atomic decompositions of lorentz martingale spaces and...

JOURNAL OF c© 2009, Scientific Horizon

FUNCTION SPACES AND APPLICATIONS http://www.jfsa.net

Volume 7, Number 2 (2009), 153-166

Atomic decompositions of Lorentz martingale spaces

and applications

Jiao Yong, Peng Lihua and Liu Peide

(Communicated by Maria Carro)

2000 Mathematics Subject Classification. Primary 60G42; Secondary

60G46.

Keywords and phrases. Atomic decompositions, martingale inequalities,

Lorentz spaces, interpolation.

Abstract. In the paper we present three atomic decomposition theorems of

Lorentz martingale spaces. With the help of atomic decomposition we obtain a

sufficient condition for sublinear operator defined on Lorentz martingale spaces

to be bounded. Using this sufficient condition, we investigate some inequalities

on Lorentz martingale spaces. Finally we discuss the restricted weak-type

interpolation, and prove the classical Marcinkiewicz interpolation theorem in the

martingale setting.

1. Introduction and Preliminaries

The idea of atomic decomposition in martingale theory is derived fromharmonic analysis. Just as it does in harmonic analysis, the methodis key ingredient in dealing with many problems including martingaleinequalities, duality, interpolation and so on, especially for small-indexmartingale and multi-parameter martingale. As well known, Weisz [8]

154 Atomic decompositions of Lorentz martingale spaces

gave some atomic decompositions on martingale Hardy spaces and provedmany important theorems by atomic decompositions; Weisz [9] made afurther study of atomic decompositions for weak Hardy spaces consistingof Vilenkin martingale, and proved a weak version of the Hardy-Littlewoodinequality; Liu and Hou [5] investigated the atomic decompositions forvector-valued martingale and some geometry properties of Banach spaceswere charactered; Hou and Ren [3] considered the vector-valued weak atomicdecompositions and weak martingale inequalities; in [10], [11], the authorsdiscussed the operator interpolation by atomic decompositions of weightedmartingale Hardy spaces.

In this paper we present three atomic decomposition theorems for Lorentzmartingale spaces Hs

p,q, Qp,q, Dp,q. Applying these theorems, a sufficientcondition for a sublinear operator defined on the Lorentz martingale spacesto be bounded is given. And then we obtain some continuous imbeddingrelationships among Lorentz martingale spaces. These are new versions ofthe basic inequalities in the classical martingale theory. Finally we alsogive a restricted weak-type interpolation theorem, and obtain the version ofclassical Marcinkiewicz interpolation theorem in the martingale setting.

Let (Ω, Σ, P ) be complete probability space and f be a measurablefunction defined on Ω. The decreasing rearrangement of f is the functionf∗ defined by

f∗(t) = inf{s > 0 : P (|f | > s) ≤ t}.We adopt the convention inf Ø = ∞. The Lorentz space Lp,q(Ω) = Lp,q, 0 0

t1/pf∗(t), q = ∞.

It will be convenient for us to use an equivalent definition of ‖f‖p,q ,namely

‖f‖p,q =(q

∫ ∞

0

[tP (|f(x)| > t)1/p]qdt

t

)1/q

, 0 < q < ∞,

‖f‖p,∞ = supt>0

tP (|f(x)| > t)1/p, q = ∞.

To check that these two expressions are the same, simply make thesubstitution y = P (|f(x)| > t) and then integrate by parts.

It is well know that if 1 < p < ∞ and 1 ≤ q ≤ ∞, or p = q = 1, then Lp,q

is a Banach space, and ‖f‖p,q is equivalent to a norm. However, for other

J. Yong, P. Lihua, L. Peide 155

values of p and q , Lp,q is only a quasi-Banach spaces. In particular,if0 < q ≤ 1 ≤ p or 0 < q ≤ p < 1 then ‖f‖p,q is equivalent to aq−norm. Recall also that a quasi-norm ‖·‖ in X is equivalent to a p−norm,0 0 such that for any xi ∈ X, i = 1, ..., n

‖x1 + · · · + xn‖p ≤ c(‖x1‖p + · · · + ‖xn‖p

).

For all these properties, and more on Lorentz spaces, see for example [1, 2, 4].Holder’s inequality for Lorentz spaces is the following, which first appearsin work of O’Neil [7],

‖fg‖p,q ≤ c‖f‖p1,q1‖g‖p2,q2

for all 0 < p, q, p1, q1, p2, q2 ≤ ∞ such that 1p = 1

p1+ 1

p2and 1

q = 1q1

+ 1q2

.

Let {Σn}n≥0 a nondecreasing sequence of sub-σ -fields of Σ such thatΣ =

∨Σn . We denote the expectation operator and the conditional

expectation operator relative to Σn by E and En , respectively. Fora martingale f = (fn)n≥0, we define Δnf = fn − fn−1, n ≥ 0 (withconvention f−1 = 0, Σ−1 = {Ω, Φ})

Mn(f) = sup0≤i≤n

|fi|, M(f) = supn≥0

|fn|,

Sn(f) = (n∑

i=0

|Δif |2)1/2, S(f) = (∞∑

n=0

|Δnf |2)1/2,

sn(f) = (n∑

i=0

Ei−1|Δif |2)1/2, s(f) = (∞∑

n=0

En−1|Δnf |2)1/2.

Denote by Λ, the set of all non-decreasing, non-negative and adaptedr.v. sequences ρ = (ρn)n≥0 with ρ∞ = limn→∞ ρn . We shall say that amartingale f = (fn)n≥0 has predictable control in Lp,q if there is a sequenceρ = (ρn)n≥0 ∈ Λ such that

|fn| ≤ ρn−1, ρ∞ ∈ Lp,q.

As usually, we define Lorentz martingale spaces(see[1]),

H∗p,q = {f = (fn)n≥0 : ‖f‖H∗

p,q= ‖M(f)‖p,q < ∞},

Hsp,q = {f = (fn)n≥0 : ‖f‖Hs

p,q= ‖s(f)‖p,q < ∞},

HSp,q = {f = (fn)n≥0 : ‖f‖HS

p,q= ‖S(f)‖p,q < ∞},

Qp,q = {f = (fn)n≥0 : ∃(ρn)n≥0 ∈ Λ, s.t.Sn(f) ≤ ρn−1, ρ∞ ∈ Lp,q},‖f‖Qp,q = inf

ρ‖ρ∞‖p,q

Dp,q = {f = (fn)n≥0 : ∃(ρn)n≥0 ∈ Λ, s.t.|fn| ≤ ρn−1, ρ∞ ∈ Lp,q},‖f‖Dp,q = inf

ρ‖ρ∞‖p,q.

If we change the Lp,q−norms in above definitions by Lp−norms, we getthe usual Hardy martingale spaces (see [6]).

Remark. The norms of Qp,q and Dp,q are attainable respectively. Forexample, there exists (ρn)n≥0 ∈ Λ, Sn(f) ≤ ρn−1, ρ∞ ∈ Lp,q such that‖f‖Qp,q = ‖ρ∞‖p,q, which is also called the optimal control.

It turns out that Lorentz spaces, as many other quasi-Banach spaces,admit some sort of atomic decomposition. Firstly we give the definition ofan atom.

Definition [8]. A measurable function a is called a (1, p,∞)-atom(or(2, p,∞)-atom or (3, p,∞)-atom,respectively) if there exists a stoppingtime τ such that

(i) an = Ena = 0, n ≤ τ,

(ii) ‖s(a)‖∞ ≤ P (τ < ∞)−1p (or (ii) ‖S(a)‖∞ ≤ P (τ < ∞)−

1p

or (ii) ‖M(a)‖∞ ≤ P (τ < ∞)−1p , respectively).

Throughout the paper, we denote the set of integers and the set of non-negative integers by Z and N , respectively. We write A � B if A ≤ cB

for some positive constant c independent of appropriate quantities involvedin the expressions A and B .

2. Atomic decompositions

Now we can present the atomic decompositions for Lorenz martingalespaces.

Theorem 2.1. If the martingale f ∈ Hsp,q, 0 < p < ∞, 0 < q ≤ ∞ then

there exist a sequence ak of (1, p,∞)-atoms and a positive real numbersequence (μk) ∈ lq such that

fn =∑k∈Z

μkakn, n ∈ N

and‖(μk)k∈Z‖lq � ‖f‖Hs

p,q.

Conversely, if 0 < q ≤ 1, q ≤ p < ∞, and the martingale f has the abovedecomposition, then f ∈ Hs

p,q and

‖f‖Hsp,q

� inf ‖(μk)k∈Z‖lq ,

where the inf is taken over all the preceding decompositions of f .

Proof. Assume that f ∈ Hsp,q, q = ∞ . Now consider the following

stopping time for all k ∈ Z :

τk = inf{n ∈ N : sn+1(f) > 2k}(inf φ = ∞).

The sequence of these stopping times is obviously non-decreasing. It easyto see that

∑k∈Z

(f τk+1n − f τk

n ) =∑k∈Z

(n∑

m=0

χ{m≤τk+1}Δmf −n∑

m=0

χ{m≤τk}Δmf)

=∑k∈Z

(n∑

m=0

χ{τk<m≤τk+1}Δmf) = fn.

Let μk = 2k3P (τk < ∞)1p , and

akn =

fτk+1n − f τk

n

μk.

If μk = 0 then we assume that akn = 0. Then for a fixed k , (ak

n) is amartingale. Since s(f τk

n ) ≤ 2k, s(f τk+1n ) ≤ 2k+1,

s(akn) ≤ s(f τk+1

n ) + s(f τkn )

μk≤ P (τk < ∞)−

1p , n ∈ N,

which implies that (akn) is a L2−bounded martingale so that there exists

ak ∈ L2 such that Enak = akn. If n ≤ τk then ak

n = 0 and we get that ak

is really a (1, p,∞) atom and

(∑k∈Z

|μk|q) 1q = 3(

∑k∈Z

(2kP (τk < ∞)1p )q)

1q = 3(

∑k∈Z

(2kP (s(f) > 2k)1p )q)

1q

� (∑k∈Z

∫ 2k

2k−1yq−1dyP (s(f) > 2k)

qp )

1q

� (∑k∈Z

∫ 2k

2k−1yq−1P (s(f) > y)

qp dy)

1q

� (∫ ∞

0

yq−1P (s(f) > y)qp dy)

1q � ‖f‖Hs

p,q

For q = ∞ , standard rectifications can be made.Conversely, if f has the above decomposition, then from ‖s(ak)‖∞ ≤

P (τk < ∞)−1p and

P (s(ak) > y) ≤ P (s(ak) = 0) ≤ P (τk < ∞),

we get

‖ak‖qHs

p,q= q

∫ ∞

0

yq−1P (s(ak) > y)qp dy

= q

∫ P (τk<∞)− 1

p

0

yq−1P (s(ak) > y)qp dy

≤ P (τk < ∞)qp

∫ P (τk<∞)− 1

p

0

yq−1dy ≤ 1q.

For 0 < q ≤ 1, q ≤ p < ∞, ‖ · ‖p,q is equivalent to a q−norm,

‖f‖qHs

p,q≤ ‖

∑k∈Z

μks(ak)‖qp,q ≤

∑k∈Z

μqk‖s(ak)‖q

p,q �∑k∈Z

μqk,

which gives the desired result. �

Theorem 2.2. If the martingale f ∈ Qp,q, 0 < p < ∞, 0 < q ≤ ∞, thenthere exist a sequence (ak) of (2, p,∞) atoms and a real number sequence(μk) ∈ lq such that

fn =∑k∈Z

μkakn, ∀n ∈ N

and(∑k∈Z

|μk|q) 1q � ‖f‖Qp,q .

Conversely, if 0 < q ≤ 1, q ≤ p < ∞, and the martingale f has the abovedecomposition, then f ∈ Qp,q and

‖f‖Qp,q � inf(∑k∈Z

|μk|q) 1q ,

where the inf is taken over all the above decompositions.

Proof. Suppose that f ∈ Qp,q. Let β = (βn)n≥0 be the optimal controlof Sn(f), i.e., β ∈ Λ, Sn(f) ≤ βn−1, ‖f‖Qp,q = ‖β∞‖p,q. The stopping timesτk are defined in this case by

τk = inf{n ∈ N : βn > 2k}(inf φ = ∞).

Let ak and μk(k ∈ Z) be defined as in the proof of Theorem 2.1. Thenfor a fixed k , (ak

n) is also a martingale. Since S(f τkn ) = Sτk

(f) ≤ βτk−1 ≤2k, S(f τk+1

n ) ≤ 2k+1,

S(akn) ≤ S(f τk+1

n ) + S(f τkn )

μk≤ P (τk < ∞)−

1p , n ∈ N.

As in Theorem 2.1, we can show that ak is a (2, p,∞)-atom. Also

(∑k∈Z

|μk|q) 1q = 3(

∑k∈Z

(2kP (τk < ∞)1p )q)

1q

= 3(∑k∈Z

(2kP (β∞ > 2k)1p )q)

1q

� ‖β∞‖p,q = ‖f‖Qp,q .

Conversely, if ak is (2, p,∞)−atom, one can show that ‖ak‖qHS

p,q≤ 1

q . Therest can be proved similar to Theorem 2.1. �

Theorem 2.3. If the martingale f ∈ Dp,q, 0 < p < ∞, 0 < q ≤ ∞, thenthere exist a sequence (ak) of (3, p,∞)-atoms and a real number sequence(μk) ∈ lq such that

fn =∑k∈Z

μkakn, n ∈ N

and(∑k∈Z

|μk|q) 1q � ‖f‖Dp,q .

Conversely, if 0 < q ≤ 1, q ≤ p < ∞, and the martingale f has the abovedecomposition, then f ∈ Dp,q and

‖f‖Dp,q � inf(∑k∈Z

|μk|q) 1q ,

where the inf is taken over all the above decompositions.

The proof of Theorem 2.3 is similar to that of Theorem 2.2 and so weomit it.

3. Bounded operators on Lorentz martingale spaces

As one of the applications of the atomic decompositions, we shall obtaina sufficient condition for a sublinear operator to be bounded from Lorentzmartingale spaces to function Lorentz spaces. Applying the condition to

Mf, Sf and sf , we deduce a series of inequalities on Lorentz martingalespaces.

An operator T : X → Y is called a sublinear operator if it satisfies

|T (f + g)| ≤ |Tf |+ |Tg|, |T (αf)| ≤ |α||Tf |,

where X is a martingale space, Y is a measurable function space.

Theorem 3.1. Let T : Hsr → Lr be a bounded sublinear operator for

some 1 ≤ r < ∞. If

P (|Ta| > 0) � P (τ < ∞)

for all (1, p,∞)−atoms a , where τ is the stopping time associated with a ,then for 0 0 choosej ∈ Z such that 2j−1 ≤ y < 2j and let

f =∑k∈Z

μkak =j−1∑

k=−∞μkak +

∞∑k=j

μkak =: g + h.

Recall that μk = 2k3P 1/p(τk < ∞) and s(ak) = 0 on the set {τk = ∞}.we have

‖g‖Hsr

≤(∫

Ω

(j−1∑

k=−∞μks(ak))rdP

)1/r

≤j−1∑

k=−∞μk

(∫Ω

(s(ak))rdP

)1/r

≤j−1∑

k=−∞μk

(∫{τk≤∞}

‖s(ak)‖r∞dP

)1/r

≤j−1∑

k=−∞μkP (τk < ∞)−

1p P (τk < ∞)

1r

=j−1∑

k=−∞2kP (τk < ∞)

1r

=j−1∑

k=−∞2kP (s(f) > 2k)

1r

It follows from the boundedness of T that

P (|Tg| > y) ≤ y−rE|Tg|r � y−r‖g‖rHs

r

� y−r(j−1∑

k=−∞2kP (s(f) > 2k)

1r )r

= y−r(j−1∑

k=−∞2k(1− p

r )2k pr P (s(f) > 2k)

1r )r

� y−r(j−1∑

k=−∞2k(1− p

r ))r‖sf‖pp,∞

≤ y−p‖sf‖pp,∞.

On the other hand, since |Th| ≤∑∞k=j μk|Tak|, we get

P (|Th| > y) ≤ P (|Th| > 0) ≤∞∑

k=j

P (|Tak| > 0)

=∞∑

k=j

P (τk < ∞) =∞∑

k=j

2−kp2kpP (sf > 2k)

≤∞∑

k=j

2−kp‖sf‖pp,∞

� y−p‖sf‖pp,∞.

Since T is subliear,

P (|Tf | > y) ≤ P (|Tg| >y

2) + P (|Th| >

y

2) � y−p‖sf‖p

p,∞,

and thus for all 0 < p < r, T : Hsp,∞ → Lp,∞ is bounded. Now for any fixed

0 < p < r , we can choose 0 < p0, p1 < r, 0 < θ < 1 satisfying 1p = 1−θ

p0+ θ

p1.

From interpolation theorem (see Theorem 5.11 [2]) and the boundedness ofsublinear is hereditary for the interpolation spaces, we obtain for 0 < q ≤ ∞

T : Hsp,q = (Hs

p0,∞, Hsp1,∞)θ,q → (Lp0,∞, Lp1,∞)θ,q = Lp,q

is bounded. Hence‖Tf‖p,q � ‖f‖Hs

p,q.

�On the lines of the proof of Theorem 3.1, we can prove the following

Theorems 3.2 and 3.3 by using Theorems 2.2 and 2.3, respectively.

Theorem 3.2. Let T : Qr → Lr be a bounded sublinear operator forsome 1 ≤ r < ∞. If

P (|Ta| > 0) � P (τ < ∞)for all (2, p,∞)−atoms a , where τ is the stopping time associated with a ,then for 0 0) � P (τ < ∞)for all (3, p,∞)−atoms a , where τ is the stopping time associated with a ,then for 0 < p < r, 0 < q ≤ ∞, we have

‖Tf‖p,q � ‖f‖Dp,q , f ∈ Dp,q.

Theorem 3.4. For all martingale f = (fn)n≥0 the following imbeddingshold:

1) For 0 2, 0 < q ≤ ∞,

H∗p,q ↪→ Hs

p,q, HSp,q ↪→ Hs

p,q.

2) For 0 < p < ∞, 0 < q ≤ ∞,

Qp,q ↪→ H∗p,q, Qp,q ↪→ HS

p,q, Qp,q ↪→ Hsp,q

Dp,q ↪→ H∗p,q, Dp,q ↪→ HS

p,q, Dp,q ↪→ Hsp,q.

Proof. 1) The maximal operator Tf = Mf is sublinear, and‖Mf‖2 ≤ ‖sf‖2. If a is any (1, p,∞)−atom and τ is the correspondingstopping time, then {|Ta| > 0} = {|Ma| > 0} ⊂ {τ < ∞} and henceP (|Ta| > 0) ≤ P (τ < ∞). It follows from Theorem 3.1 that

‖Mf‖p,q � ‖f‖Hsp,q

, (0 2, 0 < q ≤ ∞.

In fact, we can regard martingale spaces as the subspaces of sequencespaces. Consider the operator Q : Lp(l∞) → Lp defined by Q(fn) = s(f) =

(∑∞

n=0 En−1|Δnf |2)1/2. For p ≥ 2, we know that ‖s(f)‖p � ‖M(f)‖p =‖ supn≥0 |fn|‖p and so Q : Lp(l∞) → Lp is bounded for all p ≥ 2. For anyfixed p > 2, we can choose p0, p1 > 2, 0 < θ < 1 satisfying 1

p = 1−θp0

+ θp1

.

Consequently Q : Lpi(l∞) → Lpi is bounded, i = 0, 1. By interpolation,for 0 < q ≤ ∞,

Q : Lp,q(l∞) = (Lp0(l∞), Lp1(l∞))θ,q → (Lp0 , Lp1)θ,q = Lp,q

is bounded. Hence we obtain

‖s(f)‖p,q � ‖ supn≥0

|fn|‖p,q = ‖M(f)‖p,q,

which gives H∗p,q ↪→ Hs

p,q . By considering Q defined on the sequence spaceLp(l2), we can similarly prove HS

p,q ↪→ Hsp,q

2) For all 0 < r < ∞ , ‖M(f)‖r, ‖S(f)‖r, ‖s(f)‖r � ‖f‖Qr and‖M(f)‖r, ‖S(f)‖r, ‖s(f)‖r � ‖f‖Dr . Note that ak

n = 0 on the set {n ≤ τk}.Thus

χ(n ≤ τk)En−1|Δnak|2 = En−1χ(n ≤ τk)|Δnak|2 = 0.

Hence s(ak) = 0 on the set {τk = ∞}. By Theorem 3.2 and 3.3, theassertion is proved. �

Remark. If we put p = q in the above embeddings, Theorem 2.11 in [8]can be deduced.

Remark. We conjecture that for 1 ≤ p < ∞, 0 < q ≤ ∞ , HSp,q = H∗

p,q ,however our method doesn’t show it.

4. Restricted weak type interpolation

We say that a sublinear operator T is of Lorentz-s restricted weak-type(p, q) if T maps Hs

p,1 to Lp,∞ . For convenience, we call T as restrictedweak-type (p, q). Then we have the next interpolation from one restrictedweak-type estimate to another.

Theorem 4.1. Let T be of restricted weak-type (pi, qi) for i = 0, 1 , and1 < pi, qi < ∞. Put

1p

=1 − θ

p0+

θ

p1,1q

=1 − θ

q0+

θ

q1, ∀0 ≤ θ ≤ 1.

Then T is also of restricted weak-type (p, q).

Proof. Suppose that f ∈ Hsp,1 . From Theorem 2.1, f =

∑k∈Z μkak, ak

is a (1, p,∞)−atom with respect to the stopping time τk, and∑

k∈Z |μk| �

‖f‖Hsp,1

. Now we can estimate ‖Tak‖q,∞ � 1. In fact

‖Tak‖q,∞ = supt>0

t1p (Tak)∗(t) = sup

t>0(t

1q0 (Tak)∗(t))1−θ(t

1q1 (Tak)∗(t))θ

≤ ‖Tak‖1−θq0,∞‖Tak‖θ

q1,∞� ‖sak‖1−θ

p0,1‖sak‖θp1,1

≤ ‖sak‖1−θ2p0,2p0

‖χ{τk<∞}‖1−θ2p0,l‖sak‖θ

2p1,2p1‖χ{τk<∞}‖θ

2p1,m

≤ P (τk < ∞)−1p (P (τk < ∞)

1−θ2p0 P (τk < ∞)

θ2p1 )2

≤ 1,

where l = 2p02p0−1 and m = 2p1

2p1−1 . Consequently,

‖Tf‖q,∞ ≤∑k∈Z

|μk|‖Tak‖q,∞ �∑k∈Z

|μk| � ‖f‖Hsp,1

and the proof is complete. �

Now we show how restricted weak-type estimate can be transferred tostrong type. It is also the version of the classical Marcinkiewicz interpolationtheorem in the martingale setting(see Theorem 4.13 in [1]).

Theorem 4.2. Let T be of restricted weak-type (pi, qi) for i = 0, 1 , and1 < pi < ∞, 1 < qi ≤ ∞, q0 = q1. Put

1p

=1 − θ

p0+

θ

p1,1q

=1 − θ

q0+

θ

q1, 0 ≤ θ ≤ 1.

Then T is of type (Hsp,r, Lq,r), for 0 < r < 1 and r ≤ q.

Proof. For 0 < r < 1 and r ≤ q, we know that ‖ · ‖q,r is equivalent toa r−norm, so it is enough to prove ‖Ta‖q,r � 1, for all (1, p,∞)−atoms.Once it is proved then from Theorem 2.1,

‖Tf‖rq,r ≤

∑k∈Z

μrk‖Ta‖r

q,r �∑k∈Z

μrk � ‖f‖r

Hsp,r

.

Now we shall show ‖Ta‖q,r � 1. Consider the case q1, q2 < ∞. From theproof of Theorem 4.1, it is easy to see that

‖a‖pi

Hspi,1

≤ P (τ < ∞)1−pip , i = 0, 1.

Thus, for q0 < q < q1, we get

1q‖Ta‖q

q,r =∫ ∞

0

yr−1P (|Ta| > y)rq dy

≤∫ δ

0

yr−1

(1y‖a‖Hs

p0,1

) q0rq

dy +∫ ∞

δ

yr−1

(1y‖a‖Hs

p1,1

) q1rq

dy

� δrq (q−q0)P (τ < ∞)

rq0q ( 1

p0− 1

p ) + δrq (q−q1)P (τ < ∞)

rq1q ( 1

p1− 1

p )

Take δ = P (τ < ∞)α with α satisfying

qα = (1p0

− 1p)/(

1q− 1

q0) = (

1p1

− 1p)/(

1q− 1

q1)

In fact, from 1p = 1−θ

p0+ θ

p1, 1

q = 1−θq0

+ θq1

we find that qα = ( 1p0

− 1p1

)/( 1q1−

1q0

), and

r

q

[α(q − q0) + q0

( 1p0

− 1p

)]=

r

q

[α(q − q1) + q1

( 1p1

− 1p

)]= 0.

Then ‖Ta‖qq,r � 1.

When one of qi is ∞, say q1 = ∞, the proof is modified. More precisely,we have

‖Ta‖∞ � ‖a‖Hsp1,1

� P (τ < ∞)1

p1− 1

p .

Thus, from 1p = 1−θ

p0+ θ

p1, 1

q = 1−θq0

1q‖Ta‖q

q,r =∫ ‖Ta‖∞

0

yr−1P (|Ta| > y)rq dy

≤∫ ‖Ta‖∞

0

yr−1(1y‖a‖Hs

p0,1)

q0rq dy

� P (τ < ∞)rq0

q ( 1p0

− 1p )

P (τ < ∞)rq (q−q0)( 1

p1− 1

p )

� 1

the assertion follows. �

Remark. From Theorems 2.2 and 2.3, we can conclude that the familiarresults hold for Qp,1 and Dp,1. We shall not state those explicitly.

Acknowledgement. The project was supported by the National NaturalScience Foundation of China (10671147) and China Scholarship Council(2007U13085). The authors would like to thank the referee and the editorfor their helpful suggestions.


References

[1] C. Bennett and R. Sharply, Interpolation of Operator, Academic press,1988.

[2] J. Bergh, J. Lofstrom, Interpolation Spaces, an Introduction, Berlin,Heidelberg, New York, Springer, 1976.

[3] Y.L. Hou and Y.B. Ren ,Vector-valued weak martingale Hardy spacesand atomic decompositions, Acta Math. Hungarica, 115 (2007), 235–246.

[4] R. Hunt, On L(p, q) spaces, L’Ens. Math. 12 (1966), 249–275.

[5] P.D. Liu and Y.L. Hou, Atomic decomposition of Banach-space-valuedmartingales, Sci in China, Serices A Math. 42 (1999), 38–47.

[6] R. L. Long, Martingale Spaces and Inequalities, Beijing, Peking UnivPress, 1993.

[7] R. O’Neil, Convolution operator and L(p, q) spaces, Duke. Math. J.,30 (1963), 129–143.

[8] F. Weisz, Martingale Hardy Spaces and Their Applications in FourierAnalysis, Lecture Notes in Math, Vol. 1568, New York:Sping-verlag,1994.

[9] F. Weisz, Bounded operator on weak Hardy spaces and applications,Acta Math. Hungarica, 80 (1998), 249–264.

[10] J. Yong, F. Li Ping and L. Peide Interpolation on weighted Lorentzmartingale spaces, Sci in China, Serices A Math., 50 (2007), 1217–1226.

[11] J. Yong, P.D. Liu and L.H. Peng, Interpolation for martingale Hardyspaces over weightedmeasure spaces, Acta Math. Hungar., 120 (2008),127-139.

Jiao Yong and Peng LihuaSchool of Mathematical Science and Computing TechnologyCentral South University, Chang Sha 410081China(E-mail : [email protected])

Liu PeideSchool of Mathematics and StatisticsWuhan UniversityWuhan 430072China

(Received : November 2007 )

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