Journal of Function Spaces

Journal of Function Spaces / 2012 / Article

Research Article | Open Access

Volume 2012 |Article ID 293613 | https://doi.org/10.1155/2012/293613

Eddy Kwessi, Paul Alfonso, Geraldo De Souza, Asheber Abebe, "A Note on Multiplication and Composition Operators in Lorentz Spaces", Journal of Function Spaces, vol. 2012, Article ID 293613, 10 pages, 2012. https://doi.org/10.1155/2012/293613

A Note on Multiplication and Composition Operators in Lorentz Spaces

Academic Editor: Gestur Ólafsson,
Received06 Feb 2012
Accepted21 Jun 2012
Published31 Jul 2012

Abstract

we revisit the Lorentz spaces 𝐿(𝑝,𝑞) for 𝑝>1,𝑞>0 defined by G. G. Lorentz in the nineteen fifties and we show how the atomic decomposition of the spaces 𝐿(𝑝,1) obtained by De Souza in 2010 can be used to characterize the multiplication and composition operators on these spaces. These characterizations, though obtained from a completely different perspective, confirm the various results obtained by S. C. Arora, G. Datt and S. Verma in different variants of the Lorentz Spaces.

1. Introduction

In the early 1950s, Lorentz introduced the now famous Lorentz spaces 𝐿(𝑝,𝑞) in his papers [1, 2] as a generalization of the 𝐿𝑝 spaces. The parameters 𝑝 and 𝑞 encode the information about the size of a function; that is, how tall and how spread out a function is. The Lorentz spaces are quasi-Banach spaces in general, but the Lorentz quasi-norm of a function has better control over the size of the function than the 𝐿𝑝 norm, via the parameters 𝑝 and 𝑞, making the spaces very useful. We are mostly concerned with studying the multiplication and composition operators on Lorentz spaces. These have been studied before by various authors in particular by Arora et al. in [36]. In this paper, the results we obtain are in accordance with what these authors have found before. We believe that the techniques and relative simplicity of our approach are worth reporting to further enrich the topic. Our results, found on the boundary of the unit disc due to the original focus by De Souza in [7], will show how one can use the atomic characterization of the Lorentz space 𝐿(𝑝,1) in the study of multiplication and composition operators in the spaces 𝐿(𝑝,𝑞).

2. Preliminaries

Let (𝑋,𝜇) be a measure space.

Definition 2.1. Let 𝑓 be a complex-valued function defined on 𝑋. The decreasing rearrangement of 𝑓 is the function 𝑓 defined on [0,) by 𝑓(𝑡)=inf{𝑦>0𝑑(𝑓,𝑦)𝑡},(2.1) where 𝑑(𝑓,𝑦)=𝜇({𝑥|𝑓(𝑥)|>𝑦}) is the distribution of the function 𝑓.

Definition 2.2. Given a measurable function 𝑓 on (𝑋,𝜇) and 0<𝑝,𝑞, define 𝑓𝐿(𝑝,𝑞)=𝑞𝑝0𝑡1/𝑝𝑓(𝑡)𝑞𝑑𝑡𝑡1/𝑞,if𝑞<,sup𝑡>0𝑡1/𝑝𝑓(𝑡),if𝑞=.(2.2) The set of all functions 𝑓 with 𝑓𝐿(𝑝,𝑞)< is called the Lorentz space with indices 𝑝 and 𝑞 and denoted by 𝐿(𝑝,𝑞)(𝑋,𝜇).

We now consider the measure 𝜇 on 𝑋 to be finite. Let 𝑔𝑋𝑋 be a 𝜇-measurable function such that 𝜇(𝑔1(𝐴))𝐶𝜇(𝐴) for a 𝜇-measurable set 𝐴[0,2𝜋] and for an absolute constant 𝐶. Here 𝑔1(𝐴) refers to the preimage of the set 𝐴.

Remark 2.3. It is important to note that 𝑔=sup𝜇(𝐴)0(𝜇(𝑔1(𝐴))/𝜇(𝐴)) is not necessarily a norm.

Definition 2.4. For a given function 𝑔, we define the multiplication operator 𝑇𝑔 on Lorentz spaces as 𝑇𝑔(𝑓)=𝑓𝑔 and the composition operator 𝐶𝑔 as 𝐶𝑔(𝑓)=𝑓𝑔.
The following two results are used in our proofs. The first is a result of De Souza [7] which gives an atomic decomposition of 𝐿(𝑝,1). The second is the Marcinkiewicz interpolation theorem (see [8]) which we state for completeness of presentation.

Theorem 2.5 (see De Souza [7]). A function 𝑓𝐿(𝑝,1) for 𝑝>1 if and only if 𝑓(𝑡)=𝑛=1𝑐𝑛𝜒𝐴𝑛(𝑡) with 𝑛=1|𝑐𝑛|𝜇1/𝑝(𝐴𝑛)<, whereas 𝜇 is measure on 𝑋 and 𝐴𝑛 are 𝜇-measurable sets in 𝑋. Moreover, 𝑓𝐿(𝑝,1)inf𝑛=1|𝑐𝑛|𝜇1/𝑝(𝐴𝑛), where the infimum is taken over all possible representations of 𝑓.

Theorem 2.6 (see Marcinkiewicz). Assume that for 0<𝑝0𝑝1, for all 𝑞>0, for all measurable subsets 𝐴 of 𝑋, there are some constants 0<𝑀0,𝑀1< such that for a linear or quasi-linear operator 𝑇𝑔(a)𝑇𝑔𝜒𝐴𝐿(𝑝0,)𝑀0𝜇1/𝑝0(𝐴). (b)𝑇𝑔𝜒𝐴𝐿(𝑝1,)𝑀1𝜇1/𝑝1(𝐴).
Then there is some 𝑀>0 such that 𝑇𝑔𝑓𝐿(𝑝,𝑞)𝑀𝑓𝐿(𝑝,𝑞) for 1/𝑝=𝜃/𝑝0+(1𝜃)/𝑝1,0<𝜃<1.

One implication of Theorem 2.5 is that it can be used to prove and justify a theorem of Stein and Weiss [9]. That is, to show that linear operators 𝑇𝐿(𝑝,1)𝐵 are bounded, where 𝐵 is Banach space closed under absolute value and satisfying 𝑓𝐵=|𝑓|𝐵, all one needs to show is that 𝑇𝜒𝐴𝐵𝑀𝜇1/𝑝(𝐴),𝑝>1. Theorem 2.6 will be used to show that valid results on 𝐿(𝑝,1) are also valid on 𝐿(𝑝,𝑞).

Definition 2.7. We denote by 𝑀𝑝𝑟 the set of real-valued functions defined on 𝑋=[0,2𝜋] such that 𝑓𝑀𝑝𝑟=sup𝑥>0𝑟𝑝𝑥1/𝑝𝑥0𝑓(𝑡)𝑡1/𝑝𝑟𝑑𝑡𝑡1/𝑟<,(2.3) where 1𝑝𝑟<.

We will show that the space 𝑀𝑝𝑟 is equivalent to a weak 𝐿𝑘 space for some 𝑘 that depends on 𝑝 and 𝑟 and 𝑀𝑝𝑟 is quasinorm.

Lemma 2.8. 𝑀𝑝𝑟 a quasinorm on 𝑀𝑝𝑟.

Proof. 𝑓0 by definition. This implies that 𝑓𝑀𝑝𝑟0. Moreover, 𝑓𝑀𝑝𝑟=0 implies that for all 0<𝑥 ≤ 2𝜋,𝑥0(𝑓(𝑡)𝑡1/𝑝)𝑟𝑑𝑡/𝑡=0. Hence, we have 𝑓=0𝜇-a.e, thus, 𝑓=0 since 𝑓 is a representative of an equivalence class. Now let 𝑘0 be a real constant, 𝑓𝑀𝑝𝑟, and 𝑥(0,2𝜋]. Noting (𝑘𝑓)=|𝑘|𝑓, the homogeneity condition 𝑘𝑓𝑀𝑝𝑟=|𝑘|𝑓𝑀𝑝𝑟 follows trivially. Let 𝑓,𝑔𝑀𝑝𝑟. Since (𝑓+𝑔)(𝑡) ≤ 𝑓(𝑡/2)  + 𝑔(𝑡/2), for any 𝑥(0,2𝜋], we have 𝑥0(𝑓+𝑔)(𝑡)𝑡1/𝑝𝑟𝑑𝑡𝑡2𝑟1𝑥0𝑓𝑡2𝑡1/𝑝𝑟𝑑𝑡𝑡+𝑥0𝑔𝑡2𝑡1/𝑝𝑟𝑑𝑡𝑡𝑟2𝑝+𝑟10(1/2)𝑥𝑓(𝑡)𝑡1/𝑝𝑟𝑑𝑡𝑡+0(1/2)𝑥𝑔(𝑡)𝑡1/𝑝𝑟𝑑𝑡𝑡.(2.4) Since (𝑎+𝑏)1/𝑟𝑎1/𝑟+𝑏1/𝑟 for 𝑎,𝑏>0, we have 𝑓+𝑔𝑀𝑝𝑟21/𝑝1/𝑟+1𝑓𝑀𝑝𝑟+𝑔𝑀𝑝𝑟,with21/𝑝1/𝑟+1>1for𝑟,𝑝>1.(2.5)

Theorem 2.9. 𝑀𝑝𝑟𝐿(𝑝𝑟,), where 𝑟,𝑟1,1/𝑟+1/𝑟=1.

Proof. Suppose 𝑔𝑀𝑝𝑟. There is an absolute constant 𝐶 such that for all 𝑥>0, 𝑟𝐶𝑝𝑥1/𝑝𝑥0𝑔(𝑡)𝑡1/𝑝𝑟𝑑𝑡𝑡1/𝑟𝑟𝑝𝑥1/𝑝𝑔(𝑥)𝑟𝑥0𝑡𝑟/𝑝1𝑑𝑡1/𝑟=𝑥1/𝑝𝑟𝑔(𝑥).(2.6) Thus, sup𝑥>0𝑥1/𝑝𝑟𝑔(𝑥)𝐶 implying that 𝑔𝐿(𝑝𝑟,).
Conversely, let 𝑔𝐿(𝑝𝑟,). Then there is an absolute constant 𝐶 such that, 𝑔(𝑡)𝐶𝑡1/𝑝𝑟. This implies that (𝑔(𝑡)𝑡1/𝑝)𝑟𝐶𝑟𝑡1/𝑝. Thus, sup𝑥>0𝑟𝑝𝑥1/𝑝𝑥0𝑔(𝑡)𝑡1/𝑝𝑟𝑑𝑡𝑡1/𝑟sup𝑥>0𝐶𝑟𝑟𝑝𝑥1/𝑝𝑥0𝑡1/𝑝𝑑𝑡𝑡1/𝑟=𝐶𝑟1/𝑟.(2.7) This implies that 𝑔𝑀𝑝𝑟.

Remark 2.10. One can easily see from Theorem 2.9 that 𝑀𝑝𝐿(𝑝,) and 𝑀𝑝1𝐿. Moreover, 𝑔𝑀𝑝1=𝑔. To see this, note that 𝑔𝑀𝑝1=sup𝑥>01𝑝𝑥1/𝑝𝑥0𝑔(𝑡)𝑡1/𝑝𝑑𝑡𝑡𝑔sup𝑥>01𝑝𝑥1/𝑝𝑥0𝑡1/𝑝1=𝑑𝑡𝑔,𝑔𝑀𝑝1𝑔(𝑥)𝑝𝑥1/𝑝𝑥0𝑡1/𝑝1𝑑𝑡=𝑔(𝑥)(2.8) for all 𝑥 since 𝑔 is decreasing. Taking the limit as 𝑥0, we see that 𝑔𝑀𝑝1𝑔(0)=𝑔.

3. Main Results

3.1. Multiplication Operators

Theorem 3.1 (see multiplication operator on 𝐿(𝑝,1)). The multiplication operator 𝑇𝑔𝐿(𝑝,1)𝐿(𝑝,1) for 𝑝𝑝>1 is bounded if and only 𝑔𝐿. Moreover, 𝑇𝑔=𝑔.

Proof. It is convenient to use 𝑀𝑝1 which is equivalent to 𝐿. Assume that =𝑇𝑔𝑓𝐿(𝑝,1)𝐶𝑓𝐿(𝑝,1). Then for 𝑓=𝜒[0,𝑥] where 𝑥(0,2𝜋], 02𝜋𝑇𝑔𝜒[0,𝑥](𝑡)𝑡1/𝑝1𝑑𝑡=𝑥0𝑔(𝑡)𝑡1/𝑝1𝑑𝑡𝐶02𝜋𝜒[]0,𝑥(𝑡)𝑡1/𝑝1𝑑𝑡=𝐶𝑝𝑥1/𝑝.(3.1) Multiplying and dividing the integrand on the left by 𝑡1/𝑝1, we get 𝑥0𝑔(𝑡)𝑡1/𝑝1𝑡𝑝𝑝/𝑝𝑝𝑑𝑡𝐶𝑝𝑥1/𝑝.(3.2) Since 𝑡𝑡(𝑝𝑝)/𝑝𝑝 is decreasing on [0,𝑥] and 0<𝑥2𝜋, we have 1𝑝𝑥1/𝑝𝑥0𝑔(𝑡)𝑡1/𝑝1𝑑𝑡𝐶(2𝜋)𝑝𝑝/𝑝𝑝.(3.3) Taking the supremum over all 𝑥>0, we have that 𝑔𝑀𝑝1.
Assume that 𝑔𝑀𝑝1 and 𝑥>0. Since 𝑝>𝑝 we have 𝑇𝑔𝜒[0,𝑥]𝐿(𝑝,1)=𝑥0𝑔(𝑡)𝑡1/𝑝1𝑑𝑡𝑥0𝑔(𝑡)𝑡1/𝑝1𝑑𝑡.(3.4) And so, 𝑇𝑔𝜒[0,𝑥]𝐿(𝑝,1)𝜒𝑀[0,𝑥]𝐿(𝑝,1),where𝑀=sup𝑥>01𝑝𝑥1/𝑝𝑥0𝑔(𝑡)𝑡1/𝑝1𝑑𝑡.(3.5) Using the atomic decomposition of 𝐿(𝑝,1), we get 𝑇𝑔𝑓𝐿(𝑝,1)𝑀𝑓𝐿(𝑝,1)forsomepositiveconstant𝑀.(3.6)
To prove the second part of the theorem, first note that the expression in (3.5) gives that 𝑇𝑔𝑔. Now take 𝑓=(1/𝑥1/𝑝)𝜒[0,𝑥]. We can easily see that 𝑓𝐿(𝑝,1)=1 and 𝑇𝑔𝑓𝐿(𝑝,1)𝑔(𝑥) for 𝑥[0,2𝜋] since 𝑔 is decreasing. Now taking the sup over 𝑓𝐿(𝑝,1)1 and the limit as 𝑥0 gives 𝑇𝑔𝑔. Thus, 𝑇𝑔=𝑔.

The following theorem, which is equivalent to Theorem 1.1 of [6], follows from Theorems 2.6 and 3.1.

Theorem 3.2 (see multiplication operator on 𝐿(𝑝,𝑞)). The multiplication operator 𝑇𝑔𝐿(𝑝,𝑞)𝐿(𝑝,𝑞) is bounded if and only if 𝑔𝐿 for 1<𝑝,1<𝑞. Moreover, 𝑇𝑔=𝑔.

Remark 3.3. Since, by Theorem 2.9, 𝑀𝑝1𝑀𝑝𝑟 for 𝑟>1, the theorem implies that if the multiplication operator 𝑇𝑔𝐿(𝑝,𝑞)𝐿(𝑝,𝑞) defined by 𝑇𝑔𝑓=𝑔𝑓 is bounded, then 𝑔𝑀𝑝𝑟 for 𝑝,𝑞>1.

Remark 3.4. It is worth observing that the norm 𝑓𝐿(𝑝,1)=sup𝐴𝑋𝜇(𝐴)01𝜇1/𝑝(𝐴)0𝜇(𝐴)𝑔(𝑡)𝑡1/𝑝1𝑑𝑡(3.7) can also be used to prove the previous theorem, first on 𝐿(𝑝,1) and then on 𝐿(𝑝,𝑞) by means of either the Marcienkiewiecz Inteporlation Theorem or Theorem 2.5. Actually, this norm was the original motivation for the introduction of the space 𝑀𝑝𝑟. For sake of simplicity and without loss of generality, we modified it by replacing 𝜇(𝐴), 𝐴𝑋 by 𝑥>0.

Theorem 3.5. If 𝑓𝐿(𝑝1,𝑞1), and 𝑔𝐿(𝑝2,𝑞2), where 1<𝑝1,𝑝2,𝑞1,𝑞2<, then 𝑔𝑓𝐿(𝑟,𝑠) where 1/𝑟 = 1/𝑝1+1/𝑝2 and 1/𝑠=1/𝑞1+1/𝑞2.

Proof. Given 1<𝑝1,𝑝2,𝑞1,𝑞2<, assume 𝑓𝐿(𝑝1,𝑞1) and 𝑔𝐿(𝑝2,𝑞2). Let 𝑟,𝑠 be such that 1/𝑟=1/𝑝1+1/𝑝2 and 1/𝑠=1/𝑞1+1/𝑞2. Since (𝑓𝑔)(𝑡)𝑓(𝑡)𝑔(𝑡), we have 02𝜋(𝑓𝑔)(𝑡)𝑡1/𝑟𝑠𝑑𝑡𝑡02𝜋𝑓(𝑡)𝑡1/𝑝1𝑠𝑔(𝑡)𝑡1/𝑝2𝑠𝑑𝑡𝑡.(3.8) Using Holder’s inequality on the RHS with 𝑠/𝑞1+𝑠/𝑞2=1, we have 02𝜋(𝑓𝑔)(𝑡)𝑡1/𝑟𝑠𝑑𝑡𝑡02𝜋𝑓(𝑡)𝑡1/𝑝1𝑞1𝑑𝑡𝑡𝑠/𝑞102𝜋𝑔(𝑡)𝑡1/𝑞2𝑞2𝑑𝑡𝑡𝑠/𝑞2.(3.9) Thus, we have 𝑔𝑓𝐿(𝑟,𝑠)𝑓𝐿(𝑝1,𝑞1)𝑔𝐿(𝑝2,𝑞2).(3.10)

Theorem 3.6. If 𝑔𝑀𝑝𝑟, then 𝑇𝑔𝐿(𝑞,𝑠)𝐿(𝑝𝑞𝑟/(𝑝𝑟+𝑞),𝑠) is bounded, where 1/𝑟+1/𝑟=1 and for 𝑠>0 and 𝑝,𝑞>1.

Proof. Let 𝑔𝑀𝑝𝑟𝐿(𝑝𝑟,)𝑇𝑔𝑓𝑠𝐿(𝑘,𝑠)02𝜋𝑔(𝑡)𝑓(𝑡)𝑡1/𝑘𝑠𝑑𝑡𝑡=02𝜋𝑔(𝑡)𝑡1/𝑝𝑟𝑓(𝑡)𝑡1/𝑞𝑠𝑑𝑡𝑡1if𝑘=1𝑞+1.𝑞𝑟(3.11) Therefore, 𝑇𝑔𝑓𝑠𝐿(𝑘,𝑠)sup𝑡>0𝑔(𝑡)𝑡1/𝑝𝑟𝑠02𝜋𝑓(𝑡)𝑡1/𝑞𝑠𝑑𝑡𝑡.(3.12) That is, 𝑇𝑔𝑓𝐿(𝑘,𝑠)𝑔𝑀𝑝𝑟𝑓𝐿(𝑞,𝑠),where𝑘=𝑝𝑟𝑞.𝑝𝑟+𝑞(3.13)

Noting that 𝑀𝑝𝑟𝐿(𝑝𝑟,), 𝑟=𝑟/(𝑟1), it is easy to see that Theorem 3.6 shows that the result of Theorem 3.5 extends to the case where 𝑞2=.

3.2. Composition Operators

Theorem 3.7. The composition operator 𝐶𝑔𝐿(𝑝,𝑞)𝐿(𝑝,𝑞) is bounded if and only if there is an absolute constant C such that 𝜇𝑔1(𝐴)𝐶𝜇(𝐴),(3.14) for all 𝜇-measurable sets 𝐴[0,2𝜋] and for 1<𝑝,1𝑞. Moreover, 𝐶𝑔=𝑔1/𝑝.

Proof. We will prove this theorem for 𝐿(𝑝,1) and use the interpolation theorem to conclude for 𝐿(𝑝,𝑞).
First assume that 𝐶𝑔𝐿(𝑝,1)𝐿(𝑝,1) is bounded that is, there is an absolute constant 𝐶 such that 𝐶𝑔𝑓𝐿(𝑝,1)𝐶𝑓𝐿(𝑝,1).(3.15) Let 𝐴 be a 𝜇-measurable set in [0,2𝜋] and let 𝑓=𝜒𝐴. Then, (3.15) is equivalent to 𝐶𝑔𝜒𝐴𝐿(𝑝,1)𝜒𝐶𝐴𝐿(𝑝,1)1𝑝02𝜋𝐶𝑔𝜒𝐴(𝑡)𝑡1/𝑝1𝐶𝑑𝑡𝑝02𝜋𝜒𝐴(𝑡)𝑡(1/𝑝)1𝑑𝑡;(3.16) that is, 1𝑝02𝜋𝜒𝐴𝑔(𝑡)𝑡1/𝑝1𝐶𝑑𝑡𝑝02𝜋𝜒[0,𝜇(𝐴)](𝑡)𝑡(1/𝑝)1𝑑𝑡.(3.17) Since (𝜒𝐴𝑔)=𝜒𝑔1(𝐴), then (𝜒𝐴𝑔)=𝜒[0,𝜇(𝑔1(𝐴))]. Therefore, the previous inequality gives 1𝑝𝜇(𝑔10(𝐴))𝑡1/𝑝1𝐶𝑑𝑡𝑝0𝜇(𝐴)𝑡1/𝑝1𝑑𝑡.(3.18) And hence, 𝜇𝑔1(𝐴)𝐶𝑝𝜇(𝐴).(3.19)
On the other hand, assume that there is some constant 𝐶>0 such that 𝜇(𝑔1(𝐴))𝐶𝜇(𝐴). Then, 𝐶𝑔𝜒𝐴𝐿(𝑝,1)=1𝑝02𝜋𝜒𝐴𝑔(𝑡)𝑡1/𝑝1=1𝑑𝑡𝑝02𝜋𝜒[0,𝜇(𝑔1(𝐴))](𝑡)𝑡1/𝑝1=𝜇𝑔𝑑𝑡1(𝐴)1/𝑝𝐶1/𝑝(𝜇(𝐴))1/𝑝.(3.20) Consequently, 𝐶𝑔𝜒𝐴𝐿(𝑝,1)𝐶1/𝑝(𝜇(𝐴))1/𝑝.(3.21) As a consequence of Theorem 2.5 or the result by Weiss and Stein in [9], we have 𝐶𝑔𝑓𝐿(𝑝,1)𝐶1/𝑝𝑓𝐿(𝑝,1).(3.22)
To prove the second part of the theorem, note that from the above, we have 𝐶𝑔=sup𝑓𝐿(𝑝,1)1𝐶𝑔𝑓𝐿(𝑝,1)𝑓𝐿(𝑝,1)𝐶1/𝑝.(3.23) But inf{𝐶𝜇(𝑔1(𝐴))𝐶𝜇(𝐴)}=𝑔. Thus, 𝐶𝑔𝑔1/𝑝. To obtain the other inequality, let 𝑓=(1/[𝜇(𝐴)]1/𝑝)𝜒𝐴. This gives 𝑓𝐿(𝑝,1)=1 and 𝐶𝑔𝑓𝐿(𝑝,1)=𝜇𝑔1(𝐴)𝜇(𝐴)1/𝑝.(3.24) Thus, 𝐶𝑔=sup𝑓𝐿(𝑝,1)1𝐶𝑔𝑓𝐿(𝑝,1)sup𝜇(𝐴)0𝜇(𝑔1(𝐴))𝜇(𝐴)1/𝑝=𝑔1/𝑝.(3.25) Now to show the result for 𝐿(𝑝,𝑞), note that the operator 𝐶𝑔 is linear on 𝐿(𝑝,𝑞) and that for 𝑝0 and 𝑝1 such that 𝑝0<𝑝<𝑝1, we have 𝐶𝑔𝜒𝐴𝐿(𝑝0,1)𝑀0(𝜇(𝐴))1/𝑝0 and 𝐶𝑔𝜒𝐴𝐿(𝑝1,1)𝑀1(𝜇(𝐴))1/𝑝1. Since 𝐿(𝑝𝑖,1)𝐿(𝑝𝑖,),𝑖=0,1, then for some absolute constants 𝐶0 and 𝐶1 we have 𝐶𝑔𝜒𝐴𝐿(𝑝0,)𝐶0(𝜇(𝐴))1/𝑝0,𝐶𝑔𝜒𝐴𝐿(𝑝1,)𝐶0(𝜇(𝐴))1/𝑝1.(3.26)
Hence, by the interpolation theorem we conclude that there is a constant 𝐶>0 such that 𝐶𝑔𝑓𝐿(𝑝,𝑞)𝐶𝑓𝐿(𝑝,𝑞)for𝑝0<𝑝<𝑝1,𝑞and𝑓𝐿(𝑝,𝑞).(3.27)

Remark 3.8. The necessary and sufficient condition (3.14) makes intuitive sense if we consider a variety of measures. Let us consider two of them. (1)If 𝜇 is the Lebesgue measure and 𝑋 happens to be an interval, then it suffices to take 𝑔 as the left multiplication by an absolute constant 𝑎 to achieve (3.14).(2)If instead 𝜇 is the Haar measure, by taking 𝑋=(0,), the locally compact topological group of nonzero real numbers with multiplication as operation, then for any Borel set 𝐴𝑋, we have 𝜇(𝐴)=𝐴|𝑡|1𝑑𝑡. Hence (3.14) is achieved for a measurable function 𝑔 such that 𝑔1(𝐴)𝐴. The left multiplication by the reciprocal of an absolute constant 𝑎 would be enough.

Remark 3.9. The results in Theorems 3.6 and 3.7 are in accordance with the results of Arora et al. in [5, 6]. In fact, even though they obtained their results in a more general version of Lorentz spaces, their necessary and sufficient conditions for boundedness of the multiplication and composition operators are the same as ours.

4. Discussion

The space 𝐿(𝑝,1),𝑝>1, seems to be underutilized in analysis despite the fact that in the 1950s Stein and Weiss [9] showed that for a sublinear operator 𝑇 and a Banach space 𝑋 if 𝑇𝜒𝐴𝑋𝑐𝜇(𝐴)1/𝑝, then 𝑇𝑓𝑋𝑐𝑓𝐿(𝑝,1); that is, 𝑇 can be extended to the whole 𝐿(𝑝,1). De Souza [7] showed that the reason for this is the nature of 𝐿(𝑝,1) in that 𝑓𝐿(𝑝,1) if and only if 𝑓(𝑡)=𝑛=0𝑐𝑛𝜒𝐴𝑛(𝑡) with 𝑛=0|𝑐𝑛|𝜇(𝐴𝑛)1/𝑝<. This “atomic decomposition of 𝐿(𝑝,1)” provides us with a technique to study operators on 𝐿(𝑝,𝑞) and in particular 𝐿𝑝. In other words, to study operators on 𝐿(𝑝,𝑞), all we need is to study the actions of the operator on characteristic functions which can then be lifted to 𝐿(𝑝,𝑞) through the use of interpolation theorems. Although we only considered multiplication and composition operators, other operators (Hardy-Littlewood maximal, Carleson maximal, Hankel, etc.) can be studies likewise.

To conclude, the goal of the present paper is to show that a simple atomic decomposition of 𝐿(𝑝,1) spaces shows the boundedness of operators on 𝐿(𝑝,𝑞) straightforward. In fact, we showed that unlike other techniques in the literature, the boundedness of these operators on characteristic functions is enough to generalize to the whole Lorentz space. This technique is not new at all, since it was first used by Stein and Weiss in [9] to extend the Marcinkiewicz interpolation theorem. The broader question is if the the same technique can be extended to Lorentz-Bochner spaces and even Lorentz-martingale spaces. If answered positively, the technique proposed in our paper will contrast the ones by Yong et al. in [10], which we believe are not as straightforward as ours. It has been shown in the literature that operators such as the centered Hardy operator, the Hilbert operator (under Δ2 condition) are all bounded on Lorentz spaces. Usually the proofs of these facts are not trivial, so by first finding an atomic decomposition on Lorentz spaces 𝐿(𝑝,𝑞),1<𝑝,𝑞<, it would be easier to get another proof of the boundedness of these operators, without having to resort to the Bennett and Sharpley inequality in [11]. It is important to note that, atomic decomposition on general Banach spaces has been found in [12], under the same line of research as ours. Because characteristic functions are easy to manipulate, an even broader question would be to find the class of Banach spaces whose atomic decomposition can be expressed in terms of characteristic functions only.

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Copyright © 2012 Eddy Kwessi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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