we revisit the Lorentz spaces 𝐿(𝑝,ğ‘ž) for 𝑝>1,ğ‘ž>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,ğ‘ž) in his papers [1, 2] as a generalization of the 𝐿𝑝 spaces. The parameters 𝑝 and ğ‘ž 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 ğ‘ž, 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 [3–6]. 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,ğ‘ž).

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>0∶𝑑(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,ğ‘žâ‰¤âˆž, define ‖𝑓‖𝐿(𝑝,ğ‘ž)=âŽ§âŽªâŽªâŽ¨âŽªâŽªâŽ©âŽ›âŽœâŽœâŽğ‘žğ‘âˆžî€œ0𝑡1/𝑝𝑓∗(𝑡)ğ‘žğ‘‘ğ‘¡ğ‘¡âŽžâŽŸâŽŸâŽ 1/ğ‘ž,ifğ‘ž<∞,sup𝑡>0𝑡1/𝑝𝑓∗(𝑡),ifğ‘ž=∞.(2.2) The set of all functions 𝑓 with ‖𝑓‖𝐿(𝑝,ğ‘ž)<∞ is called the Lorentz space with indices 𝑝 and ğ‘ž and denoted by 𝐿(𝑝,ğ‘ž)(𝑋,𝜇).

We now consider the measure 𝜇 on 𝑋 to be finite. Let 𝑔∶𝑋→𝑋 be a 𝜇-measurable function such that 𝜇(𝑔−1(𝐴))≤𝐶𝜇(𝐴) for a 𝜇-measurable set 𝐴⊆[0,2𝜋] and for an absolute constant 𝐶. Here 𝑔−1(𝐴) refers to the preimage of the set 𝐴.

Remark 2.3. It is important to note that ‖𝑔‖=sup𝜇(𝐴)≠0(𝜇(𝑔−1(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)=𝑓⋅𝑔 and the composition operator 𝐶𝑔 as 𝐶𝑔(𝑓)=𝑓∘𝑔.
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 𝑓∈𝐿(𝑝,1) for 𝑝>1 if and only if ∑𝑓(𝑡)=âˆžğ‘›=1𝑐𝑛𝜒𝐴𝑛(𝑡) with âˆ‘âˆžğ‘›=1|𝑐𝑛|𝜇1/𝑝(𝐴𝑛)<∞, whereas 𝜇 is measure on 𝑋 and 𝐴𝑛 are 𝜇-measurable sets in 𝑋. Moreover, ‖𝑓‖𝐿(𝑝,1)∑≅infâˆžğ‘›=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<𝑝0≠𝑝1≤∞, for all ğ‘ž>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)‖𝑇𝑔𝜒𝐴‖𝐿(𝑝0,∞)≤𝑀0𝜇1/𝑝0(𝐴). (b)‖𝑇𝑔𝜒𝐴‖𝐿(𝑝1,∞)≤𝑀1𝜇1/𝑝1(𝐴).
Then there is some 𝑀>0 such that ‖𝑇𝑔𝑓‖𝐿(𝑝,ğ‘ž)≤𝑀‖𝑓‖𝐿(𝑝,ğ‘ž) for 1/𝑝=𝜃/𝑝0+(1−𝜃)/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 𝑇∶𝐿(𝑝,1)→𝐵 are bounded, where 𝐵 is Banach space closed under absolute value and satisfying ‖𝑓‖𝐵=‖|𝑓|‖𝐵, all one needs to show is that ‖𝑇𝜒𝐴‖𝐵≤𝑀𝜇1/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,ğ‘ž).

Definition 2.7. We denote by 𝑀𝑝𝑟 the set of real-valued functions defined on 𝑋=[0,2𝜋] such that ‖𝑓‖𝑀𝑝𝑟=sup𝑥>0𝑟𝑝𝑥1/𝑝𝑥0𝑓∗(𝑡)𝑡1/𝑝𝑟𝑑𝑡𝑡1/𝑟<∞,(2.3) where 1≤𝑝≤𝑟<∞.

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. 𝑓∗≥0 by definition. This implies that ‖𝑓‖𝑀𝑝𝑟≥0. Moreover, ‖𝑓‖𝑀𝑝𝑟=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 𝑘≠0 be a real constant, 𝑓∈𝑀𝑝𝑟, and 𝑥∈(0,2𝜋]. Noting (𝑘𝑓)∗=|𝑘|𝑓∗, the homogeneity condition ‖𝑘𝑓‖𝑀𝑝𝑟=|𝑘|‖𝑓‖𝑀𝑝𝑟 follows trivially. Let 𝑓,𝑔∈𝑀𝑝𝑟. Since (𝑓+𝑔)∗(𝑡) ≤ 𝑓∗(𝑡/2)  + 𝑔∗(𝑡/2), for any 𝑥∈(0,2𝜋], we have 𝑥0(𝑓+𝑔)∗(𝑡)𝑡1/𝑝𝑟𝑑𝑡𝑡≤2𝑟−1𝑥0𝑓∗𝑡2𝑡1/𝑝𝑟𝑑𝑡𝑡+𝑥0𝑔∗𝑡2𝑡1/𝑝𝑟𝑑𝑡𝑡𝑟≤2𝑝+𝑟−10(1/2)𝑥𝑓∗(𝑡)𝑡1/𝑝𝑟𝑑𝑡𝑡+0(1/2)𝑥𝑔∗(𝑡)𝑡1/𝑝𝑟𝑑𝑡𝑡.(2.4) Since (ğ‘Ž+𝑏)1/ğ‘Ÿâ‰¤ğ‘Ž1/𝑟+𝑏1/𝑟 for ğ‘Ž,𝑏>0, we have ‖𝑓+𝑔‖𝑀𝑝𝑟≤21/𝑝−1/𝑟+1‖𝑓‖𝑀𝑝𝑟+‖𝑔‖𝑀𝑝𝑟,with21/𝑝−1/𝑟+1>1for𝑟,𝑝>1.(2.5)

Theorem 2.9. 𝑀𝑝𝑟≅𝐿(ğ‘ğ‘Ÿî…ž,∞), where 𝑟,𝑟′≥1,1/𝑟+1/ğ‘Ÿî…ž=1.

Proof. Suppose 𝑔∈𝑀𝑝𝑟. There is an absolute constant 𝐶 such that for all 𝑥>0, 𝑟𝐶≥𝑝𝑥1/𝑝𝑥0𝑔∗(𝑡)𝑡1/𝑝𝑟𝑑𝑡𝑡1/𝑟≥𝑟𝑝𝑥1/𝑝𝑔∗(𝑥)𝑟𝑥0𝑡𝑟/𝑝−1𝑑𝑡1/𝑟=𝑥1/𝑝𝑟′𝑔∗(𝑥).(2.6) Thus, sup𝑥>0𝑥1/𝑝𝑟′𝑔∗(𝑥)≤𝐶 implying that 𝑔∈𝐿(𝑝𝑟′,∞).
Conversely, let 𝑔∈𝐿(ğ‘ğ‘Ÿî…ž,∞). Then there is an absolute constant 𝐶 such that, 𝑔∗(𝑡)≤𝐶𝑡−1/𝑝𝑟′. This implies that (𝑔∗(𝑡)𝑡1/𝑝)𝑟≤𝐶𝑟𝑡1/𝑝. Thus, sup𝑥>0𝑟𝑝𝑥1/𝑝𝑥0𝑔∗(𝑡)𝑡1/𝑝𝑟𝑑𝑡𝑡1/𝑟≤sup𝑥>0𝐶𝑟𝑟𝑝𝑥1/𝑝𝑥0𝑡1/𝑝𝑑𝑡𝑡1/𝑟=𝐶𝑟1/𝑟.(2.7) This implies that 𝑔∈𝑀𝑝𝑟.

Remark 2.10. One can easily see from Theorem 2.9 that ğ‘€ğ‘âˆžâ‰…ğ¿(𝑝,∞) and 𝑀𝑝1â‰…ğ¿âˆž. Moreover, ‖𝑔‖𝑀𝑝1=â€–ğ‘”â€–âˆž. To see this, note that ‖𝑔‖𝑀𝑝1=sup𝑥>01𝑝𝑥1/𝑝𝑥0𝑔∗(𝑡)𝑡1/ğ‘ğ‘‘ğ‘¡ğ‘¡î‚¶â‰¤â€–ğ‘”â€–âˆžsup𝑥>01𝑝𝑥1/𝑝𝑥0𝑡1/𝑝−1=ğ‘‘ğ‘¡â€–ğ‘”â€–âˆž,‖𝑔‖𝑀𝑝1≥𝑔∗(𝑥)𝑝𝑥1/𝑝𝑥0𝑡1/𝑝−1𝑑𝑡=𝑔∗(𝑥)(2.8) for all 𝑥 since 𝑔∗ is decreasing. Taking the limit as 𝑥→0, we see that ‖𝑔‖𝑀𝑝1≥𝑔∗(0)=â€–ğ‘”â€–âˆž.

3. Main Results

3.1. Multiplication Operators

Theorem 3.1 (see multiplication operator on 𝐿(𝑝,1)). The multiplication operator 𝑇𝑔∶𝐿(𝑝,1)→𝐿(ğ‘î…ž,1) for ğ‘î…žâ‰¥ğ‘>1 is bounded if and only ğ‘”âˆˆğ¿âˆž. Moreover, ‖𝑇𝑔‖=â€–ğ‘”â€–âˆž.

Proof. It is convenient to use 𝑀𝑝1 which is equivalent to ğ¿âˆž. Assume that =‖𝑇𝑔𝑓‖𝐿(ğ‘î…ž,1)≤𝐶‖𝑓‖𝐿(𝑝,1). Then for 𝑓=𝜒[0,𝑥] where 𝑥∈(0,2𝜋], 02𝜋𝑇𝑔𝜒[0,𝑥]∗(𝑡)𝑡1/𝑝′−1𝑑𝑡=𝑥0𝑔∗(𝑡)𝑡1/𝑝′−1𝑑𝑡≤𝐶02𝜋𝜒∗[]0,𝑥(𝑡)𝑡1/𝑝−1𝑑𝑡=𝐶𝑝𝑥1/𝑝.(3.1) Multiplying and dividing the integrand on the left by 𝑡1/𝑝−1, we get 𝑥0𝑔∗(𝑡)𝑡1/𝑝−1𝑡𝑝−𝑝′/ğ‘ğ‘î…žğ‘‘ğ‘¡â‰¤ğ¶ğ‘ğ‘¥1/𝑝.(3.2) Since 𝑡↦𝑡(ğ‘âˆ’ğ‘î…ž)/ğ‘ğ‘î…ž is decreasing on [0,𝑥] and 0<𝑥≤2𝜋, we have 1𝑝𝑥1/𝑝𝑥0𝑔∗(𝑡)𝑡1/𝑝−1𝑑𝑡≤𝐶(2𝜋)𝑝−𝑝′/𝑝𝑝′.(3.3) Taking the supremum over all 𝑥>0, we have that 𝑔∈𝑀𝑝1.
Assume that 𝑔∈𝑀𝑝1 and 𝑥>0. Since ğ‘î…ž>𝑝 we have ‖‖𝑇𝑔𝜒[0,𝑥]‖‖𝐿(ğ‘î…ž,1)=𝑥0𝑔∗(𝑡)𝑡1/𝑝′−1𝑑𝑡≤𝑥0𝑔∗(𝑡)𝑡1/𝑝−1𝑑𝑡.(3.4) And so, ‖‖𝑇𝑔𝜒[0,𝑥]‖‖𝐿(𝑝′,1)‖‖𝜒≤𝑀[0,𝑥]‖‖𝐿(𝑝,1),where𝑀=sup𝑥>01𝑝𝑥1/𝑝𝑥0𝑔∗(𝑡)𝑡1/𝑝−1𝑑𝑡.(3.5) Using the atomic decomposition of 𝐿(𝑝,1), we get ‖‖𝑇𝑔𝑓‖‖𝐿(𝑝′,1)â‰¤ğ‘€î…žâ€–ğ‘“â€–ğ¿(𝑝,1)forsomepositiveconstantğ‘€î…ž.(3.6)
To prove the second part of the theorem, first note that the expression in (3.5) gives that â€–ğ‘‡ğ‘”â€–â‰¤â€–ğ‘”â€–âˆž. Now take 𝑓=(1/𝑥1/𝑝)𝜒[0,𝑥]. We can easily see that ‖𝑓‖𝐿(𝑝,1)=1 and ‖𝑇𝑔𝑓‖𝐿(ğ‘î…ž,1)≥𝑔∗(𝑥) for 𝑥∈[0,2𝜋] since 𝑔∗ is decreasing. Now taking the sup over ‖𝑓‖𝐿(𝑝,1)≤1 and the limit as 𝑥→0 gives â€–ğ‘‡ğ‘”â€–â‰¥â€–ğ‘”â€–âˆž. Thus, ‖𝑇𝑔‖=â€–ğ‘”â€–âˆž.

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,ğ‘ž)). The multiplication operator 𝑇𝑔∶𝐿(𝑝,ğ‘ž)→𝐿(𝑝,ğ‘ž) is bounded if and only if ğ‘”âˆˆğ¿âˆž for 1<ğ‘â‰¤âˆž,1<ğ‘žâ‰¤âˆž. Moreover, ‖𝑇𝑔‖=â€–ğ‘”â€–âˆž.

Remark 3.3. Since, by Theorem 2.9, 𝑀𝑝1⊆𝑀𝑝𝑟 for 𝑟>1, the theorem implies that if the multiplication operator 𝑇𝑔∶𝐿(𝑝,ğ‘ž)→𝐿(𝑝,ğ‘ž) defined by 𝑇𝑔𝑓=𝑔⋅𝑓 is bounded, then 𝑔∈𝑀𝑝𝑟 for 𝑝,ğ‘ž>1.

Remark 3.4. It is worth observing that the norm ‖𝑓‖𝐿(𝑝,1)=sup𝐴⊆𝑋𝜇(𝐴)≠01𝜇1/𝑝(𝐴)0𝜇(𝐴)𝑔∗(𝑡)𝑡1/𝑝−1𝑑𝑡(3.7) can also be used to prove the previous theorem, first on 𝐿(𝑝,1) and then on 𝐿(𝑝,ğ‘ž) 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), 𝐴⊂𝑋 by 𝑥>0.

Theorem 3.5. If 𝑓∈𝐿(𝑝1,ğ‘ž1), and 𝑔∈𝐿(𝑝2,ğ‘ž2), where 1<𝑝1,𝑝2,ğ‘ž1,ğ‘ž2<∞, then 𝑔⋅𝑓∈𝐿(𝑟,𝑠) where 1/𝑟 = 1/𝑝1+1/𝑝2 and 1/𝑠=1/ğ‘ž1+1/ğ‘ž2.

Proof. Given 1<𝑝1,𝑝2,ğ‘ž1,ğ‘ž2<∞, assume 𝑓∈𝐿(𝑝1,ğ‘ž1) and 𝑔∈𝐿(𝑝2,ğ‘ž2). Let 𝑟,𝑠 be such that 1/𝑟=1/𝑝1+1/𝑝2 and 1/𝑠=1/ğ‘ž1+1/ğ‘ž2. Since (𝑓⋅𝑔)∗(𝑡)≤𝑓∗(𝑡)𝑔∗(𝑡), we have 02𝜋(𝑓⋅𝑔)∗(𝑡)𝑡1/𝑟𝑠𝑑𝑡𝑡≤02𝜋𝑓∗(𝑡)𝑡1/𝑝1𝑠⋅𝑔∗(𝑡)𝑡1/𝑝2𝑠𝑑𝑡𝑡.(3.8) Using Holder’s inequality on the RHS with 𝑠/ğ‘ž1+𝑠/ğ‘ž2=1, we have 02𝜋(𝑓⋅𝑔)∗(𝑡)𝑡1/𝑟𝑠𝑑𝑡𝑡≤02𝜋𝑓∗(𝑡)𝑡1/𝑝1î€¸ğ‘ž1𝑑𝑡𝑡𝑠/ğ‘ž1⋅02𝜋𝑔∗(𝑡)𝑡1/ğ‘ž2î€¸ğ‘ž2𝑑𝑡𝑡𝑠/ğ‘ž2.(3.9) Thus, we have ‖𝑔⋅𝑓‖𝐿(𝑟,𝑠)≤‖𝑓‖𝐿(𝑝1,ğ‘ž1)⋅‖𝑔‖𝐿(𝑝2,ğ‘ž2).(3.10)

Theorem 3.6. If 𝑔∈𝑀𝑝𝑟, then 𝑇𝑔∶𝐿(ğ‘ž,𝑠)→𝐿(ğ‘ğ‘žğ‘Ÿî…ž/(ğ‘ğ‘Ÿî…ž+ğ‘ž),𝑠) is bounded, where 1/𝑟+1/𝑟′=1 and for 𝑠>0 and 𝑝,ğ‘ž>1.

Proof. Let 𝑔∈𝑀𝑝𝑟≅𝐿(ğ‘ğ‘Ÿî…ž,∞)‖‖𝑇𝑔𝑓‖‖𝑠𝐿(𝑘,𝑠)≤02𝜋𝑔∗(𝑡)⋅𝑓∗(𝑡)𝑡1/𝑘𝑠𝑑𝑡𝑡=02𝜋𝑔∗(𝑡)𝑡1/𝑝𝑟′⋅𝑓∗(𝑡)𝑡1/ğ‘žî‚ğ‘ ğ‘‘ğ‘¡ğ‘¡1if𝑘=1ğ‘ž+1.ğ‘žğ‘Ÿâ€²(3.11) Therefore, ‖‖𝑇𝑔𝑓‖‖𝑠𝐿(𝑘,𝑠)≤sup𝑡>0𝑔∗(𝑡)𝑡1/𝑝𝑟′𝑠⋅02𝜋𝑓∗(𝑡)𝑡1/ğ‘žî€¸ğ‘ ğ‘‘ğ‘¡ğ‘¡.(3.12) That is, ‖‖𝑇𝑔𝑓‖‖𝐿(𝑘,𝑠)≤‖𝑔‖𝑀𝑝𝑟⋅‖𝑓‖𝐿(ğ‘ž,𝑠),where𝑘=ğ‘ğ‘Ÿâ€²ğ‘ž.𝑝𝑟′+ğ‘ž(3.13)

Noting that 𝑀𝑝𝑟≅𝐿(𝑝𝑟′,∞), 𝑟′=𝑟/(𝑟−1), it is easy to see that Theorem 3.6 shows that the result of Theorem 3.5 extends to the case where ğ‘ž2=∞.

3.2. Composition Operators

Theorem 3.7. The composition operator 𝐶𝑔∶𝐿(𝑝,ğ‘ž)→𝐿(𝑝,ğ‘ž) is bounded if and only if there is an absolute constant C such that 𝜇𝑔−1(𝐴)≤𝐶𝜇(𝐴),(3.14) for all 𝜇-measurable sets 𝐴⊆[0,2𝜋] and for 1<ğ‘â‰¤âˆž,1â‰¤ğ‘žâ‰¤âˆž. Moreover, ‖𝐶𝑔‖=‖𝑔‖1/𝑝.

Proof. We will prove this theorem for 𝐿(𝑝,1) and use the interpolation theorem to conclude for 𝐿(𝑝,ğ‘ž).
First assume that 𝐶𝑔∶𝐿(𝑝,1)→𝐿(𝑝,1) is bounded that is, there is an absolute constant 𝐶 such that ‖‖𝐶𝑔𝑓‖‖𝐿(𝑝,1)≤𝐶‖𝑓‖𝐿(𝑝,1).(3.15) Let 𝐴 be a 𝜇-measurable set in [0,2𝜋] and let 𝑓=𝜒𝐴. Then, (3.15) is equivalent to ‖‖𝐶𝑔𝜒𝐴‖‖𝐿(𝑝,1)‖‖𝜒≤𝐶𝐴‖‖𝐿(𝑝,1)⟺1𝑝02𝜋𝐶𝑔𝜒𝐴∗(𝑡)𝑡1/𝑝−1≤𝐶𝑑𝑡𝑝02𝜋𝜒∗𝐴(𝑡)𝑡(1/𝑝)−1𝑑𝑡;(3.16) that is, 1𝑝02𝜋𝜒𝐴∘𝑔∗(𝑡)𝑡1/𝑝−1𝐶𝑑𝑡≤𝑝02𝜋𝜒[0,𝜇(𝐴)](𝑡)𝑡(1/𝑝)−1𝑑𝑡.(3.17) Since (𝜒𝐴∘𝑔)=𝜒𝑔−1(𝐴), then (𝜒𝐴∘𝑔)∗=𝜒[0,𝜇(𝑔−1(𝐴))]. Therefore, the previous inequality gives 1𝑝𝜇(𝑔−10(𝐴))𝑡1/𝑝−1𝐶𝑑𝑡≤𝑝0𝜇(𝐴)𝑡1/𝑝−1𝑑𝑡.(3.18) And hence, 𝜇𝑔−1(𝐴)≤𝐶𝑝𝜇(𝐴).(3.19)
On the other hand, assume that there is some constant 𝐶>0 such that 𝜇(𝑔−1(𝐴))≤𝐶𝜇(𝐴). Then, ‖‖𝐶𝑔𝜒𝐴‖‖𝐿(𝑝,1)=1𝑝02𝜋𝜒𝐴∘𝑔∗(𝑡)𝑡1/𝑝−1=1𝑑𝑡𝑝02𝜋𝜒[0,𝜇(𝑔−1(𝐴))](𝑡)𝑡1/𝑝−1=𝜇𝑔𝑑𝑡−1(𝐴)1/𝑝≤𝐶1/𝑝(𝜇(𝐴))1/𝑝.(3.20) Consequently, ‖‖𝐶𝑔𝜒𝐴‖‖𝐿(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 ‖‖𝐶𝑔𝑓‖‖𝐿(𝑝,1)≤𝐶1/𝑝‖𝑓‖𝐿(𝑝,1).(3.22)
To prove the second part of the theorem, note that from the above, we have ‖‖𝐶𝑔‖‖=sup‖𝑓‖𝐿(𝑝,1)≤1‖‖𝐶𝑔𝑓‖‖𝐿(𝑝,1)‖𝑓‖𝐿(𝑝,1)≤𝐶1/𝑝.(3.23) But inf{𝐶∶𝜇(𝑔−1(𝐴))≤𝐶𝜇(𝐴)}=‖𝑔‖. Thus, ‖𝐶𝑔‖≤‖𝑔‖1/𝑝. To obtain the other inequality, let 𝑓=(1/[𝜇(𝐴)]1/𝑝)𝜒𝐴. This gives ‖𝑓‖𝐿(𝑝,1)=1 and ‖‖𝐶𝑔𝑓‖‖𝐿(𝑝,1)=𝜇𝑔−1(𝐴)𝜇(𝐴)1/𝑝.(3.24) Thus, ‖‖𝐶𝑔‖‖=sup‖𝑓‖𝐿(𝑝,1)≤1‖‖𝐶𝑔𝑓‖‖𝐿(𝑝,1)≥sup𝜇(𝐴)≠0𝜇(𝑔−1(𝐴))𝜇(𝐴)1/𝑝=‖𝑔‖1/𝑝.(3.25) Now to show the result for 𝐿(𝑝,ğ‘ž), note that the operator 𝐶𝑔 is linear on 𝐿(𝑝,ğ‘ž) and that for 𝑝0 and 𝑝1 such that 𝑝0<𝑝<𝑝1, we have ‖𝐶𝑔𝜒𝐴‖𝐿(𝑝0,1)≤𝑀0(𝜇(𝐴))1/𝑝0 and ‖𝐶𝑔𝜒𝐴‖𝐿(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 ‖‖𝐶𝑔𝜒𝐴‖‖𝐿(𝑝0,∞)≤𝐶0(𝜇(𝐴))1/𝑝0,‖‖𝐶𝑔𝜒𝐴‖‖𝐿(𝑝1,∞)≤𝐶0(𝜇(𝐴))1/𝑝1.(3.26)
Hence, by the interpolation theorem we conclude that there is a constant 𝐶>0 such that ‖‖𝐶𝑔𝑓‖‖𝐿(𝑝,ğ‘ž)≤𝐶‖𝑓‖𝐿(𝑝,ğ‘ž)for𝑝0<𝑝<𝑝1,âˆ€ğ‘žand∀𝑓∈𝐿(𝑝,ğ‘ž).(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 ğ‘Ž 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 𝐴⊆𝑋, we have ∫𝜇(𝐴)=𝐴|𝑡|−1𝑑𝑡. Hence (3.14) is achieved for a measurable function 𝑔 such that 𝑔−1(𝐴)⊆𝐴. The left multiplication by the reciprocal of an absolute constant ğ‘Ž 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 ‖𝑇𝜒𝐴‖𝑋≤𝑐𝜇(𝐴)1/𝑝, then ‖𝑇𝑓‖𝑋≤𝑐‖𝑓‖𝐿(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 𝑓∈𝐿(𝑝,1) if and only if ∑𝑓(𝑡)=âˆžğ‘›=0𝑐𝑛𝜒𝐴𝑛(𝑡) with âˆ‘âˆžğ‘›=0|𝑐𝑛|𝜇(𝐴𝑛)1/𝑝<∞. This “atomic decomposition of 𝐿(𝑝,1)” provides us with a technique to study operators on 𝐿(𝑝,ğ‘ž) and in particular 𝐿𝑝. In other words, to study operators on 𝐿(𝑝,ğ‘ž), all we need is to study the actions of the operator on characteristic functions which can then be lifted to 𝐿(𝑝,ğ‘ž) 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,ğ‘ž) 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,ğ‘ž),1<𝑝,ğ‘ž<∞, 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.