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BMO-Lorentz martingale spaces

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 1951 when they were first introduced by Lorentz in [1], Lorentz spaces have attracted more and more attention. A lot of results were obtained such as normability, duality, interpolation, and so on [27].

We know that martingale theory is intimately related to harmonic analysis. In martingale case, Weisz [8] and Long [9] considered the spaces H p , 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 of X.

It was exactly with this in mind that Xu [10] developed the vector-valued Littlewood-Paley theory, which was inspired by Pisier’s celebrated work [8] on martingale inequalities in uniformly convex spaces. Very recently, Ouyang and Xu [11] studied the endpoint case of the main results of [10] by means of the classical relationship between BMO functions and Carleson measures. Jiao [12] discussed the relationship between Carleson measures and vector-valued martingales.

Let (Ω,μ) be a nonatomic σ-finite measure space. Suppose that f is a measurable function on a measure space (Ω,μ). We define its distribution function

λ f (t)=μ { ω : f ( ω ) > t } ,t0,

and its decreasing rearrangement function

f (t)=inf { s > 0 : λ f ( s ) t } .

Given a measurable function f on a measure space (Ω,μ) and 0<p,q, define

f L p , q ={ ( 0 ( t 1 p f ( t ) ) q d t t ) 1 q if  q < , sup t > 0 t 1 p f ( t ) if  q = .

Remark 1.1 Observe that for all 0<p,r< and 0<q we have

| g | r L p , q = g L p r , q r r .
(1.1)

Unfortunately, the functions L p , q do not satisfy the triangle inequality. However, since for all t>0, ( f + g ) (t) f (t/2)+ g (t/2), we have

f + g L p , q c p , q ( f L p , q + g L p , q ) ,
(1.2)

where c p , q = 2 1 / p max{1, 2 ( 1 q ) / q }.

The set of all f with f L p , q < is denoted by L p , q (X,μ) and is called the Lorentz space with indices p and q. Observe that the definition implies that L , = L .

Let (Ω,Σ,P) be a complete probability space, and let ( Σ n ) n 0 be a nondecreasing sequence of sub-σ-algebras of Σ with Σ=σ( n 0 Σ n ). We denote by E and E n the expectation and conditional expectation with respect to Σ and Σ n , respectively. For a martingale f= ( f n ) n 0 with martingale difference d f n = f n f n 1 , n0, f 1 0, we define its maximal function, p-square function, respectively, as usual:

Mf= sup n f n , S ( p ) (f)= ( n = 1 d f n p ) 1 / p .

We say that an X-valued martingale f= ( f n ) n 0 L p , q (X) if sup n f n L p , q <.

The space BMO L p , q ( X ) a (a1, 1<p,q) consists of all martingale f L p , q (X) such that

f BMO L p , q ( X ) a = sup n ( E ( f f n 1 a | Σ n ) ) 1 / a L p , q ( X ) <.

Remark 1.2 The spaces BMO L p , q ( X ) a are independent of a and all corresponding norms are equivalent. This allows us to denote any of them by BMO L p , q ( X ) .

Proof If a1, φ(x)= x a is a convex function, by Jensen’s inequality, we have E(f f n 1 | Σ n ) ( E ( f f n 1 a | Σ n ) ) 1 / a , which implies E ( f f n 1 | Σ n ) (t) ( ( E ( f f n 1 a | Σ n ) ) 1 / a ) (t), i.e., BMO L p , q ( X ) a BMO L p , q ( X ) 1 .

On the contrary, let g n =E(f f n 1 | Σ n ), Mg= sup n g n , h n =E ( f f n 1 a | Σ n ) 1 / a , Mh= sup n h n . Now we set A t ={ω:Mh>t}. Then Mg>t a.e. on A t . (Factually, if there is a B A t with μ(B)>0 such that Mgt on B, then E(f f n 1 | Σ n )t=E(t| Σ n ) a.e. on B, which implies f f n 1 t a.e. on B. This is a contradiction for A t .) So, P{ω:Mh>t}cP{ω:Mg>t}. Then

f BMO L p , q ( X ) a ( q 0 [ t P ( M h ( ω ) > t ) 1 / p ] q d t t ) 1 / q c ( q 0 [ t P ( M g ( ω ) > t ) 1 / p ] q d t t ) 1 / q c f BMO L p , q ( X ) 1 ,

i.e., BMO L p , q ( X ) 1 BMO L p , q ( X ) a . Thus we complete the proof. □

Remark 1.3 If p=q=, BMO L p , q ( X ) is the classical BMO space.

Remark 1.4 For the classical BMO space, we have the following statement (see [12]):

f BMO sup τ μ ( τ < ) 1 / p f f τ 1 L p ,1p<.
(1.3)

It is well known that L p , q is a subspace of L p , r for 0<p, 0<q<r and L p 2 , q 2 is a subspace of L p 1 , q 1 for 1< p 1 p 2 , 1< q 1 , q 2 . Thus we have the following proposition.

Proposition 1.5 (1) If 0<p, 0<q<r, BMO L p , q BMO L p , r .

  1. (2)

    If 1< p 1 p 2 , 1< q 1 , q 2 , BMO L p 2 , q 2 BMO L p 1 , q 1 .

Theorem 1.6 Let f= ( f n ) n 0 be an X-valued martingale in L p , q (X), 1<p,q. Then f BMO L p , q ( X ) if and only if there exists an adapted process θ= ( θ ) n 0 such that

C θ = sup n E ( f θ n 1 | Σ n ) L p , q ( X ) <.

And, in any case, we have

f BMO L p , q ( X ) := inf θ C θ f BMO L p , q ( X ) cf BMO L p , q ( X ) .

Proof Assume f BMO L p , q ( X ) . Then, obviously,

f BMO L p , q ( X ) f BMO L p , q ( X ) .

Now let f BMO L p , q ( X ) < and θ= ( θ ) n 0 be any one such that C θ <. Then we have

E ( f f n 1 | Σ n ) E ( f θ n 1 | Σ n ) + θ n 1 f n 1 = E ( f θ n 1 | Σ n ) + E ( ( f θ n 1 ) | Σ n 1 ) E ( f θ n 1 | Σ n ) + E ( f θ n 1 | Σ n 1 ) = E ( f θ n 1 | Σ n ) + E ( E ( f θ n 1 | Σ n ) | Σ n 1 ) .

Taking ‘inf’ over all possible θ and (1.2), we get the desired inequality. □

Theorem 1.7 Let f= ( f n ) n 0 be an X-valued martingale in BMO L p , q ( X ) , where 1 p + 1 s 1 = 1 r and 1 q + 1 s 2 = 1 s , 1<p,q,r, s 1 , s 2 . Then

f BMO L p , q ( X ) sup τ μ ( { τ < } ) 1 s 1 f f τ 1 L r , s ( X ) ,

where ‘sup’ is taken over all stopping times τ.

Proof Assume that f BMO L p , q ( X ) <, τ is any stopping time. Then, by Hölder’s inequality, we have

f f τ 1 L r , s ( X ) = ( f f τ 1 ) χ { τ < } L r , s ( X ) c f f τ 1 L p , q ( X ) χ { τ < } L s 1 , s 2 ( X ) c sup g ( L p , q ) 1 | { τ < } ( f f τ 1 ) g d μ | χ { τ < } L s 1 , s 2 ( X ) = c ( q p ) 1 / q sup g ( L p , q ) 1 | { τ < } E ( f f τ 1 g | Σ τ ) d μ | μ ( τ < ) 1 s 1 c ( q p ) 1 / q f BMO L p , q μ ( τ < ) 1 s 1 .

This proves one half of the assertion. Conversely, assume that β= sup τ μ ( { τ < } ) 1 s 1 f f τ 1 L r , s ( X ) <, and τ is any stopping time, F Σ τ , F{τ<}. Define

τ F ={ τ if  ω F , if  ω F .

Thus we get

β μ ( { τ < } ) 1 s 1 f f τ 1 L r , s ( X ) = 1 μ ( F ) 1 / s 1 f f τ F 1 L r , s ( X ) 1 μ ( F ) 1 / s 1 f f τ F 1 L r 1 , r 1 ( X ) 1 μ ( F ) f f τ F 1 L 1 , 1 ( X ) = 1 μ ( F ) F f f τ F 1 d u .

That is, E(f f τ F 1 | Σ τ F )β. By Remark 1.2 we have

f BMO L p , q ( X ) cβ.

Thus we complete the proof of the theorem. □

Proposition 1.8 Particularly, if s=r and p=q=, we get Remark 1.4.

By Theorem 1.6 and Theorem 1.7, we have the following proposition.

Proposition 1.9 Let f= ( f n ) n 0 be an X-valued martingale in BMO L p , q ( X ) , where 1 p + 1 s 1 = 1 r and 1 q + 1 s 2 = 1 s , 1<p,q,r, s 1 , s 2 . Then

f BMO L p , q ( X ) inf θ sup τ μ ( { τ < } ) 1 s 1 f θ τ 1 L r , s ( X ) ,

where ‘sup’ is taken over all stopping times τ and ‘inf’ is taken over all adapted process θ= ( θ ) n 0 .

2 Carleson measure and BMO-Lorentz martingale spaces

Definition 2.1 Let ν be a nonnegative measure on Ω×N, where N is equipped with the counting measure dm. Let τ ˆ denote the tent over τ:

τ ˆ = { ( ω , k ) : k τ ( ω ) , τ ( ω ) < } .

ν is said to be an s-Carleson measure if

ν= sup τ ν ( τ ˆ ) μ ( { τ < } ) s <,

where τ runs through all stopping times.

Theorem 2.2 Let 1<p,q, and g= ( g n ) n 0 be a real-valued martingale and dν= | Δ k g | 2 δ k dμ, where δ k is the Dirac measure centered at k. So, if g BMO L 2 p , 2 q 2 , ν is a 1/ p -Carleson measure. Moreover, if 1<p,q<, the converse is also true, where 1 p + 1 p =1, 1 q + 1 q =1.

Proof Let g= ( g n ) n 0 be a real-valued martingale, let ν be generated by g as above, and let τ be any stopping time. Then, for 1<q<,

ν ( τ ˆ ) = E ( k = 0 | Δ k g | 2 χ { τ ( ω ) k } ) = E ( E ( k = τ ( ω ) | Δ k g | 2 | Σ τ ) χ { τ ( ω ) k } ) = E ( E ( | g g τ 1 | 2 | Σ τ ) χ { τ ( ω ) k } ) c E ( | g g τ 1 | 2 | Σ τ ) L p , q χ { τ ( ω ) k } L p , q = c E ( | g g τ 1 | 2 | Σ τ ) 1 2 2 L p , q χ { τ ( ω ) k } L p , q c E ( | g g τ 1 | 2 | Σ τ ) 1 / 2 L 2 p , 2 q 2 μ ( { τ ( ω ) k } ) 1 / p c g BMO L 2 p , 2 q 2 2 μ ( { τ ( ω ) k } ) 1 / p .

So, g BMO L 2 p , 2 q 2 implies that ν is a 1/ p -Carleson measure and ν g BMO L 2 p , 2 q 2 2 .

Conversely, for any n and any F Σ n , we define

τ={ n if  ω F , if  ω F .

Since ν is a 1/ p -Carleson measure, we have

ν 1 μ ( { τ < } ) 1 / p ν ( { ( ω , k ) : k τ ( ω ) , τ ( ω ) < } ) = 1 μ ( F ) 1 / p F k = n | Δ k g | 2 d μ 1 μ ( F ) F k = n | Δ k g | 2 d μ = 1 μ ( F ) F | g g n 1 | 2 d μ .

That is, νE( | g g n 1 | 2 | Σ n ). Then we have g BMO L 2 p , 2 q 2 2 cν. Thus, we complete the proof of the theorem. □

3 The characterization of Banach space’s geometrical properties

Let 1<q<. Then a Banach space X has an equivalent q-uniformly convex norm if and only if for one 1<p< (or equivalently, for every 1<p<) there exists a positive constant c such that

S ( q ) ( f ) p c sup n f n p

for all finite L p -martingales f with values in X. Again, the validity of the converse inequality amounts to saying that X has an equivalent q-uniformly smooth norm.

Definition 3.1 Let X 1 and X 2 be two Banach spaces. Let L( X 1 , X 2 ) denote the space of all bounded linear operators from X 1 to X 2 . Let v= ( v n ) n 1 be an adapted sequence such that v n L (L( X 1 , X 2 )) and sup n 1 v n L ( L ( X 1 , X 2 ) ) 1. Then the martingale transform T associated to v is defined as follows. For any X 1 -valued martingale f= ( f n ) n 1 ,

( T f ) n = k = 1 n v k d f k .

We get the following results from [13, 14].

Lemma 3.2 With the assumptions above, the following statements are equivalent:

  1. (1)

    There exists a positive constant c such that

    T f BMO ( X 2 ) c f BMO ( X 1 ) ,fBMO( X 1 ).
  2. (2)

    There exists a positive constant c such that

    ( T f ) BMO ( X 2 ) c f BMO ( X 1 ) ,fBMO( X 1 ).
  3. (3)

    For some 1p< (or equivalently, for every 1p<), there exists a positive constant c such that

    T f L p ( X ) c f L p ( X ) ,f L p (X).

Theorem 3.3 Let X be a Banach space, 2q<, 1<p<. Then the following statements are equivalent:

  1. (1)

    There exists a positive constant c such that for any finite X-valued martingale,

    S ( q ) ( f ) BMO L p , q ( X ) c f BMO L p , q ( X ) .
  2. (2)

    X has an equivalent norm which is q-uniformly convex.

Proof (1)(2) Let 1 p + 1 s 1 = 1 r , 1 q + 1 s 2 = 1 r , 1<p,q,r, s 1 , s 2 By Theorem 1.7 we have

S ( q ) ( f ) BMO L p , q ( X ) sup τ μ ( { τ < } ) 1 s 1 S ( q ) ( f ) S τ 1 ( q ) ( f ) L r , r ( X ) ,
(3.1)
f BMO L p , q ( X ) a sup τ μ ( { τ < } ) 1 s 1 f f τ 1 L r , r ( X ) .
(3.2)

So, if (1) holds, we have

S ( q ) ( f ) S τ 1 ( q ) ( f ) L r , r ( X ) c f f τ 1 L r , r ( X ) .

By Remark 1.4 we have

S ( q ) ( f ) BMO c f BMO .
(3.3)

We now consider a martingale transform operator Q from the family of X-valued martingales to that of l q (X)-valued martingales. Let vL(X, l q (X)) be the operator defined by v k (x)= { x j } j = 1 for xX, where x j =x if j=k and x j =0 otherwise. Q is the martingale transform associated to the sequence ( v k ):

( Q f ) n = k = 1 n v k d f k =(d f 1 ,d f 2 ,,d f n ,0,).

Then

( Q f ) = S ( q ) (f).
(3.4)

By (3.3) and (3.4) we have

( Q f ) BMO c f BMO .

By Lemma 3.2 we have

S ( q ) ( f ) q = ( Q f ) q c f q .

Thus, by Pisier’s theorem, X has an equivalent q-uniformly convex norm.

(2)(1) Suppose that X has an equivalent q-uniformly convex norm. By Pisier’s theorem [15], we find, for any 1n<,

E ( i = n d f i q | Σ n ) cE ( f f n 1 q | Σ n ) .

Since E( | S ( q ) ( f ) S n 1 ( q ) ( f ) | q | Σ n )cE(| ( S ( q ) ( f ) ) q ( S n 1 ( q ) ( f ) ) q || Σ n ), we have

E ( | S ( q ) ( f ) S n 1 ( q ) ( f ) | q | Σ n ) cE ( f f n 1 q | Σ n ) .

Thus

S ( q ) ( f ) BMO L p , q ( X ) c f BMO L p , q ( X ) .

We complete the proof. □

Theorem 3.4 Let X be a Banach space and 1<p2, 1<q<. If there exists a positive constant c such that for any finite X-valued martingale

f BMO L p ( X ) c S ( p ) ( f ) BMO L p ( X ) ,
(3.5)

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

f BMO L p , q ( X ) a c S ( p ) ( f ) BMO L p , q ( X ) a

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, where p is the conjugate index of p. By Pisier’s theorem, this is equivalent to showing that

S ( p ) ( g ) L p c g L p =c g H p ( X ) .
(3.6)

Recall that H p ( X ) is defined by

H p ( X ) = { X -valued martingale  g = ( g n ) : g L p } .

It is well known that BMO L p ( X ) can be identified as a subspace of H p ( X ). Thus, for any finite martingale, f= ( f n ) n 0 BMO L p ( X ) and g= ( g n ) n 0 H p ( X ).

| g , f | = Ω g ( ω ) , f ( ω ) dPc f BMO L p ( X ) g H p ( X ) .

On the other hand, S ( p ) ( g ) L p is the norm of the difference sequence (d g n ) in L p ( l p ( X )).

Thus

S ( p ) ( g ) L p = sup ( a k ) { | d g k , a k | : ( a k ) L p ( l p ( X ) ) 1 } = sup ( a k ) { | d g k , E k ( a k ) E k 1 ( a k ) | : ( a k ) L p ( l p ( X ) ) 1 } .

Set d f k = E k ( a k ) E k 1 ( a k ) and f=d f k . Then f is an X-valued martingale. We have

S ( p ) ( g ) L p = sup ( a k ) { | d g k , d f k | } c f BMO L p ( X ) g H p ( X ) .
(3.7)

It remains to estimate f BMO L p ( X ) . Since ( a k ) L p ( l p ( X ) ) 1, we can also get the conditional case E(( k = n a k p )| Σ n )1. Then by (3.5)

f BMO L p ( X ) c S ( p ) ( f ) BMO L p ( X ) c sup n E ( S ( p ) ( f ) S n 1 ( p ) ( f ) p | Σ n ) L p c sup n E ( S ( p ) ( f ) p S n 1 ( p ) ( f ) p | Σ n ) L p c sup n E ( k = n E k ( a k ) E k 1 ( a k ) p | Σ n ) L p c E ( ( k = n a k p ) | Σ n ) L p c .

On the contrary, we define f ˆ τ =( f ˆ i τ ,), where Σ ˆ i = Σ τ + i , f ˆ i τ = f τ + i f τ , i0. So, we have

( S ( p ) ( f ˆ τ ) ) p = S ( p ) ( f ) p S τ ( p ) ( f ) p .

Suppose that X has an equivalent p-uniformly smooth norm. Then by Pisier’s theorem, we have

f ˆ τ p c ( S ( p ) ( f ˆ τ ) p ,

i.e.,

E ( f f τ p ) E ( S ( p ) ( f ) p S τ ( p ) ( f ) p ) .

Moreover, we conditionalize it, we will get

E ( f f τ p | Σ τ + 1 ) E ( S ( p ) ( f ) p S τ ( p ) ( f ) p | Σ τ + 1 ) .

By the definition and Theorem 1.7, we get

f BMO L p , q ( X ) a c S ( p ) ( f ) BMO L p , q ( X ) a .

Thus we complete the proof. □

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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.

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Zhang, X., Yi, X. & Zhang, C. BMO-Lorentz martingale spaces. J Inequal Appl 2013, 371 (2013). https://doi.org/10.1186/1029-242X-2013-371

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Keywords

  • Banach Space
  • Equivalent Norm
  • Lorentz Space
  • Carleson Measure
  • Martingale Inequality