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BMO-Lorentz martingale spaces
Journal of Inequalities and Applications volume 2013, Article number: 371 (2013)
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 , Lorentz spaces have attracted more and more attention. A lot of results were obtained such as normability, duality, interpolation, and so on [2–7].
We know that martingale theory is intimately related to harmonic analysis. In martingale case, Weisz  and Long  considered the spaces 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  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 be a nonatomic σ-finite measure space. Suppose that f is a measurable function on a measure space . We define its distribution function
and its decreasing rearrangement function
Given a measurable function f on a measure space and , define
Remark 1.1 Observe that for all and we have
Unfortunately, the functions do not satisfy the triangle inequality. However, since for all , , we have
The set of all f with is denoted by and is called the Lorentz space with indices p and q. Observe that the definition implies that .
Let be a complete probability space, and let be a nondecreasing sequence of sub-σ-algebras of Σ with . We denote by E and the expectation and conditional expectation with respect to Σ and , respectively. For a martingale with martingale difference , , , we define its maximal function, p-square function, respectively, as usual:
We say that an X-valued martingale if .
The space (, ) consists of all martingale such that
Remark 1.2 The spaces are independent of a and all corresponding norms are equivalent. This allows us to denote any of them by .
Proof If , is a convex function, by Jensen’s inequality, we have , which implies , i.e., .
On the contrary, let , , , . Now we set . Then a.e. on . (Factually, if there is a with such that on B, then a.e. on B, which implies a.e. on B. This is a contradiction for .) So, . Then
i.e., . Thus we complete the proof. □
Remark 1.3 If , is the classical BMO space.
Remark 1.4 For the classical BMO space, we have the following statement (see ):
It is well known that is a subspace of for , and is a subspace of for , . Thus we have the following proposition.
Proposition 1.5 (1) If , , .
If , , .
Theorem 1.6 Let be an X-valued martingale in , . Then if and only if there exists an adapted process such that
And, in any case, we have
Proof Assume . Then, obviously,
Now let and be any one such that . Then we have
Taking ‘inf’ over all possible θ and (1.2), we get the desired inequality. □
Theorem 1.7 Let be an X-valued martingale in , where and , . Then
where ‘sup’ is taken over all stopping times τ.
Proof Assume that , τ is any stopping time. Then, by Hölder’s inequality, we have
This proves one half of the assertion. Conversely, assume that , and τ is any stopping time, , . Define
Thus we get
That is, . By Remark 1.2 we have
Thus we complete the proof of the theorem. □
Proposition 1.8 Particularly, if and , we get Remark 1.4.
By Theorem 1.6 and Theorem 1.7, we have the following proposition.
Proposition 1.9 Let be an X-valued martingale in , where and , . Then
where ‘sup’ is taken over all stopping times τ and ‘inf’ is taken over all adapted process .
2 Carleson measure and BMO-Lorentz martingale spaces
Definition 2.1 Let ν be a nonnegative measure on , where N is equipped with the counting measure dm. Let denote the tent over τ:
ν is said to be an s-Carleson measure if
where τ runs through all stopping times.
Theorem 2.2 Let , and be a real-valued martingale and , where is the Dirac measure centered at k. So, if , ν is a -Carleson measure. Moreover, if , the converse is also true, where , .
Proof Let be a real-valued martingale, let ν be generated by g as above, and let τ be any stopping time. Then, for ,
So, implies that ν is a -Carleson measure and .
Conversely, for any n and any , we define
Since ν is a -Carleson measure, we have
That is, . Then we have . Thus, we complete the proof of the theorem. □
3 The characterization of Banach space’s geometrical properties
Let . Then a Banach space X has an equivalent q-uniformly convex norm if and only if for one (or equivalently, for every ) there exists a positive constant c such that
for all finite -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 and be two Banach spaces. Let denote the space of all bounded linear operators from to . Let be an adapted sequence such that and . Then the martingale transform T associated to v is defined as follows. For any -valued martingale ,
Lemma 3.2 With the assumptions above, the following statements are equivalent:
There exists a positive constant c such that
There exists a positive constant c such that
For some (or equivalently, for every ), there exists a positive constant c such that
Theorem 3.3 Let X be a Banach space, , . Then the following statements are equivalent:
There exists a positive constant c such that for any finite X-valued martingale,
X has an equivalent norm which is q-uniformly convex.
Proof Let , , By Theorem 1.7 we have
So, if (1) holds, we have
By Remark 1.4 we have
We now consider a martingale transform operator Q from the family of X-valued martingales to that of -valued martingales. Let be the operator defined by for , where if and otherwise. Q is the martingale transform associated to the sequence :
By (3.3) and (3.4) we have
By Lemma 3.2 we have
Thus, by Pisier’s theorem, X has an equivalent q-uniformly convex norm.
Suppose that X has an equivalent q-uniformly convex norm. By Pisier’s theorem , we find, for any ,
Since , we have
We complete the proof. □
Theorem 3.4 Let X be a Banach space and , . If there exists a positive constant c such that for any finite X-valued martingale
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
for every martingale f.
Proof Let be the dual space of X. It suffices to prove that has an equivalent -uniformly convex norm, where is the conjugate index of p. By Pisier’s theorem, this is equivalent to showing that
Recall that is defined by
It is well known that can be identified as a subspace of . Thus, for any finite martingale, and .
On the other hand, is the norm of the difference sequence in .
Set and . Then f is an X-valued martingale. We have
It remains to estimate . Since , we can also get the conditional case . Then by (3.5)
On the contrary, we define , where , , . So, we have
Suppose that X has an equivalent p-uniformly smooth norm. Then by Pisier’s theorem, we have
Moreover, we conditionalize it, we will get
By the definition and Theorem 1.7, we get
Thus we complete the proof. □
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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.
The authors declare that they have no competing interests.
All authors contributed equally to the manuscript and read and approved the final manuscript.
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Cite this article
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
- Banach Space
- Equivalent Norm
- Lorentz Space
- Carleson Measure
- Martingale Inequality