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Atomic decompositions of weak Orlicz-Lorentz martingale spaces
Journal of Inequalities and Applications volume 2014, Article number: 66 (2014)
Abstract
In this paper we establish atomic decompositions of some weak Orlicz-Lorentz martingale spaces which are generalization of Orlicz martingale spaces and of Lorentz martingale spaces. With the help of atomic decompositions, the boundedness of sublinear operators is obtained.
MSC:60G46, 47A30.
1 Introduction and preliminaries
The idea of atomic decomposition in martingale theory is derived from harmonic analysis [1]. Just as it does in harmonic analysis, the method is a key ingredient in dealing with many problems including martingale inequalities, duality, interpolation, and so on, especially for small-index martingale and multi-parameter martingales. As is well known, Weisz [2] gave some atomic decompositions on martingale Hardy spaces and proved many important theorems by atomic decompositions; Weisz [3] made a further study of atomic decompositions for weak Hardy spaces consisting of Vilenkin martingales, and he proved a weak version of the Hardy-Littlewood inequality; Liu and Hou [4] investigated the atomic decompositions for vector-valued martingales and some geometry properties of Banach spaces were characterized; Hou and Ren [5] considered the vector-valued weak atomic decompositions and weak martingale inequalities. Orlicz Hardy martingale spaces are also studied by some authors such as Miyamoto, Nakai, Sadasue and Jiao [6–8]. At the same time, the Lorentz spaces are discussed (see [9–13]). For example, the atomic decompositions of Lorentz martingales are first studied by Jiao et al. in [10], and in 2013 Ho investigated the atomic decomposition of Lorentz-Karamata martingale spaces similarly to the idea of [10]. As the generalization of Orlicz and Lorentz spaces, the Orlicz-Lorentz spaces attract more attention. Montgomery-Smith [14] discuss the comparison of Orlicz-Lorentz spaces. Rajeev and Romesh [15] studied composition operators on Orlicz-Lorentz spaces. Echandia [16] discussed the interpolation of Orlicz-Lorentz spaces.
Let be a measure space, be a space of all Σ-measurable functions. Let there be given an Orlicz function (i.e., it is a convex function and takes value zero only at zero) and a weight function (i.e., it is a non-increasing function and locally integrable and ). The Orlicz-Lorentz space on is the set of all functions such that
for some , where is the non-increasing rearrangement of f defined by
We shall not work with this definition of the Orlicz-Lorentz space, however, but with a different, equivalent definition. A Young function F is an even continuous and non-negative function in , increasing on , such that , , iff . A Young function F is said to satisfy the global â–³2-condition if there is such that for all . We define to be if and 0 if .
We define the Orlicz-Lorentz space as the set of all measurable f’s on Ω for which the Orlicz-Lorentz functional
is finite.
Similarly, by means of the weak Orlicz-Lorentz functional
we define the Orlicz-Lorentz space .
Remark 1.1 By the fact , we have . We see that , where is Orlicz space, and that if and , then . If A is any measurable set, then .
Let be a complete probability space, and a non-decreasing 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 differences , , , denote
Denote by Λ the collection of all sequences of non-decreasing, non-negative and adapted functions and set . Thus we can define some weak Orlicz-Lorentz martingale spaces as follows:
Definition 1.2 A measurable function a is called a weak atom of the first category (or of the second category, of the third category, respectively) if there exists a stopping time ν (ν is called the stopping time associated with a) such that
-
(i)
if ;
-
(ii)
(or (ii) , (ii) , respectively).
These three category weak atoms are briefly called w-1-atom, w-2-atom, and w-3-atom, respectively.
Throughout this article, we denote the set of integers and the set of non-negative integers by ℤ and ℕ, respectively. We use c to denote constants and may denote different constants at different occurrences.
2 Weak atomic decompositions
Weak atomic decompositions of some weak martingale Hardy spaces were studied in [5, 7]. In this section, we will consider weak atomic decompositions of some weak Orlicz-Lorentz martingale spaces.
Theorem 2.1 Let . Then if and only if there exist a sequence of w-1-atoms and the corresponding stopping times such that
-
(1)
;
-
(2)
, for some constant , and .
Moreover the following equivalence of norms holds:
where the infimum is taken over all the preceding decompositions of f.
Proof Assume . Let us consider the following stopping time for all :
Then the sequence of these stopping times is non-decreasing and . Let be the stopping martingale. It is easy to see that
Now let . Then for a fixed (, ) is a martingale and
Thus and is bounded. So there exists an integrable function such that . If , then , so we get is really a w-1-atom. Moreover, we have
Hence we get (1). As for any , we have
which implies .
Conversely, assume that has a decomposition of the form (1). Let . For any fixed choose such that .
Let
Thus by the fact that , we have
Since , , we have
Then .
On the other hand, since if , thus on the set . Moreover and . Since , then . Moreover is c-subadditive, i.e.,
Consequently,
which implies
We combine (2.5) and (2.9) to obtain (2.1). Thus we prove Theorem 2.1. □
Theorems similar to Theorem 2.1 hold for the spaces and .
Theorem 2.2 Let . Then if and only if there exist a sequence of w-2-atoms and the corresponding stopping times such that
-
(1)
;
-
(2)
, for some constant , and .
Moreover the following equivalence of norms holds:
where the infimum is taken over all the preceding decompositions of f.
Theorem 2.3 Let . Then if and only if there exist a sequence of w-3-atoms and the corresponding stopping times such that
-
(1)
;
-
(2)
, for some constant , and .
Moreover the following equivalence of norms holds:
where the infimum is taken over all the preceding decompositions of f.
We sketch the proofs of Theorem 2.2 and Theorem 2.3 and omit the details since they are similar to that of Theorem 2.1. Let in these cases where is the sequence in the definitions of and , respectively. Let be defined as in the proof of Theorem 2.1. Equation (1) and the analogs of (2.5) can be proved in the same way as in Theorem 2.1. For the converse parts of the proof of Theorem 2.2 assume that has a decomposition of the form (1) and let . Then is a non-negative, non-decreasing and adapted sequence with . For any fixed choose such that , then with
Similarly to the argument of (2.8) (replacing and by and , respectively) we have
It follows that , which shows that and (2.10) holds. As for the converse parts of the proof of Theorem 2.3 we let
3 Sublinear operators on weak Orlicz-Lorentz martingale spaces
As one of the applications of the atomic decompositions, we shall obtain a sufficient condition for a sublinear operator to be bounded from weak Orlicz-Lorentz martingale spaces to weak Orlicz-Lorentz function spaces.
An operator is called a sublinear operator if it satisfies , , where X is a martingale space, Y is a measurable function space. In this paper, we will add some restrictions to the function F.
Definition 3.1 A strict concave function F is said to obey the â–³-condition written often as , if there exists a positive constant b such that for arbitrary ; and it obeys the â–½-condition denoted symbolically as , if there exists a positive constant B such that for arbitrary , where (see [17]).
Here we should notice that:
-
(1)
Any strict concave function since , ;
-
(2)
Not only the power function , for example .
Proof By the definition of , we have . Thus we have
which means is a strict concave function. Now we will prove that , . Since
we have . Then , . Thus we complete the proof of (2). □
Theorem 3.2 Let concave function and be a bounded sublinear operator. If
for all w-1-atom a, where Ï„ is the stopping time associated with a, then
Proof By Theorem 2.1, f can be decomposed into the sum of a sequence of w-1-atoms and , for some constant A. For any fixed choose such that and let
Recall that on the set , we have
Since , we have for any
It follows from the boundedness of T and that
which implies
On the other hand, by the assumption (3.1) we have
which implies
By (3.4) and (3.5),
Thus we complete the proof. □
Similarly to the proof of Theorem 3.2, we can prove the following two theorems. In the proof we need Theorem 2.2 and Theorem 2.3 instead of Theorem 2.1, respectively. Here we only give the two theorems and omit the proofs
Theorem 3.3 Let the concave function and be a bounded sublinear operator. If
for all w-2-atom a, where Ï„ is the stopping time associated with a, then
Theorem 3.4 Let concave function and be a bounded sublinear operator. If
for all w-3-atom a, where Ï„ is the stopping time associated with a, then
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Acknowledgements
This work was Supported by 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) (Y201321) and by National Natural Science Foundation of Pre-Research Item (2011XG005).
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Zhang, C., Zhong, M. & Zhang, X. Atomic decompositions of weak Orlicz-Lorentz martingale spaces. J Inequal Appl 2014, 66 (2014). https://doi.org/10.1186/1029-242X-2014-66
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DOI: https://doi.org/10.1186/1029-242X-2014-66