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Hardy-Littlewood maximal function on noncommutative Lorentz spaces
Journal of Inequalities and Applications volume 2013, Article number: 384 (2013)
This paper is mainly devoted to the study of the Hardy-Littlewood maximal function on noncommutative Lorentz spaces and to obtaining --type inequality for the Hardy-Littlewood maximal function on noncommutative Lorentz spaces.
MSC: 47A30, 47L05.
In  Nelson defined the measure topology of τ-measurable operators affiliated with a semi-finite von Neumann algebra. Fack and Kosaki  investigated generalized s-numbers of τ-measurable operators and proved dominated convergence theorems for a gage and convexity (or concavity) inequality.
As for noncommutative maximal inequalities, a version of ergodic theory was given by Junge  and Junge, Xu . In 2007, Mei  presented a version of noncommutative Hardy-Littlewood maximal inequality for an operator-valued function. In this paper, we study another version of Hardy-Littlewood maximal inequality introduced by Bekjan . In , Bekjan defined the Hardy-Littlewood maximal function for τ-measurable operators and, among other things, obtained weak -type and -type inequalities for the Hardy-Littlewood maximal function. In , for an operator T affiliated with a semi-finite von Neumann algebra, the Hardy-Littlewood maximal function of T is defined by
The classical Hardy-Littlewood maximal function of a Lebesgue measurable function denoted by is defined as
where m is a Lebesgue measure on (cf. ). Moreover, a natural generalization of this is the case and μ, a Borel measure on , where
As discussed by Bekjan in , let , where A is a Borel subset of . Then μ is a Borel measure and
i.e., is the Hardy-Littlewood maximal function of defined by
with respect to μ.
In view of spectral theory, is represented as
and is represented as . Thus, for T, is considered as the operator analogue of the Hardy-Littlewood maximal function in the classical case. Therefore, roughly speaking, stands in relation to T as stands in relation to f in classical analysis.
In this paper, we study the Hardy-Littlewood maximal function on noncommutative Lorentz spaces. By primarily adapting the techniques in , we obtain the --type inequality for the Hardy-Littlewood maximal function on noncommutative Lorentz spaces.
The remainder of this paper is organized as follows. Section 2 consists of some notations and preliminaries, including the noncommutative Lorentz spaces and their properties. In Section 3, we present the main result of this paper.
Throughout the paper, let ℳ be a finite von Neumann algebra acting on the Hilbert space ℋ with a normal faithful tracial state τ, and C will be a numerical constant not necessarily the same in each instance. The identity in ℳ is denoted by 1, and we denote by the lattice of (orthogonal) projections in ℳ. A linear operator , with domain , is said to be affiliated with ℳ if for all unitary u in the commutant of ℳ. The closed densely defined linear operator T affiliated with ℳ is called τ-measurable if for every there exists an orthogonal projection such that and . The collection of all τ-measurable operators is denoted by . With the sum and product defined as the respective closures of the algebraic sum and product, is an ∗-algebra. For a positive self-adjoint operator T affiliated with ℳ, we set
where is the spectral decomposition of T.
Let T be a τ-measurable operator and . The ‘t th singular number (or generalized s-number) of T’ is defined by
By Proposition 2.2 of , we have
where () and is the spectral projection of corresponding to the interval . The reader is referred to  for basic properties and detailed information on generalized s-numbers and the distribution function of τ-measurable operators.
Definition 2.1 (See, e.g., )
Let T be a τ-measurable operator affiliated with a finite von Neumann algebra ℳ, and let . Define
The set of all with is called the noncommutative Lorentz space, denoted by with indices p and q.
For convenience, we need the following Hardy inequalities in .
Lemma 2.2 If , and , then
Lemma 2.3 Let and , then
Let be the set of all τ-measurable operators such that
for all bounded intervals .
Definition 2.4 (See, e.g., )
Let , the maximal function of T is defined by
(let ). M is called the Hardy-Littlewood maximal operator.
Remark 2.5 By the introduction of , we know that is represented as . Hence, for , is considered as the operator analogue of the Hardy-Littlewood maximal function in the classical case. Therefore, roughly speaking, stands in relation to T as stands in relation to f in classical analysis. Also, in , it was proved that defined in Definition 2.4 was weak -type and -type. We refer the readers to  for more details and basic properties of .
3 Main result
Lemma 3.1 Let , and such that
Assume that ℳ has no minimal projection, then there exists a constant C such that we have
Proof We assume that . Theorem 2 of  and Lemma 2.3 imply that
By Lemma 1.8 of , for all , we can take such that and . Set , , it is easy to check that and . Indeed, we see that and . Thus we obtain
taking supremum, we get
which implies that
We estimate each term separately. For the first term, using (3.2) we get
After replacing r and q respectively with and in the first inequality in Lemma 2.2, we see that the last expression is estimated as follows:
i.e., . To estimate the second term, by applying (3.3) we obtain
For the second term , replace r and q respectively with and in the second inequality in Lemma 2.2, and we estimate the last expression as follows:
For the case of , we may simply reverse the roles of and in the above proof.
We have now shown that
Theorem 3.2 Let , and be such that
Assume that ℳ has minimal projections, then there exists a constant C such that for all we have
Proof Since ℳ has minimal projections, we consider the von Neumann algebra tensor product denoted by , equipped with the tensor product trace , where dm is the Lebesgue measure on , then has no minimal projection.
Let be the spectral decomposition of T. Since
It is easy to check that is a spectral series for each . Hence, for any interval
by the uniqueness of the spectral decomposition, we see that
For , since
which implies that
By an adaptation of the proof of Lemma 3.1, we deduce that
With the trivial fact , we know
Combing this result with , we infer that
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The author would like to thank the editor and anonymous referees for their helpful comments and suggestions on the quality improvement of the manuscript. This research is supported by the National Natural Science Foundation of China (No. 11071204) and XJUBSCX-2012002.
The author declares that she has no competing interests.
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Cite this article
Shao, J. Hardy-Littlewood maximal function on noncommutative Lorentz spaces. J Inequal Appl 2013, 384 (2013). https://doi.org/10.1186/1029-242X-2013-384
- noncommutative Lorentz spaces
- Hardy-Littlewood maximal function
- von Neumann algebra