Abstract

We obtain an atomic decomposition of weighted Lorentz spaces for a class of weights satisfying the Δ₂ condition. Consequently, we study operators such as the multiplication and composition operators and also provide Hölder’s-type and duality-Riesz type inequalities on these weighted Lorentz spaces.

1. Introduction

Weighted spaces are studied in most cases as a generalization of a special case. The Lorentz spaces, introduced by Lorentz in [1, 2], are no exception to this. The first version of the weighted Lorentz spaces was provided by Lorentz himself and was defined as , where is the decreasing rearrangement of and is a weight function. He proved that, for , is a norm if and only if the weight is decreasing. Carro and Soria in [3] proved that is a quasi-norm in general provided that satisfies the condition, that is, , for some constant . In this paper, we study that we denote by , with , in the sense that belongs to if and only if . Our interest in this special space stems from the fact that, as demonstrated in [4] with , this space has some interesting properties that allow an easy study of operators on via the Interpolation Theorem. The atomic decomposition of Banach spaces has been studied by many authors before: the Fourier transform of a function over the space can be thought of as an atomic decomposition of the space . Coifman in [5] gave the unifying definition of an atom and showed that Hardy’s spaces , the spaces of holomorphic functions on the unit disc , have an atomic decomposition and he used the latter result to prove that the dual spaces of these spaces are equivalent to the spaces of functions of bounded means oscillations. In [6], Jiao et al. proved that the Lorentz-Matingales spaces also have an atomic decomposition. In an attempt to give a different proof of the acclaimed Carleson Theorem (see, e.g., [7]), de Souza [8] showed that the Lorentz spaces have an atomic decomposition. In this paper, we continue the ideas in [8] and show that the weighted Lorentz spaces also admit an atomic decomposition, for a certain class of weights.

The remainder of the paper is organized as follows. In the preliminaries section, we introduce the necessary notions needed; namely, we define the conditions on our weight functions, and provide some preliminary definitions and results. In the second section, we prove that the weighted Lorentz spaces have an atomic decomposition and in the third section, we utilize this atomic decomposition to show the boundedness of some operators on theses weighted spaces. The last section opens up a discussion about the relevance of this line of research.

2. Preliminaries

We begin with some preliminary definitions and results (proofs can be found in the appendices) that will be helpful throughout the paper.

Definition 1. Define as the space of weights so that,(1),(2) is increasing,(3) is decreasing,(4)there is a positive constant such that, , (Dini’s Condition),(5) satisfies the condition; that is, there is a an constant such that .

Note that the space is nonempty since , for , belongs to . Hereafter, will denote a nonatomic measure defined on .

Definition 2. One defines the weighted Lorentz space as where is the decreasing rearrangement of defined as , and is a nonatomic measure defined on .

Remark 3. For , is identical to the classical Lorentz space .

Definition 4. One will also consider the following space: where the ’s are -measurable sets in and represents the characteristic function on the set .

We will show in Theorem 14 that this space is an atomic decomposition of the space .

Put where the infimum is taken over all possible representations of . The next result is proved in the appendix.

Proposition 5. If one endows with , then(1) is a norm,(2) is a Banach space.

Definition 6. For , define for a measurable function the quantity
The space is the set of measurable functions for which . This space generalizes the space introduced in [4]. In the next theorem and remark, we give further properties of theses spaces in the weighted case.

Theorem 7. For , one has that(1)if , then is a quasi-norm on ,(2) with .

The remark below is stated only for completeness and the proof can be found in [4].

Remark 8. For and , we have , where .

Definition 9. For , define , . For a measure defined on , consider the following spaces

The first space was basically introduced by Lorentz in [2] for . We will prove in Theorem 10 that these spaces are norm-equivalent.

Theorem 10. Let and , . For a measurable function , one has

Theorem 11 (Hölder’s type inequalities). Let and , .(1)For and , one has (2)For and , one has

3. Atomic Decomposition

We start with this important result on the dual of the spaces and .

Theorem 12. Let and , . Then one has the following.(1); that is, if and only if there is a unique so that for all (2)Likewise, one has ; that is, if and only if there is a unique so that for all

Proof. Let . Define . By Theorem 11, we have
Thus, using the linearity of the integral, we conclude that . On the other hand, let . For a -measurable set , define . Then there is a constant such that
Since then using Dini’s condition (4) in Definition 1, we have
It follows from (11) and (13) that and condition 1 in Definition 1 yield . By the Radon-Nikodym Theorem and the definition of functions in , there is an integrable function on such that, for all ,
To prove that , observe that
Thus taking the supremum over -measurable sets such that , we have
The proof is complete using the equivalence in Theorem 10. The proof of the second part is very similar to that of the first part and uses the second part of Theorem 12.

The following result is a classical result in function analysis. (See, e.g., [9, page 160], for a proof.)

Theorem 13. Let and be two vector normed spaces and let , the space of bounded linear operators from onto . Let be the adjoint operator of defined by for all , the dual space of . Then(a) and ;(b) is injective if and only if the range of is dense in .

The next result is the most important of the present paper and gives an equivalent representation of functions in as “linear” combinations of simple functions.

Theorem 14 (atomic decomposition of ). For , one has

Proof. Let us show first that . Take . Then using Dini’s condition (4) in Definition 1, we have
Thus if for , then
And (18) implies
Taking the infimum over all representations of , we have
To prove the other direction, we can use either Theorem  1 in [10] or Theorem 13. In this paper, we will use the latter. Note that we have the following:: and , by inequality (21).: is dense in , see [11].: since by Theorems 10 and 12  .
Using Theorem 13, we conclude from that the inclusion map is a bounded linear map and that where is the duality map . From , it follows that duality map is injective and from that is the identity map. Therefore, we have that is an isomorphism and thus .

Remark 15. The space is called an atomic decomposition of in the sense that each function of coincides with a function of and thus can be written as a “linear” combination of atoms, where the atoms are the “simple” functions .

4. Operators on Weighted Lorentz Spaces

In this section, we study two types of operators: the multiplication and composition operators of weighted Lorentz spaces .

Theorem 16 (multiplication operator). For and for , define the multiplication operator as . Then is bounded if and only if . Moreover, .

Proof. If is bounded, then there is an absolute constant such that
Take for . Then from (22), it follows that which after simplification is equivalent to
Using Dini’s condition (4) in Definition 1, we have and hence
Thus,
This completes the proof that .
On the other hand, if , then for , we have
Therefore,
Since from Theorem 14, ; then for , we have , for some ’s . And so,
Taking the infimum over all representations of and using the equivalence between and , we have
To prove the second statement of the theorem, observe that (31) implies that
If we take for some , then and since and are decreasing, we have
Consequently, and so . Thus
Taking the limit as , we have
From the inequality in the proof of Theorem 7 (Appendix B), it follows that
The result then follows by combining (32) and (36).

Definition 17. Let be a nonsingular measurable transformation on , and let . We define as where is a -measurable set in . is an absolute real constant and is the inverse image of the -measurable subset of . Put .

Theorem 18 (composition operator). For and for , define the composition operator as . Then is bounded if and only if . Moreover, .

Proof. The technique of this proof mirrors that of Theorem 16. For, assume that the operator is bounded; that is, there is some absolute constant such that
Taking for some -measurable set , the inequality (38) implies
Since , (39) entails
Using the fact that is decreasing and Dini’s condition (4) in Definition 1, respectively, on the LHS and the RHS of (40), we have
This proves that .
On the other hand, suppose that . Then for a -measurable subset of we have
Using Dini’s condition (4) and the fact that , there is an absolute constant such that
Now let . Then since . Then, using (43), it follows that
Taking the infimum over all representations of and using the equivalence we have showing that the operator is bounded on .
Note that, from (45), we get . Without loss of generality, consider the constant to be such that . Then
Moreover, if we take for a -measurable subset in , we have and using the fact that is decreasing
Hence
Taking the supremum over such that , we have that
The proof of the second part is complete by combining (46) and (49).