#### Abstract

We obtain an atomic decomposition of weighted Lorentz spaces for a
class of weights satisfying the Δ_{2} 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).

*Remark 19. *The previous result in part shows that boundedness of operators other than the aforementioned ones on weighted Lorentz spaces is possible if their action on characteristic functions can be controlled. In particular, the centered Hardy-Littlewood Maximal operator, the Hilbert operator (under Sawyer’s type condition) are bounded on .

#### 5. Discussion

The special atoms spaces originally introduced by de Souza in [12] for seem to have an interesting role in analysis with its connection to Lipschitz spaces (see [13]) through Hölder’s inequality and duality. These spaces allow for simple characterization of the Bergman-Besov-Lipschitz spaces (see [11]), that is, spaces of functions defined on such that

Another interesting use of the special atoms space is the real characterization of some spaces of analytic functions in the unit disc such that where represents the derivative of (see [11, 14]). The special atom spaces have been generalized in a couple of different ways: one is the weighted case with its connections to weighted Lipschitz spaces and other weighted spaces of analytic functions. The other is that, unlike in the original definition of special atoms spaces where the atoms were intervals, the atoms can now be replaced with measurable sets for general measures. This last generalization has led to the study of Lorentz spaces , and the weak- spaces also known as , . Indeed in [8], we show that for if and only if where , the ’s are -measurable sets in . It was also shown in [8] that (52) is equivalent to where and for -measurable set , in .

What makes (52) and (53) remarkable is that they help to prove and generalize a result by Weiss and Stein ([15]) which states that a linear operator is bounded, where is a Banach space closed under absolute value and satisfying if , for an absolute constant .

Another interesting observation is that the dual of can be identified as the set of measurable functions such that either of the following is satisfied, for -measurable subsets of ,

In fact, (54) and (55) provide a natural generalization of Lipschitz spaces. Indeed in (54), letting for a differentiable function on , , and be the Lebesgue measure yields

Also in (55), letting , , , and be the Lebesgue measure yields

In [4], Kwessi et al. use this new representation of to study operators such as the multiplication and composition operators on via interpolation. The key part is to show that the study of the boundedness of such operators on and in particular on amounts to the study of the action of such operators on characteristic functions of sets. The present paper follows the same idea on weighted Lorentz spaces .

#### Appendices

#### A. Proof of Proposition 5

(a) We first prove that is a norm on . Let such that . Then for arbitrary, we have that

Thus

Since is arbitrary, it follows that either or , . Since and are increasing on , it follows that is equivalent to , . The latter implies that the ’s are atoms of in . But since is nonatomic, this is impossible. Hence implies that which in turn implies that . The homogeneity condition follows directly from the fact that . Now let such that , where and , for some arbitrary . Put

Note that we can write with .

It follows that

Since is arbitrary, it finishes the proof that is a norm on .

(b) We now prove that is a Banach space. It suffices to show that, for any sequence , we have .

Let then be a sequence of functions in . Given and an integer , let be a real number and let be a -measurable set in such that with . It follows that

Taking the infimum over all possible representations of and since is arbitrary we get that and this completes the proof.

#### B. Proof of Theorem 7

For , implies , on . Therefore, since is not identically zero, we have -a.e. Since we can choose equivalence classes, it follows that . Similarly, if , implies that . The homogeneity condition for follows trivially from the fact that . Finally, consider . Since , we have used the properties of the decreasing rearrangement that

Likewise, for , we have

This proves that is a quasi-norm on , since and .

Now suppose that . If , then and , so

Using (4) in Definition 1, we get

On the other hand, using (2) and (3) in Definition 1, we have

So

The result then follows by combining inequalities (B.4) and (B.6).

#### C. Proof of Theorem 10

It easy to show that

Let be -measurable subset of . For a -measurable set , we have (see [7, exercise 1.4.5, page 65])

Therefore using (C.1) and (C.2) we can show easily that

We only need to prove that to conclude.

Suppose that with , . Then, for all , . Integrating both sides on the interval , we have

Using Dini’s conditions above and taking the supremum over , we have

On the other hand, since is decreasing, for , we have

Taking the supremum over , we have

The equivalence follows by combining (C.5) and (C.7).

#### D. Proof of Theorem 11

First consider and . Using a result by Hardy and Littlewood (see, e.g., Exercise 1.4.1(b) in [7]), we have

For the second part, we start with for some -measurable subset of and . Then

Thus

So if , the linearity of the integral gives us

Thus, taking the infimum over all the representations of , we have

#### Conflict of Interests

The authors declare that they have no conflict of interests regarding the publication of this paper.

#### Acknowledgment

The authors are grateful to the anonymous referees for their comments and suggestions that helped improve the quality of this paper.