BMO Lorentz martingale spaces

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R E S E A R C H

Open Access

BMO-Lorentz martingale spaces

Xueying Zhang

1*

_{, Xuming Yi}

1

_{and Chuanzhou Zhang}

2

*_{Correspondence:}

[email protected]

1_{School of Mathematics and}

Statistics, Wuhan University, Wuhan, Hubei 430072, China

Full list of author information is available at the end of the article

Abstract

In this paper BMO-Lorentz martingale spaces are investigated. We give the characterization of BMO-Lorentz martingale spaces. Moreover, we discuss the relationship between the Carleson measure and BMO-Lorentz martingales. As a consequence, we ﬁnd a new way to characterize the geometrical properties of a Banach space.

1 Introduction and preliminaries

Since  when they were ﬁrst introduced by Lorentz in [], Lorentz spaces have attracted more and more attention. A lot of results were obtained such as normability, duality, in-terpolation, and so on [–].

We know that martingale theory is intimately related to harmonic analysis. In martingale

case, Weisz [] and Long [] considered the spacesHp,q and the interpolations between

them, respectively. It is well known that the validity of a classical (scalar-valued) result in

the vector-valued setting,i.e., for functions or martingales with values in a Banach space

X, depends on the geometrical or topological properties ofX.

It was exactly with this in mind that Xu [] developed the vector-valued Littlewood-Paley theory, which was inspired by Pisier’s celebrated work [] on martingale inequalities in uniformly convex spaces. Very recently, Ouyang and Xu [] studied the endpoint case of the main results of [] by means of the classical relationship between BMO functions and Carleson measures. Jiao [] discussed the relationship between Carleson measures and vector-valued martingales.

Let (,μ) be a nonatomicσ-ﬁnite measure space. Suppose thatf is a measurable

func-tion on a measure space (,μ). We deﬁne its distribution function

λf(t) =μ

ω:f(ω)>t, t≥,

and its decreasing rearrangement function

f∗(t) =infs>  :λf(s)≤t

.

Given a measurable functionf on a measure space (,μ) and  <p,q≤ ∞, deﬁne

fLp,q=

⎧ ⎨ ⎩

(_∞(tp_f∗₍_t₎₎q dt t)



q _if_q_<∞_,

sup_t_>tp_f∗₍_t₎ _if_q₌∞_.

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Remark . Observe that for all  <p,r<∞and  <q≤ ∞we have

_|g|r_Lp,q =grLpr,qr. (.)

Unfortunately, the functions·Lp,qdo not satisfy the triangle inequality. However, since

for allt> , (f+g)∗(t)≤f∗(t/) +g∗(t/), we have

f+gLp,q≤c_p_,_q f_Lp,q+g_Lp,q, (.)

wherecp,q= /pmax{, (–q)/q}.

The set of allf withfLp,q<∞is denoted byLp,q(X,μ) and is called the Lorentz space

with indicespandq. Observe that the deﬁnition implies thatL∞,∞₌_L∞_.

Let (,,P) be a complete probability space, and let (n)n≥be a nondecreasing

se-quence of sub-σ-algebras ofwith=σ(_n_≥_n). We denote byEandEnthe

expecta-tion and condiexpecta-tional expectaexpecta-tion with respect toandn, respectively. For a martingale

f = (fn)n≥with martingale diﬀerencedfn=fn–fn–,n≥,f–≡, we deﬁne its maximal

function,p-square function, respectively, as usual:

Mf =sup

n

fn, S(p)(f) =

_∞

n=

dfnp

/p

.

We say that anX-valued martingalef = (fn)n≥∈Lp,q(X) ifsupnfnLp,q<∞.

The spaceBMOa

Lp,q₍_X₎(a≥,  <p,q≤ ∞) consists of all martingalef ∈Lp,q(X) such that

fBMOa_Lp,q₍_X₎=sup n

E f–fn–a|n

/a

Lp,q₍_X₎<∞.

Remark . The spacesBMOa_Lp,q₍_X₎are independent ofaand all corresponding norms are

equivalent. This allows us to denote any of them byBMOLp,q₍_X₎.

Proof If a≥, ϕ(x) =xa _{is a convex function, by Jensen’s inequality, we have} _E₍_f _– fn–|n)≤(E(f –fn–a|n))/a, which impliesE(f –fn–|n)∗(t)≤((E(f –fn–a| n))/a)∗(t),i.e.,BMOaLp,q₍_X₎⊂BMO_Lp,q₍_X₎.

On the contrary, letgn=E(f–fn–|n),Mg=supngn,hn=E(f–fn–a|n)/a,Mh=

sup_nhn. Now we setAt ={ω:Mh>t}. ThenMg>ta.e. onAt. (Factually, if there is a

B⊂Atwithμ(B) >  such thatMg≤tonB, thenE(f–fn–|n)≤t=E(t|n) a.e. onB,

which impliesf –fn– ≤ta.e. onB. This is a contradiction forAt.) So,P{ω:Mh>t} ≤

cP{ω:Mg>t}. Then

fBMOa_Lp,q₍_X₎≤

q

∞



tP Mh(ω) >t/pqdt t

/q

≤c

q

∞



tP Mg(ω) >t/pqdt t

/q

≤cf_BMO Lp,q(X),

i.e.,BMO_Lp,q₍_X₎⊂BMOa_Lp,q₍_X₎. Thus we complete the proof.

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Remark . For the classical BMO space, we have the following statement (see []):

fBMO∼sup τ

μ(τ<∞)–/pf–fτ–Lp, ≤p<∞. (.)

It is well known thatLp,q_{is a subspace of}_Lp,r_{for  <}_p_{≤ ∞}_{,  <}_q_<_r_{≤ ∞}_and_Lp,q

is a subspace ofLp,q _{for  <}_p ≤_p≤ ∞_{,  <}_q_,_q≤ ∞_{. Thus we have the following}

proposition.

Proposition . ()If <p≤ ∞,  <q<r≤ ∞,BMOLp,q⊆BMO_Lp,r.

()If <p≤p≤ ∞,  <q,q≤ ∞,BMOLp,q ⊆BMOLp,q.

Theorem . Let f = (fn)n≥be an X-valued martingale in Lp,q(X),  <p,q≤ ∞.Then f ∈BMOLp,q₍_X₎if and only if there exists an adapted processθ= (θ)_n_≥_such that

Cθ=sup

n

E f –θn–|n_Lp,q₍_X₎<∞.

And,in any case,we have

|f|BMO_Lp,q₍_X₎:=inf

θ Cθ≤ fBMOLp,q(X)≤c|f|BMOLp,q(X).

Proof Assumef ∈BMOLp,q₍_X₎. Then, obviously,

|f|BMOLp,q(X)≤ fBMOLp,q(X).

Now let|f|BMOLp,q(X)<∞andθ= (θ)n≥be any one such thatCθ<∞. Then we have

E f –fn–|n

≤E f –θn–|n

+θn––fn–

=E f –θn–|n

+E (f –θn–)|n–

≤E f –θn–|n

+E f–θn–|n–

=E f –θn–|n

+E E f–θn–|n

|n–

.

Taking ‘inf’ over all possibleθ and (.), we get the desired inequality.

Theorem . Let f = (fn)n≥be an X-valued martingale in BMOLp,q₍_X₎,where  p +

 s =

 r and_q+_s

 = 

s,  <p,q,r,s,s≤ ∞.Then

fBMOLp,q(X)∼sup τ

μ {τ<∞}–s__f _–_fτ

–Lr,s₍_X₎,

where‘sup’is taken over all stopping timesτ.

Proof Assume thatfBMO_Lp,q₍_X₎<∞,τis any stopping time. Then, by Hölder’s inequality,

we have

f–fτ–Lr,s₍_X₎ =(f–fτ–)χ_{_τ_<_∞} Lr,s₍_X₎

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≤c sup g(Lp,q)∗≤

_{_τ_<_∞} f–fτ–g dμ· χ{τ<∞}Ls,s₍_X₎

=c

q p

/q

sup g(Lp,q)∗≤

_{_τ_<_∞}E f –fτ– ·g|τdμ·μ(τ<∞)s

≤c

q p

/q

fBMOLp,q ·μ(τ <∞)  s_.

This proves one half of the assertion. Conversely, assume thatβ=sup_τμ({τ <∞})–s__f_–

fτ–Lr,s₍_X₎<∞, andτis any stopping time,F∈τ,F⊂ {τ<∞}. Deﬁne

τF=

⎧ ⎨ ⎩

τ ifω∈F,

∞ ifω∈/F.

Thus we get

β≥μ {τ<∞}–

 s_f _–_fτ

–Lr,s₍_X₎

= 

μ(F)/sf –fτF–L r,s₍_X₎

≥ 

μ(F)/sf –fτF–Lr∧,r∧(X)

≥ 

μ(F)f –fτF–L,(X)

= 

μ(F)

F

f–fτF–du.

That is,E(f–fτF–|τF)≤β. By Remark . we have

fBMOLp,q(X)≤cβ.

Thus we complete the proof of the theorem.

Proposition . Particularly,if s=r and p=q=∞,we get Remark..

By Theorem . and Theorem ., we have the following proposition.

Proposition . Let f = (fn)n≥be an X-valued martingale in BMOLp,q₍_X₎,where p+

 s=

 r and_q+_s

 = 

s,  <p,q,r,s,s≤ ∞.Then

fBMO_Lp,q₍_X₎∼inf θ supτ

μ {τ <∞}–s__f _–_θ

τ–Lr,s₍_X₎,

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2 Carleson measure and BMO-Lorentz martingale spaces

Deﬁnition . Letνbe a nonnegative measure on×N, whereNis equipped with the

counting measuredm. Letτˆdenote the tent overτ:

ˆ

τ=(ω,k) :k≥τ(ω),τ(ω) <∞.

νis said to be ans-Carleson measure if

|ν|=sup

τ

ν(τˆ)

μ({τ<∞})s <∞,

whereτ runs through all stopping times.

Theorem . Let  <p,q≤ ∞,and g= (gn)n≥be a real-valued martingale and dν=

|kg|δkdμ,whereδkis the Dirac measure centered at k.So,if g∈BMO_Lp,q,νis a/p -Carleson measure.Moreover,if <p,q<∞,the converse is also true,where _p +_p = ,

 q +

 q = .

Proof Letg= (gn)n≥be a real-valued martingale, letνbe generated bygas above, and let

τ be any stopping time. Then, for  <q<∞,

ν(τˆ) =E

_∞

k=

|kg|χ{τ(ω)≤k}

=E

E

_∞

k=τ(ω)

|kg|τ

χ_{τ(ω)≤k}

=E E |g–gτ–||τ

χ{τ(ω)≤k}

≤cE |g–gτ–||τ_Lp,q· χ{τ(ω)≤k}_Lp,q

=cE |g–gτ–||τ

 ·

Lp,q· χ{τ(ω)≤k}Lp,q

≤cE |g–gτ–||τ

/

Lp,q·μ τ(ω)≤k

/p

≤cg_BMO Lp,q

·μ τ(ω)≤k/p.

So,g∈BMO

Lp,q implies thatνis a /p-Carleson measure and|ν| ≤ g_BMO

Lp,q

.

Conversely, for anynand anyF∈n, we deﬁne

τ= ⎧ ⎨ ⎩

n ifω∈F,

∞ ifω∈/F.

Sinceνis a /p-Carleson measure, we have

|ν| ≥ 

μ({τ<∞})/pν (ω,k) :k≥τ(ω),τ(ω) <∞

= 

μ(F)/p

F

∞

k=n

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≥ 

μ(F)

F

∞

k=n

|kg|dμ

= 

μ(F)

F

|g–gn–|dμ.

That is,|ν| ≥E(|g–gn–||n). Then we haveg_BMO

Lp,q

≤c|ν|. Thus, we complete

the proof of the theorem.

3 The characterization of Banach space’s geometrical properties

Let  <q<∞. Then a Banach spaceXhas an equivalentq-uniformly convex norm if and

only if for one  <p<∞(or equivalently, for every  <p<∞) there exists a positive

con-stantcsuch that

_S(q)₍_f₎

p≤csup n

fnp

for all ﬁniteLp-martingalesf with values inX. Again, the validity of the converse inequality

amounts to saying thatXhas an equivalentq-uniformly smooth norm.

Deﬁnition . LetX andX be two Banach spaces. LetL(X,X) denote the space of

all bounded linear operators fromXtoX. Letv= (vn)n≥ be an adapted sequence such

thatvn∈L∞(L(X,X)) andsupn≥vnL∞(L(X,X))≤. Then the martingale transformT

associated tovis deﬁned as follows. For anyX-valued martingalef = (fn)n≥,

(Tf)n= n

k= vkdfk.

We get the following results from [, ].

Lemma . With the assumptions above,the following statements are equivalent: () There exists a positive constantcsuch that

TfBMO(X)≤cfBMO(X), ∀f ∈BMO(X).

() There exists a positive constantcsuch that

₍_Tf₎∗

BMO(X)≤cfBMO(X), ∀f∈BMO(X).

() For some≤p<∞(or equivalently,for every≤p<∞),there exists a positive constantcsuch that

TfLp(X)≤cfLp(X), ∀f ∈Lp(X).

Theorem . Let X be a Banach space, ≤q<∞,  <p<∞.Then the following state-ments are equivalent:

() There exists a positive constantcsuch that for any ﬁniteX-valued martingale, S(q)(f)_BMO

Lp,q(X)≤cfBMOLp,q(X).

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Proof ()⇒() Let_p+_s

=  r,

 q+

 s =



r,  <p,q,r,s,s≤ ∞By Theorem . we have

S(q)(f)_BMO

Lp,q(X)∼sup_τ μ {τ<∞}

–s__S(q)₍_f_{) –}_S(q)

τ–(f)Lr,r₍_X₎, (.)

fBMOa_Lp,q₍_X₎∼sup τ

μ {τ<∞}–s_f _–_fτ

–Lr,r₍_X₎. (.)

So, if () holds, we have

S(q)(f) –Sτ(q–)(f)Lr,r₍_X₎≤cf –fτ–Lr,r₍_X₎.

By Remark . we have

S(q)(f)_BMO≤cfBMO. (.)

We now consider a martingale transform operatorQfrom the family ofX-valued

mar-tingales to that oflq(X)-valued martingales. Letv∈L(X,lq(X)) be the operator deﬁned

byvk(x) ={xj}∞j=forx∈X, wherexj=xifj=kandxj=  otherwise.Qis the martingale

transform associated to the sequence (vk):

(Qf)n= n

k=

vkdfk= (df,df, . . . ,dfn, , . . .).

Then

(Qf)∗=S(q)(f). (.)

By (.) and (.) we have

(Qf)∗_BMO≤cfBMO.

By Lemma . we have

S(q)(f)_q=(Qf)∗_q≤cfq.

Thus, by Pisier’s theorem,Xhas an equivalentq-uniformly convex norm.

()⇒() Suppose thatXhas an equivalentq-uniformly convex norm. By Pisier’s

theo-rem [], we ﬁnd, for any ≤n<∞,

E

_∞

i=n

dfiqn

≤cE f–fn–q|n

.

SinceE(|S(q)₍_f_{) –}_S(q)

n–(f)|q|n)≤cE(|(S(q)(f))q– (Sn(q–)(f))q||n), we have

E S(q)(f) –S(_nq_–)(f)q|n

≤cE f –fn–q|n

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Thus

S(q)(f)_BMO

Lp,q(X)≤cfBMOLp,q(X).

We complete the proof.

Theorem . Let X be a Banach space and <p≤,  <q<∞.If there exists a positive constant c such that for any ﬁnite X-valued martingale

fBMOLp(X)≤cS (p)₍_f₎

BMO_Lp₍_X₎, (.)

then X has an equivalent norm which is p-uniformly smooth.

On the contrary,if X has an equivalent norm which is p-uniformly smooth,then

fBMOa_Lp,q₍_X₎≤cS (p)₍_f₎

BMOa_Lp,q₍_X₎

for every martingale f.

Proof Let X∗ be the dual space of X. It suﬃces to prove that X∗ has an equivalent

p-uniformly convex norm, wherepis the conjugate index ofp. By Pisier’s theorem, this

is equivalent to showing that

S(p)(g)_L

p ≤c

g∗_L

p =cgH

∗

p(X∗). (.)

Recall thatH_p∗(X∗) is deﬁned by

H_p∗ X∗=X∗-valued martingaleg= (gn) :g∗∈Lp

.

It is well known thatBMOLp₍_X₎can be identiﬁed as a subspace ofH∗

p(X∗). Thus, for any

ﬁnite martingale,f = (fn)n≥∈BMOLp₍_X₎andg= (g_n)_n_≥_∈H∗

p(X∗).

g,f=

g(ω),f(ω)dP≤cfBMOLp(X)· gHp∗(X∗).

On the other hand,S(p)(g)Lp is the norm of the diﬀerence sequence (dgn) inLp(lp(X

∗_)).

Thus

_S(p)₍_g₎

L_p =sup

(ak)

dgk,ak:(ak)_L_p(_l_p(_X₎₎≤

=sup

(ak)

dgk,Ek(ak) –Ek–(ak):(ak)_L_p(_l_p(_X₎₎≤

.

Setdfk=Ek(ak) –Ek–(ak) andf =

dfk. Thenf is anX-valued martingale. We have

S(p)(g)_L

p =sup₍_a k)

dgk,dfk

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It remains to estimatefBMOLp(X). Since(ak)Lp(lp(X))≤, we can also get the conditional

caseE((∞_k₌_nakp)|n)≤. Then by (.)

fBMOLp(X)≤cS (p)₍_f₎

BMO_Lp₍_X₎

≤csup

n

_E _S(p)₍_f_{) –}_S(p) n–(f)

p

|n_L_p

≤csup

n

E S(p)(f)p–S(_np_–)(f)p|n_L_p

≤csup

n

E

_∞

k=n

_E_k₍_a_k_{) –}_E_k_–₍_a_k₎p

n

Lp

≤c

E

_∞

k=n

akp

n

Lp

≤c.

On the contrary, we deﬁnefˆτ_{= (}ˆ_fτ

i , ), whereˆi=τ+i,fˆiτ=fτ+i–fτ,i≥. So, we have

S(p) fˆτp=S(p)(f)p–S(τp)(f)p.

Suppose thatXhas an equivalentp-uniformly smooth norm. Then by Pisier’s theorem,

we have

fˆτ_p≤c(S(p) fˆτ_p,

i.e.,

E f –fτp≤E S(p)(f)p–S(_τp)(f)p.

Moreover, we conditionalize it, we will get

E f –fτp|τ+

≤E S(p)(f)p–S(τp)(f)p|τ+

.

By the deﬁnition and Theorem ., we get

fBMOa_Lp,q₍_X₎≤cS (p)₍_f₎

BMOa_Lp,q₍_X₎.

Thus we complete the proof.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the manuscript and read and approved the ﬁnal manuscript.

Author details

1_{School of Mathematics and Statistics, Wuhan University, Wuhan, Hubei 430072, China.}2_{College of Science, Wuhan}

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant No. 11201354), by Hubei Province Key Laboratory of Systems Science in Metallurgical Process (Wuhan University of Science and Technology) (Y201121), (Y201321) and by the National Natural Science Foundation of Pre-Research Item (2011XG005). The authors would like to deeply thank all the reviewers for their insightful and constructive comments.

Received: 25 April 2013 Accepted: 18 July 2013 Published: 7 August 2013 References

1. Lorentz, GG: On the theory of spaces. Pac. J. Math.1, 411-429 (1951)

2. Neugebauer, CJ: Some classical operators on Lorentz spaces. Forum Math.4, 135-146 (1992) 3. Soria, J: Lorentz spaces of weak-type. Q. J. Math.49(2), 93-103 (1998)

4. Munckenhoupt, AMA: Maximal functions on classical Lorentz spaces and Hardy’s inequality with weights for nonincreasing functions. Trans. Am. Math. Soc.320, 727-735 (1990)

5. Cerdà, J, Martín, J: Interpolation restricted to decreasing function and Lorentz spaces. Proc. Edinb. Math. Soc.42, 243-256 (1999)

6. Boza, S, Martín, J: Equality of some classical Lorentz spaces. Positivity9, 225-232 (2005)

7. Carro, MJ, Raposo, JA, Soria, J: Recent developments in the theory of Lorentz spaces and weighted inequalities. Mem. Am. Math. Soc.187, 877 (2007)

8. Weisz, F: Martingale Hardy Spaces and Their Applications in Fourier Analysis. Springer, Berlin (1994) 9. Long, RL: Martingale Spaces and Inequalities. Peking University Press, Beijing (1993)

10. Xu, Q: Littlewood-Paley theory for functions with values in uniformly convex spaces. J. Reine Angew. Math.504, 195-226 (1998)

11. Ouyang, C, Xu, Q: BMO functions and Carleson measures with values in uniformly convex spaces. Can. J. Math.62, 827-844 (2010)

12. Yong, J: Carleson measures and vector-valued BMO martingales. Probab. Theory Relat. Fields145, 421-434 (2009) 13. Martínez, T: Uniform convexity in terms of martingaleH1and BMO spaces. Houst. J. Math.27, 461-478 (2001)

14. Martínez, T, Torrea, JL: Operator-valued martingale transforms. Tohoku Math. J.52(2), 449-474 (2000) 15. Pisier, G: Martingales with values in uniformly convex spaces. Isr. J. Math.20, 326-350 (1975)

doi:10.1186/1029-242X-2013-371