A Note on Multiplication and Composition Operators in Lorentz Spaces

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 𝐿^𝑝 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 𝐿^𝑝 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 [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,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>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,𝑞≤∞, define ‖𝑓‖_{𝐿(𝑝,𝑞)}=⎧⎪⎪⎨⎪⎪⎩⎛⎜⎜⎝𝑞𝑝^∞₀𝑡^1/𝑝𝑓^∗(𝑡)^𝑞𝑑𝑡𝑡⎞⎟⎟⎠^1/𝑞sup_𝑡>0,if𝑞<∞,𝑡^1/𝑝𝑓^∗(𝑡),if𝑞=∞.(2.2) The set of all functions 𝑓 with ‖𝑓‖_{𝐿(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 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‖=sup_{𝜇(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 𝑇_𝑔 on Lorentz spaces as 𝑇_𝑔(𝑓)=𝑓⋅𝑔 and the composition operator 𝐶_𝑔 as 𝐶_𝑔(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)=∑^∞𝑛=1𝑐_𝑛𝜒_𝐴𝑛(𝑡) with ∑^∞𝑛=1|𝑐_𝑛|𝜇^1/𝑝(𝐴_𝑛)<∞, whereas 𝜇 is measure on 𝑋 and 𝐴_𝑛 are 𝜇-measurable sets in 𝑋. Moreover, ‖𝑓‖_𝐿(𝑝,1)≅inf∑^∞𝑛=1|𝑐_𝑛|𝜇^1/𝑝(𝐴_𝑛), where the infimum is taken over all possible representations of 𝑓.

Theorem 2.6 (see Marcinkiewicz). Assume that for 0<𝑝₀≠𝑝₁≤∞, 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 𝑇_𝑔(a)‖𝑇_𝑔𝜒_𝐴‖_{𝐿(𝑝0,∞)}≤𝑀₀𝜇^1/𝑝0(𝐴). (b)‖𝑇_𝑔𝜒_𝐴‖_{𝐿(𝑝1,∞)}≤𝑀₁𝜇^1/𝑝1(𝐴).
Then there is some 𝑀>0 such that ‖𝑇_𝑔𝑓‖_{𝐿(𝑝,𝑞)}≤𝑀‖𝑓‖_{𝐿(𝑝,𝑞)} 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 ‖𝑇𝜒_𝐴‖_𝐵≤𝑀𝜇^1/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 𝑀^𝑝𝑟 the set of real-valued functions defined on 𝑋=[0,2𝜋] such that ‖𝑓‖_𝑀^𝑝𝑟=sup_𝑥>0𝑝𝑥𝑟^1/𝑝^𝑥0𝑓^∗(𝑡)𝑡^1/𝑝^𝑟𝑑𝑡𝑡^1/𝑟<∞,(2.3) where 1≤𝑝≤𝑟<∞.

We will show that the space 𝑀^𝑝𝑟 is equivalent to a weak 𝐿^𝑘 space for some 𝑘 that depends on 𝑝 and 𝑟 and ‖⋅‖_𝑀^𝑝𝑟 is quasinorm.

Lemma 2.8. ‖⋅‖_𝑀^𝑝𝑟 a quasinorm on 𝑀^𝑝𝑟.

Proof. 𝑓^∗≥0 by definition. This implies that ‖𝑓‖_𝑀^𝑝𝑟≥0. Moreover, ‖𝑓‖_𝑀^𝑝𝑟=0 implies that for all 0<𝑥 ≤ 2𝜋,∫^𝑥0(𝑓^∗(𝑡)𝑡^1/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∈𝑀^𝑝𝑟, and 𝑥∈(0,2𝜋]. Noting (𝑘𝑓)^∗=|𝑘|𝑓^∗, the homogeneity condition ‖𝑘𝑓‖_𝑀^𝑝𝑟=|𝑘|‖𝑓‖_𝑀^𝑝𝑟 follows trivially. Let 𝑓,𝑔∈𝑀^𝑝𝑟. Since (𝑓+𝑔)^∗(𝑡) ≤ 𝑓^∗(𝑡/2)  + 𝑔^∗(𝑡/2), for any 𝑥∈(0,2𝜋], we have ^𝑥0(𝑓+𝑔)^∗(𝑡)𝑡^1/𝑝^𝑟𝑑𝑡𝑡≤2^𝑟−1^𝑥0𝑓^∗2𝑡𝑡^1/𝑝^𝑟𝑑𝑡𝑡+^𝑥0𝑔^∗2𝑡𝑡^1/𝑝^𝑟𝑑𝑡≤2𝑟𝑡𝑝^+𝑟−1^0(1/2)𝑥𝑓^∗(𝑡)𝑡^1/𝑝^𝑟𝑑𝑡𝑡+^0(1/2)𝑥𝑔^∗(𝑡)𝑡^1/𝑝^𝑟𝑑𝑡𝑡.(2.4) Since (𝑎+𝑏)^1/𝑟≤𝑎^1/𝑟+𝑏^1/𝑟 for 𝑎,𝑏>0, we have ‖𝑓+𝑔‖_𝑀^𝑝𝑟≤2^{1/𝑝−1/𝑟+1}‖𝑓‖_𝑀^𝑝𝑟+‖𝑔‖_𝑀^𝑝𝑟,with2^{1/𝑝−1/𝑟+1}>1for𝑟,𝑝>1.(2.5)

Theorem 2.9. 𝑀^𝑝𝑟≅𝐿(𝑝𝑟^,∞), where 𝑟,𝑟′≥1,1/𝑟+1/𝑟^=1.

Proof. Suppose 𝑔∈𝑀^𝑝𝑟. There is an absolute constant 𝐶 such that for all 𝑥>0, 𝐶≥𝑟𝑝𝑥^1/𝑝^𝑥0𝑔^∗(𝑡)𝑡^1/𝑝^𝑟𝑑𝑡𝑡^1/𝑟≥𝑟𝑝𝑥^1/𝑝𝑔^∗(𝑥)^𝑟^𝑥0𝑡^{𝑟/𝑝−1}𝑑𝑡^1/𝑟=𝑥^{1/𝑝𝑟′}𝑔^∗(𝑥).(2.6) Thus, sup_𝑥>0𝑥^{1/𝑝𝑟′}𝑔^∗(𝑥)≤𝐶 implying that 𝑔∈𝐿(𝑝𝑟′,∞).
Conversely, let 𝑔∈𝐿(𝑝𝑟^,∞). Then there is an absolute constant 𝐶 such that, 𝑔^∗(𝑡)≤𝐶𝑡^{−1/𝑝𝑟′}. This implies that (𝑔^∗(𝑡)𝑡^1/𝑝)^𝑟≤𝐶^𝑟𝑡^1/𝑝. Thus, sup_𝑥>0𝑝𝑥𝑟^1/𝑝^𝑥0𝑔^∗(𝑡)𝑡^1/𝑝^𝑟𝑑𝑡𝑡^1/𝑟≤sup_𝑥>0𝐶^𝑟𝑟𝑝𝑥^1/𝑝^𝑥0𝑡^1/𝑝𝑑𝑡𝑡^1/𝑟=𝐶𝑟^1/𝑟.(2.7) This implies that 𝑔∈𝑀^𝑝𝑟.

Remark 2.10. One can easily see from Theorem 2.9 that 𝑀^𝑝∞≅𝐿(𝑝,∞) and 𝑀^𝑝1≅𝐿^∞. Moreover, ‖𝑔‖_𝑀^𝑝1=‖𝑔‖_∞. To see this, note that ‖𝑔‖_𝑀^𝑝1=sup_𝑥>0𝑝𝑥1^1/𝑝^𝑥0𝑔^∗(𝑡)𝑡^1/𝑝𝑑𝑡𝑡≤‖𝑔‖^∞sup_𝑥>0𝑝𝑥1^1/𝑝^𝑥0‖𝑔‖_𝑀𝑝1𝑡^1/𝑝−1𝑑𝑡=‖𝑔‖_∞,≥𝑔^∗(𝑥)𝑝𝑥^1/𝑝^𝑥0𝑡^1/𝑝−1𝑑𝑡=𝑔^∗(𝑥)(2.8) for all 𝑥 since 𝑔^∗ is decreasing. Taking the limit as 𝑥→0, we see that ‖𝑔‖_𝑀^𝑝1≥𝑔^∗(0)=‖𝑔‖_∞.

3. Main Results

3.1. Multiplication Operators

Theorem 3.1 (see multiplication operator on 𝐿(𝑝,1)). The multiplication operator 𝑇_𝑔∶𝐿(𝑝,1)→𝐿(𝑝^,1) for 𝑝^≥𝑝>1 is bounded if and only 𝑔∈𝐿^∞. Moreover, ‖𝑇_𝑔‖=‖𝑔‖_∞.

Proof. It is convenient to use 𝑀^𝑝1 which is equivalent to 𝐿^∞. Assume that =‖𝑇_𝑔𝑓‖_{𝐿(𝑝,1)}≤𝐶‖𝑓‖_𝐿(𝑝,1). Then for 𝑓=𝜒_[0,𝑥] where 𝑥∈(0,2𝜋], ^02𝜋𝑇_𝑔𝜒_[0,𝑥]^∗(𝑡)𝑡^{1/𝑝′−1}𝑑𝑡=^𝑥0𝑔^∗(𝑡)𝑡^{1/𝑝′−1}𝑑𝑡≤𝐶^02𝜋𝜒^∗[]0,𝑥(𝑡)𝑡^1/𝑝−1𝑑𝑡=𝐶𝑝𝑥^1/𝑝.(3.1) Multiplying and dividing the integrand on the left by 𝑡^1/𝑝−1, we get ^𝑥0𝑔^∗(𝑡)𝑡^1/𝑝−1𝑡^{𝑝−𝑝′/𝑝𝑝}𝑑𝑡≤𝐶𝑝𝑥^1/𝑝.(3.2) Since 𝑡↦𝑡^{(𝑝−𝑝)/𝑝𝑝} is decreasing on [0,𝑥] and 0<𝑥≤2𝜋, we have 1𝑝𝑥^1/𝑝^𝑥0𝑔^∗(𝑡)𝑡^1/𝑝−1𝑑𝑡≤𝐶(2𝜋)^{𝑝−𝑝′/𝑝𝑝′}.(3.3) Taking the supremum over all 𝑥>0, we have that 𝑔∈𝑀^𝑝1.
Assume that 𝑔∈𝑀^𝑝1 and 𝑥>0. Since 𝑝^>𝑝 we have ‖‖𝑇_𝑔𝜒_[0,𝑥]‖‖_{𝐿(𝑝,1)}=^𝑥0𝑔^∗(𝑡)𝑡^{1/𝑝′−1}𝑑𝑡≤^𝑥0𝑔^∗(𝑡)𝑡^1/𝑝−1𝑑𝑡.(3.4) And so, ‖‖𝑇_𝑔𝜒_[0,𝑥]‖‖_{𝐿(𝑝′,1)}≤𝑀‖‖𝜒_[0,𝑥]‖‖_𝐿(𝑝,1),where𝑀=sup_𝑥>01𝑝𝑥^1/𝑝^𝑥0𝑔^∗(𝑡)𝑡^1/𝑝−1𝑑𝑡.(3.5) Using the atomic decomposition of 𝐿(𝑝,1), we get ‖‖𝑇_𝑔𝑓‖‖_{𝐿(𝑝′,1)}≤𝑀^‖𝑓‖_𝐿(𝑝,1)forsomepositiveconstant𝑀^.(3.6)
To prove the second part of the theorem, first note that the expression in (3.5) gives that ‖𝑇_𝑔‖≤‖𝑔‖_∞. Now take 𝑓=(1/𝑥^1/𝑝)𝜒_[0,𝑥]. We can easily see that ‖𝑓‖_𝐿(𝑝,1)=1 and ‖𝑇_𝑔𝑓‖_{𝐿(𝑝,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 ‖𝑇_𝑔‖≥‖𝑔‖_∞. Thus, ‖𝑇_𝑔‖=‖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 𝑇_𝑔∶𝐿(𝑝,𝑞)→𝐿(𝑝,𝑞) is bounded if and only if 𝑔∈𝐿^∞ for 1<𝑝≤∞,1<𝑞≤∞. Moreover, ‖𝑇_𝑔‖=‖𝑔‖_∞.

Remark 3.3. Since, by Theorem 2.9, 𝑀^𝑝1⊆𝑀^𝑝𝑟 for 𝑟>1, the theorem implies that if the multiplication operator 𝑇_𝑔∶𝐿(𝑝,𝑞)→𝐿(𝑝,𝑞) defined by 𝑇_𝑔𝑓=𝑔⋅𝑓 is bounded, then 𝑔∈𝑀^𝑝𝑟 for 𝑝,𝑞>1.

Remark 3.4. It is worth observing that the norm ‖𝑓‖_𝐿(𝑝,1)=sup_{𝐴⊆𝑋𝜇(𝐴)≠0}1𝜇^1/𝑝(𝐴)^0𝜇(𝐴)𝑔^∗(𝑡)𝑡^1/𝑝−1𝑑𝑡(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 𝑀^𝑝𝑟. 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,𝑞₁), and 𝑔∈𝐿(𝑝₂,𝑞₂), where 1<𝑝₁,𝑝₂,𝑞₁,𝑞₂<∞, then 𝑔⋅𝑓∈𝐿(𝑟,𝑠) where 1/𝑟 = 1/𝑝₁+1/𝑝₂ and 1/𝑠=1/𝑞₁+1/𝑞₂.

Proof. Given 1<𝑝₁,𝑝₂,𝑞₁,𝑞₂<∞, assume 𝑓∈𝐿(𝑝₁,𝑞₁) and 𝑔∈𝐿(𝑝₂,𝑞₂). Let 𝑟,𝑠 be such that 1/𝑟=1/𝑝₁+1/𝑝₂ and 1/𝑠=1/𝑞₁+1/𝑞₂. Since (𝑓⋅𝑔)^∗(𝑡)≤𝑓^∗(𝑡)𝑔^∗(𝑡), we have ^02𝜋(𝑓⋅𝑔)^∗(𝑡)𝑡^1/𝑟^𝑠𝑑𝑡𝑡≤^02𝜋𝑓^∗(𝑡)𝑡^1/𝑝1^𝑠⋅𝑔^∗(𝑡)𝑡^1/𝑝2^𝑠𝑑𝑡𝑡.(3.8) Using Holder’s inequality on the RHS with 𝑠/𝑞₁+𝑠/𝑞₂=1, we have ^02𝜋(𝑓⋅𝑔)^∗(𝑡)𝑡^1/𝑟^𝑠𝑑𝑡𝑡≤^02𝜋𝑓^∗(𝑡)𝑡^1/𝑝1^𝑞1𝑑𝑡𝑡^𝑠/𝑞1⋅^02𝜋𝑔^∗(𝑡)𝑡^1/𝑞2^𝑞2𝑑𝑡𝑡^𝑠/𝑞2.(3.9) Thus, we have ‖𝑔⋅𝑓‖_{𝐿(𝑟,𝑠)}≤‖𝑓‖_{𝐿(𝑝1,𝑞1)}⋅‖𝑔‖_{𝐿(𝑝2,𝑞2)}.(3.10)

Theorem 3.6. If 𝑔∈𝑀^𝑝𝑟, then 𝑇_𝑔∶𝐿(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 𝑔∈𝑀^𝑝𝑟≅𝐿(𝑝𝑟^,∞)‖‖𝑇_𝑔𝑓‖‖^{𝑠𝐿(𝑘,𝑠)}≤^02𝜋𝑔^∗(𝑡)⋅𝑓^∗(𝑡)𝑡^1/𝑘^𝑠𝑑𝑡=_02𝜋𝑡𝑔^∗(𝑡)𝑡^{1/𝑝𝑟′}⋅𝑓^∗(𝑡)𝑡^1/𝑞^𝑠𝑑𝑡𝑡if𝑘1=1𝑞+𝑞1𝑟′.(3.11) Therefore, ‖‖𝑇_𝑔𝑓‖‖^{𝑠𝐿(𝑘,𝑠)}≤sup_𝑡>0𝑔^∗(𝑡)𝑡^{1/𝑝𝑟′}^𝑠⋅^02𝜋𝑓^∗(𝑡)𝑡^1/𝑞^𝑠𝑑𝑡𝑡.(3.12) That is, ‖‖𝑇_𝑔𝑓‖‖_{𝐿(𝑘,𝑠)}≤‖𝑔‖_𝑀^𝑝𝑟⋅‖𝑓‖_{𝐿(𝑞,𝑠)},where𝑘=𝑝𝑟′𝑞𝑝𝑟′+𝑞.(3.13)

Noting that 𝑀^𝑝𝑟≅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 𝑞₂=∞.

3.2. Composition Operators

Theorem 3.7. The composition operator 𝐶_𝑔∶𝐿(𝑝,𝑞)→𝐿(𝑝,𝑞) 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<𝑝≤∞,1≤𝑞≤∞. Moreover, ‖𝐶_𝑔‖=‖𝑔‖^1/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 𝐶_𝑔∶𝐿(𝑝,1)→𝐿(𝑝,1) is bounded that is, there is an absolute constant 𝐶 such that ‖‖𝐶_𝑔𝑓‖‖_𝐿(𝑝,1)≤𝐶‖𝑓‖_𝐿(𝑝,1).(3.15) Let 𝐴 be a 𝜇-measurable set in [0,2𝜋] and let 𝑓=𝜒_𝐴. Then, (3.15) is equivalent to ‖‖𝐶_𝑔𝜒_𝐴‖‖_𝐿(𝑝,1)≤𝐶‖‖𝜒_𝐴‖‖_𝐿(𝑝,1)⟺1𝑝^02𝜋𝐶_𝑔≤𝐶𝜒_𝐴^∗(𝑡)𝑡^1/𝑝−1𝑑𝑡𝑝^02𝜋𝜒^∗𝐴(𝑡)𝑡^(1/𝑝)−1𝑑𝑡;(3.16) that is, 1𝑝^02𝜋𝜒_𝐴∘𝑔^∗(𝑡)𝑡^1/𝑝−1𝑑𝑡≤𝐶𝑝^02𝜋𝜒_{[0,𝜇(𝐴)]}(𝑡)𝑡^(1/𝑝)−1𝑑𝑡.(3.17) Since (𝜒_𝐴∘𝑔)=𝜒_{𝑔−1(𝐴)}, then (𝜒_𝐴∘𝑔)^∗=𝜒_{[0,𝜇(𝑔−1(𝐴))]}. Therefore, the previous inequality gives 1𝑝^{𝜇(𝑔−10(𝐴))}𝑡^1/𝑝−1𝑑𝑡≤𝐶𝑝^0𝜇(𝐴)𝑡^1/𝑝−1𝑑𝑡.(3.18) And hence, 𝜇𝑔⁻¹(𝐴)≤𝐶^𝑝𝜇(𝐴).(3.19)
On the other hand, assume that there is some constant 𝐶>0 such that 𝜇(𝑔⁻¹(𝐴))≤𝐶𝜇(𝐴). Then, ‖‖𝐶_𝑔𝜒_𝐴‖‖_𝐿(𝑝,1)=1𝑝^02𝜋𝜒_𝐴=1∘𝑔^∗(𝑡)𝑡^1/𝑝−1𝑑𝑡𝑝^02𝜋=𝜇𝑔𝜒_{[0,𝜇(𝑔−1(𝐴))]}(𝑡)𝑡^1/𝑝−1𝑑𝑡₋₁(𝐴)^1/𝑝≤𝐶^1/𝑝(𝜇(𝐴))^1/𝑝.(3.20) Consequently, ‖‖𝐶_𝑔𝜒_𝐴‖‖_𝐿(𝑝,1)≤𝐶^1/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 ‖‖𝐶_𝑔𝑓‖‖_𝐿(𝑝,1)≤𝐶^1/𝑝‖𝑓‖_𝐿(𝑝,1).(3.22)
To prove the second part of the theorem, note that from the above, we have ‖‖𝐶_𝑔‖‖=sup_{‖𝑓‖𝐿(𝑝,1)≤1}‖‖𝐶_𝑔𝑓‖‖_𝐿(𝑝,1)‖𝑓‖_𝐿(𝑝,1)≤𝐶^1/𝑝.(3.23) But inf{𝐶∶𝜇(𝑔⁻¹(𝐴))≤𝐶𝜇(𝐴)}=‖𝑔‖. Thus, ‖𝐶_𝑔‖≤‖𝑔‖^1/𝑝. To obtain the other inequality, let 𝑓=(1/[𝜇(𝐴)]^1/𝑝)𝜒_𝐴. This gives ‖𝑓‖_𝐿(𝑝,1)=1 and ‖‖𝐶_𝑔𝑓‖‖_𝐿(𝑝,1)=𝜇𝑔⁻¹(𝐴)𝜇(𝐴)^1/𝑝.(3.24) Thus, ‖‖𝐶_𝑔‖‖=sup_{‖𝑓‖𝐿(𝑝,1)≤1}‖‖𝐶_𝑔𝑓‖‖_𝐿(𝑝,1)≥sup_{𝜇(𝐴)≠0}𝜇(𝑔⁻¹(𝐴))𝜇(𝐴)^1/𝑝=‖𝑔‖^1/𝑝.(3.25) Now to show the result for 𝐿(𝑝,𝑞), note that the operator 𝐶_𝑔 is linear on 𝐿(𝑝,𝑞) and that for 𝑝₀ and 𝑝₁ such that 𝑝₀<𝑝<𝑝₁, we have ‖𝐶_𝑔𝜒_𝐴‖_{𝐿(𝑝0,1)}≤𝑀₀(𝜇(𝐴))^1/𝑝0 and ‖𝐶_𝑔𝜒_𝐴‖_{𝐿(𝑝1,1)}≤𝑀₁(𝜇(𝐴))^1/𝑝1. Since 𝐿(𝑝_𝑖,1)⊆𝐿(𝑝_𝑖,∞),𝑖=0,1, then for some absolute constants 𝐶₀ and 𝐶₁ we have ‖‖‖‖𝐶𝐶_𝑔𝑔𝜒_𝐴‖‖_{𝐿(𝑝0,∞)}≤𝐶₀(𝜇(𝐴))^1/𝑝0,𝜒_𝐴‖‖_{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 ‖‖𝐶_𝑔𝑓‖‖_{𝐿(𝑝,𝑞)}≤𝐶‖𝑓‖_{𝐿(𝑝,𝑞)}for𝑝₀<𝑝<𝑝₁,∀𝑞and∀𝑓∈𝐿(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 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 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 ‖𝑇𝜒_𝐴‖_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)=∑^∞𝑛=0𝑐_𝑛𝜒_𝐴𝑛(𝑡) with ∑^∞𝑛=0|𝑐_𝑛|𝜇(𝐴_𝑛)^1/𝑝<∞. This “atomic decomposition of 𝐿(𝑝,1)” provides us with a technique to study operators on 𝐿(𝑝,𝑞) and in particular 𝐿_𝑝. 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.