BURKHOLDER-GUNDY-DAVIS INEQUALITY IN MARTINGALE HARDY SPACES WITH VARIABLE EXPONENT?

Peide LIU(劉培德) Maofa WANG(王茂發(fā))

School of Mathematics and Statistics,Wuhan University,Wuhan 430072,China

E-mail:pdliu@whu.edu.cn;mfwang.math@whu.edu.cn

Abstract In this article,by extending classical Dellacherie’s theorem on stochastic sequences to variable exponent spaces,we prove that the famous Burkholder-Gundy-Davis inequality holds for martingales in variable exponent Hardy spaces.We also obtain the variable exponent analogues of several martingale inequalities in classical theory,including convexity lemma,Chevalier’s inequality and the equivalence of two kinds of martingale spaces with predictable control.Moreover,under the regular condition on σ-algebra sequence we prove the equivalence between five kinds of variable exponent martingale Hardy spaces.

Key words variable exponent Lebesgue space;martingale inequality;Dellacherie theorem;Burkholder-Gundy-Davis inequality;Chevalier inequality

1 Introduction

Due to their important role in elasticity, fluid dynamics,calculus of variations,differential equations and so on,Musielak-Orlicz spaces and their special case,variable exponent Lebesgue spaces,have attracted more and more attention in modern analysis and functional space theory.In particular,Musielak-Orlicz spaces were studied by Orlicz and Musielak,see[20].Hudzik and Kowalewski[14]studied some geometry properties of Musielak-Orlicz spaces.Kovacik and Rakosnik[16],Fan and Zhao[11]investigated various properties of variable exponent Lebesgue spaces and Sobolev spaces.Diening[10]and Cruz-Uribe et al.[6,7]proved the boundedness of Hardy-Littlewood maximal operator on variable exponent Lebesgue function spaces Lp(x)(Rn)under the conditions that the exponent p(x)satisfies so called log-H?lder continuity and decay restriction.Many other authors studied its applications to harmonic analysis and some other subjects.

As we have known,the situation of martingale spaces is different from function spaces.For example,the log-H?lder continuity of a measurable function on a probability space can’t be defined.Moreover,generally speaking,the “good-λ” inequality method used in classical martingale theory can’t be used in variable exponent case.However,recently,variable exponent martingale spaces were paid more attention too.Among others,Aoyama[1]proved weak-type Doob’s maximal inequalities under some restrictions about exponent p.Under the condition that every σ-algebra is generated by countable atoms,Nakai and Sadasue[21]proved the boundedness of strong-type Doob’s maximal operator.Using different methods,Jiao et al.[15]also proved the weak-type and strong-type Doob’s maximal inequalities in discrete σ-algebra case.

The aim of this article is to establish some variable exponent analogues of several famous inequalities in classical martingale theory.In specially,by extending Dellacherie’s theorem on stochastic sequences to variable exponent Lebesgue spaces,we prove Burkholder-Gundy-Davis’inequality,convexity lemma and Chevalier’s inequality for variable exponent martingale Hardy spaces.Moreover,we investigate some equivalent relations between several variable exponent martingale Hardy spaces:we prove that two kinds of martingale spaces Dp(·)and Qp(·)with predictable control are equivalent,and under regular condition on σ?algebra sequence,all five martingale Hardy spaceswith variable exponent 1≤p?≤p+<∞are equivalent(for their definitions,see below).

Let(?,Σ,μ)be a non-atomic complete probability space,L0(?)the set of all measurable functions(i.e.,r.v.)on ?,and E the expectation with respect to Σ.We say that p ∈ P,if p ∈ L0(?)with 1 ≤ p(ω)≤ ∞.For p ∈ P,we denote ?∞={ω ∈ ?,p(ω)= ∞},and define variable exponent Lebesgue space

where the modular

For every u ∈ Lp(·),its Luxemburg norm is defined by

We denote by p?and p+the below index and the upper index of p,that is

and p’s conjugate index is p′(ω),that is

Here we mention some basic properties of Lp(·).Their proofs are standard and similar to of variable exponent function spaces,for example,see[11,16].

Lemma 1.1 Let p∈P with p+<∞.Then

(1) ρp(·)(u)<1(=1,>1)if and only if kukp(·)<1(=1,>1).

(2) ρp(·)(u) ≤ kukp(·),if kukp(·)≤ 1; ρp(·)(u) ≥ kukp(·),if kukp(·)>1.

(3)(Lp(·),k ·kp(·))is a Banach space.

(4)If u ∈ Lp(·),v ∈ Lp′(·),then

where C is a positive constant depending only on p.

(5)If un∈ Lp(·),then kun? ukp(·)→ 0 if and only if ρp(·)(un? u) → 0.

(6)If p∈P and s>0 with sp?≥1,then

(7)If p,q ∈ P,then Lp(·)? Lq(·)if and only if p(ω)≥ q(ω)a.e..In this case the embedding is continuous with

Let us fix some notations in martingale theory.

Let(Σn)n≥0be a stochastic basis,that is a nondecreasing sequence of sub-σ-algebras of Σ with Σ =WΣn,f=(fn)n≥0a martingale adapted to(Σn)n≥0with its difference sequence(dfn)n≥0,where dfn=fn?fn?1(with convention f?1≡ 0 and Σ?1={?,?}).We denote by Enthe conditional expectation with respect to Σn.For a martingale f=(fn)n≥0,we define its maximal function,square function and conditional square function as usual

For p ∈ P,the variable exponent martingale Lebesgue space Lp(·)and the martingale Hardy spaces,,andare defined as follows

The structure of this article is as follows.After some preliminaries about variable exponent Lebesgue spaces over a probability space,in section 2 we mainly deal with the extension of Dellacherie’s theorem and the convexity lemma to variable exponent case.In section 3,we establish the variable exponent analogues of Burkholder-Gundy-Davis’inequality and Chevalier’s inequality.In the last section,we first prove the equivalence of two martingale spaces with predictable control,then prove the equivalence between five variable exponent martingale Hardy spaces under regular condition.

Through this article,we always denote by C some positive constant,it may be different in each appearance,and denote by Cpa constant depending only on p.Moreover,we say that two norms on X are equivalent,if the identity is continuous in double directions,that is,there is a constant C>0 such that

2Some Lemmas

We first prove several lemmas,which will be used in the sequel.

Lemma 2.1 Let p∈P.Then every martingale or nonnegative submartingale f=(fn)satisfying supkfnkp(·)< ∞ converges a.e.to a measurable function f∞∈ Lp(·).

Proof As p(ω)≥ 1,from Lemma 1.1(7)we have

By Doob’s martingale convergence theorem,fn→ f∞a.e..In this case,|fn|p→ |f∞|pa.e.,by Fatou lemma,f∞∈ Lp(·). ?

Lemma 2.2 If p∈P,then there is an increasing sequence pnof simple functions with p?≤pnsuch that(pn)is adapted to(Σn)and pn→p a.e..

Proof Indeed,for p ∈ P,we can take a simple function sequence{gk},which is Σ-measurable and gk↑p.Due tofor every A∈Σ,there is a sequencesuch thatμ(A△Ak)→ 0.Because(Σn)is increasing,for every gk,there is a simple functionsuch thatis Σnk-measurable,≤p,andμ{gk6=}<2?k.Without loss of generality,we assume that nk↑∞ and define

Then{pn}is desired.?

In classical martingale theory,Dellacherie exploited a special approach to prove convex Φ?function inequalities for martingales.It was first formulated in[9],also see[19].The following lemma generalizes Dellacherie’s theorem to variable exponent case.

Lemma 2.3 Let p∈ P with p+< ∞,v be a non-negative r.v.,(un)n≥0a nonnegative,nondecreasing adapted sequence satisfying

or a nonnegative,nondecreasing predictable sequence satisfying

and in both cases u0=0.Then

where C depends only on p.

Proof (1)We first assume that p is Σn-measurable for some n,then there is a simple function sequence{si}such that all siare Σn-measurable,si≥ 1 and si↑p.

Similarly,(2.2)becomes

By classical Dellacherie’s theorem,we obtain that

and

From Lebesgue dominated convergence theorem and Levi monotonic convergence theorem we obtain Evuas i→∞.Similarly,from

Now suppose that p∈P,from Lemma 2.2,there is a sequence{pn}of simple functions such that pnis Σn-measurable and pn↑ p.From previous proof,(1)holds for every pn.Then we obtain(1)for the general case by taking limit.

(2)Notice that for p,q∈P withBy Young’s inequality,we have

Letting a=u∞(ω),b=v(ω)in these inequalities and taking integrals on both sides,then(2)immediately follows from(1).

(3)To prove(3),we assume that ku∞kp(·)=1.Due to the factLemma 1.1(4)and the proof of(1)above show that

The proof is completed.?

Lemma 2.4 Let p ∈ P with p+< ∞,(ξn)n≥0be a non-negative r.v.sequence and(En)n≥0be nondecreasing.Then there is a constant C=Cp>0 such that

therefore(2.6)follows from Lemma 2.3.

The following lemma is so-called convexitylemma whose classical version belongs to Burkholder,Davis and Gundy,see[3].

Lemma 2.5 Let p∈P with 2≤p?≤p+<∞.Then there is a C=Cp>0 such that for every martingale f=(fn),

3 B-G-D Inequality and Chevalier Inequality

Let us first extend Burkholder-Gundy-Davis’inequality for martingales to variable exponent case.As we have known,Burkholder-Gundy-Davis’inequality is one of the most fundamental theorems in classical martingale theory,see[3].

Theorem 3.1 Let p ∈ P with p+< ∞.Then there is a C=Cp(·)such that for every martingale f=(fn),

Proof Here we use Davis’method.For a martingale f=(fn),we define

From classical Burkholder-Gundy-Davis inequality(in conditioned version),we have

By Lemma 2.3,inequality(3.1)follows from(3.2)and(3.3). ?

Now we prove a shaper inequality:Chevalier’s inequality.For a martingale f=(fn),as usual we consider the functions M(f)and m(f):

Theorem 3.2 Let p∈P with p+<∞.Then there is a C=Cp>0 such that for every martingale f=(fn),

Proof We begin with a well known result.Let g=(gm)be as in the proof of Theorem 3.1 and(Dn)a predictable control of difference sequence(dfn),that is(Dn)an increasing adapted r.v.sequence with|dfn|≤ Dn?1,n ≥ 0,then there is a C>0 such that

(see[19],Theorem 3.5.5).Now for any fixed n,we have

Using(3.5)we obtain

where D′is a predictable control of g withLemma 2.3 guarantees that the following inequality holds

Making f’s Davis decomposition f=g+h with|dgn|≤ 4d?n?1and

Using again Lemma 2.3,we have

Using(3.6),we obtain that there should be a constant C=Cpsuch that

This completes the proof.

4 Some Equivalent Relations Between Martingale Spaces

Let p∈ P with p+< ∞,λ =(λn)be a nonnegative and increasing adapted sequence withWe denote by Λ the set of all such sequences and define two martingale spaces as follows

It is easy to check that both two martingale spaces are Banach spaces,and as in the classical case the norms of Qp(·),Dp(·)can be reached by some λ,respectively.We call such λ an optimal predictable control of f.We also introduce the following martingale space Ap(·):

To prove the equivalence between Qp(·)and Dp(·),we first need the following theorem.

Theorem 4.1 Let p∈P with p+<∞.Then

and there are C=Cp>0 such that

Proof The two inequalities in(4.1)are obvious due to their definitions.The two inequalities in(4.2)come from(3.1)and(4.1). ?

We now prove a Davis’decomposition theorem for martingales inand

Theorem 4.2 Let p∈P with p+<∞.Then

(1)Every f=(fn)∈has a decomposition f=g+h with g ∈ Qp(·),h ∈ Ap(·)such that

(2)Every f=(fn)∈has a decomposition f=g+h with g ∈ Dp(·),h ∈ Ap(·)such that

Proof Here we only prove(4.3),the proof of(4.4)is similar.

Let λ =(λn)be an adapted control of(Sn(f))n≥0:|Sn(f)|≤ λn,λ∞∈ Lp(·).We define

thus

Lemma 2.4 shows that

Because|dfk|χ{λk≤2λk?1}≤ 2λk?1and|dgk|≤ 4λk?1,we have

so g ∈ Qp(·).It follows from Lemma 2.4 that

Then(4.3)follows from(4.5)and(4.6).?

The following theorem shows the equivalence of Qp(·)and Dp(·).For the classcial version,we refer to Chao and Long[4],also see[23].

Theorem 4.3 Let p∈P with p+<∞.Then there is a C=Cp>0 such that for every martingale f=(fn),

Proof Let f=(fn) ∈ Dp(·)and λ =(λn)be its optimal predictable control:|fn|≤ λn?1,kfkDp(·)=kλ∞kp(·).Because

namely,(Sn?1(f)+2λn?1)n≥0is a predictable control of(Sn(f))n≥0,then f ∈ Qp(·)and it follows from(4.2)that

Conversely,if f=(fn) ∈ Qp(·)and λ =(λn)is its optimal predictable control.Since

then f ∈ Dp(·).Using(4.2)again we obtain

Theorem 4.4 If p∈P with p+<∞,then there is a C=Cp>0 such that for every martingale f=(fn)with f0=0,

Proof Here we use Garsia’s idea which was used to prove Theorem 4.1.2 in[12].

Let f ∈ Dp(·)and λ =(λn)be its optimal predictable control:|fn|≤ λn?1and kλ∞kp(·)=kfkDp(·).Define

A simple computation shows that

and thus

So g=(gn)is an L2-bounded martingale,it converges toa.e.and in L2.Notice that

and similarly

By classical H?lder’s inequality we have

Using Lemma 2.5 and inequalities(3.1),(4.9),(4.12),we obtain

This implies the first inequality of(4.8).

A similar argument gives the second inequality of(4.8). ?

At last we consider some equivalent relations between five martingale spaces under regular condition.In[23],Weisz called a martingale f=(fn)is previsible,if there is a real number R>0 such that

and proved that if it holds for all martingale with the same constant R,then the stochastic basis(Σn)is regular(refer to Garsia[12]for its definition).He also proved that if the sequence of σ?algebras(Σn)n≥0is regular,then the spacesare all equivalent for 0

Theorem 4.5 If p∈ P with p+< ∞ and(Σn)n≥0is regular,then the variable exponent martingale Hardy spacesare all equivalent.

Proof Under the regular condition,we have

that is to say,(Sn?1(f)+REn?1Sn(f))n≥0is a predictable control of(Sn(f))n≥0.Since

Lemma 2.4 shows that

where C depends only on p and R.Then from Theorems 3.1,4.1 and 4.3 we obtain

It remains to prove

In fact,the first inequality comes from Theorem 4.4 and(4.13).Due to the regularity,we have Sn(f)≤Rsn(f)and S(f)≤Rs(f),so the second inequality follows directly.The proof completes. ?

AcknowledgementsThe authors would like to thank the referees for their valuable suggestions.By the way,the earlier version of this article was appeared in arXiv.org,2014.(arXiv:1412.8146)