P(λ∞ > 2k−1 ))
≤ Cp kf kpwQp ,
which proves (2.2).
P
(ii)⇒(i). Let f = (fn )≥0 ∈ wQp . Then f can be decomposed as in (ii) fn = k∈Z µk akn of
(3, p, ∞) atoms such that (2.2) holds. For any fixed y > 0 choose j ∈ Z such that 2j ≤ y < 2j+1
and let
j−1
∞
X
X
X
k
f=
µk a =
µ k ak +
µk ak =: g + h.
k∈Z
k=−∞
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).
J. Inequal. Pure and Appl. Math., 9(4) (2008), Art. 101, 5 pp.
http://jipam.vu.edu.au/
4
YANBO R EN AND D EWU YANG
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
j−1
X
k∗
µk ka kq =
k=−∞
µk kak∗ χ(νk 6= ∞)kq
k=−∞
j−1
≤
p
X
kp
1
A · 2k(1− q ) 2 q P(νk 6= ∞) q
k=−∞
≤ Cp
j−1
X
p
p
q
A · 2k(1− q ) kf kwQ
p
k=−∞
p
p
q
.
≤ Cp y 1− q kf kwQ
p
It follows that
P(g ∗ > y) ≤ y −q E[g ∗q ] ≤ Cp y −p kf kpwQp .
(2.7)
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= ∞)
2−kp · 2kp P(νk 6= ∞)
k=j
≤ Cp y −p kf kpwQp .
Combining (2.7) with (2.8), we get P(f ∗ > y) ≤ Cp y −p kf kpwQp . Hence
kf ∗ kwLp ≤ Cp kf kwQp .
The proof is completed.
Theorem 2.2. Let 0 < p < ∞. Then the following statements are equivalent:
(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
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 .
J. Inequal. Pure and Appl. Math., 9(4) (2008), Art. 101, 5 pp.
http://jipam.vu.edu.au/
W EAK H ARDY S PACES
5
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 .
i=0
Define stopping times for all k ∈ Z:
νk = inf{n ≥ 0 : Sn (f ) + 2λn > 2k },
(inf ∅ = ∞),
and akn and µk are as in the proof of Theorem 2.1. Then by (2.11) we have
1
νk+1
) + S(f νk )) ≤ P(νk 6= ∞)− p .
S(ak ) ≤ µ−1
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.
R EFERENCES
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H1 LNM, 581, Springer-Verlag, Berlin, 1977, 303–323.
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[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.
[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.
J. Inequal. Pure and Appl. Math., 9(4) (2008), Art. 101, 5 pp.
http://jipam.vu.edu.au/