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Journal of Function Spaces and Applications
VolumeΒ 2012, Article IDΒ 293613, 10 pages
Research Article

A Note on Multiplication and Composition Operators in Lorentz Spaces

1Trinity University, One Trinity Place, San Antonio, TX 78212, USA
2US Air Force Academy, Colorado Springs, CO 80840, USA
3Department of Mathematics and Statistics, Auburn University, 221 Parker Hall, Auburn, AL 36849, USA

Received 6 February 2012; Accepted 21 June 2012

Academic Editor: GesturΒ Γ“lafsson

Copyright Β© 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.


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

2. Preliminaries

Let (𝑋,πœ‡) 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 π‘“βˆ— defined on [0,∞) by π‘“βˆ—(𝑑)=inf{𝑦>0βˆΆπ‘‘(𝑓,𝑦)≀𝑑},(2.1) where 𝑑(𝑓,𝑦)=πœ‡({π‘₯∢|𝑓(π‘₯)|>𝑦}) is the distribution of the function 𝑓.

Definition 2.2. Given a measurable function 𝑓 on (𝑋,πœ‡) and 0<𝑝,π‘žβ‰€βˆž, define ‖𝑓‖𝐿(𝑝,π‘ž)=⎧βŽͺβŽͺ⎨βŽͺβŽͺβŽ©βŽ›βŽœβŽœβŽπ‘žπ‘βˆžξ€œ0𝑑1/π‘π‘“βˆ—ξ€Έ(𝑑)π‘žπ‘‘π‘‘π‘‘βŽžβŽŸβŽŸβŽ 1/π‘ž,ifπ‘ž<∞,sup𝑑>0𝑑1/π‘π‘“βˆ—(𝑑),ifπ‘ž=∞.(2.2) The set of all functions 𝑓 with ‖𝑓‖𝐿(𝑝,π‘ž)<∞ is called the Lorentz space with indices 𝑝 and π‘ž and denoted by 𝐿(𝑝,π‘ž)(𝑋,πœ‡).

We now consider the measure πœ‡ on 𝑋 to be finite. Let π‘”βˆΆπ‘‹β†’π‘‹ be a πœ‡-measurable function such that πœ‡(π‘”βˆ’1(𝐴))β‰€πΆπœ‡(𝐴) for a πœ‡-measurable set π΄βŠ†[0,2πœ‹] and for an absolute constant 𝐢. Here π‘”βˆ’1(𝐴) refers to the preimage of the set 𝐴.

Remark 2.3. It is important to note that ‖𝑔‖=supπœ‡(𝐴)β‰ 0(πœ‡(π‘”βˆ’1(𝐴))/πœ‡(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 𝐢𝑔 as 𝐢𝑔(𝑓)=π‘“βˆ˜π‘”.
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 π‘“βˆˆπΏ(𝑝,1) for 𝑝>1 if and only if βˆ‘π‘“(𝑑)=βˆžπ‘›=1π‘π‘›πœ’π΄π‘›(𝑑) with βˆ‘βˆžπ‘›=1|𝑐𝑛|πœ‡1/𝑝(𝐴𝑛)<∞, whereas πœ‡ is measure on 𝑋 and 𝐴𝑛 are πœ‡-measurable sets in 𝑋. Moreover, ‖𝑓‖𝐿(𝑝,1)βˆ‘β‰…infβˆžπ‘›=1|𝑐𝑛|πœ‡1/𝑝(𝐴𝑛), where the infimum is taken over all possible representations of 𝑓.

Theorem 2.6 (see Marcinkiewicz). Assume that for 0<𝑝0≠𝑝1β‰€βˆž, for all π‘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)β€–π‘‡π‘”πœ’π΄β€–πΏ(𝑝0,∞)≀𝑀0πœ‡1/𝑝0(𝐴). (b)β€–π‘‡π‘”πœ’π΄β€–πΏ(𝑝1,∞)≀𝑀1πœ‡1/𝑝1(𝐴).
Then there is some 𝑀>0 such that ‖𝑇𝑔𝑓‖𝐿(𝑝,π‘ž)≀𝑀‖𝑓‖𝐿(𝑝,π‘ž) for 1/𝑝=πœƒ/𝑝0+(1βˆ’πœƒ)/𝑝1,0<πœƒ<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 π‘‡βˆΆπΏ(𝑝,1)→𝐡 are bounded, where 𝐡 is Banach space closed under absolute value and satisfying ‖𝑓‖𝐡=β€–|𝑓|‖𝐡, all one needs to show is that β€–π‘‡πœ’π΄β€–π΅β‰€π‘€πœ‡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,π‘ž).

Definition 2.7. We denote by π‘€π‘π‘Ÿ the set of real-valued functions defined on 𝑋=[0,2πœ‹] such that β€–π‘“β€–π‘€π‘π‘Ÿ=supπ‘₯>0ξ‚΅π‘Ÿπ‘π‘₯1/π‘ξ€œπ‘₯0ξ€·π‘“βˆ—(𝑑)𝑑1/π‘ξ€Έπ‘Ÿπ‘‘π‘‘π‘‘ξ‚Ά1/π‘Ÿ<∞,(2.3) where 1β‰€π‘β‰€π‘Ÿ<∞.

We will show that the space π‘€π‘π‘Ÿ is equivalent to a weak πΏπ‘˜ space for some π‘˜ that depends on 𝑝 and π‘Ÿ and β€–β‹…β€–π‘€π‘π‘Ÿ is quasinorm.

Lemma 2.8. β€–β‹…β€–π‘€π‘π‘Ÿ a quasinorm on π‘€π‘π‘Ÿ.

Proof. π‘“βˆ—β‰₯0 by definition. This implies that β€–π‘“β€–π‘€π‘π‘Ÿβ‰₯0. Moreover, β€–π‘“β€–π‘€π‘π‘Ÿ=0 implies that for all 0<π‘₯β€‰β‰€β€‰βˆ«2πœ‹,π‘₯0(π‘“βˆ—(𝑑)𝑑1/𝑝)π‘Ÿπ‘‘π‘‘/𝑑=0. Hence, we have π‘“βˆ—=0πœ‡-a.e, thus, 𝑓=0 since 𝑓 is a representative of an equivalence class. Now let π‘˜β‰ 0 be a real constant, π‘“βˆˆπ‘€π‘π‘Ÿ, and π‘₯∈(0,2πœ‹]. Noting (π‘˜π‘“)βˆ—=|π‘˜|π‘“βˆ—, the homogeneity condition β€–π‘˜π‘“β€–π‘€π‘π‘Ÿ=|π‘˜|β€–π‘“β€–π‘€π‘π‘Ÿ follows trivially. Let 𝑓,π‘”βˆˆπ‘€π‘π‘Ÿ. Since (𝑓+𝑔)βˆ—(𝑑)β€‰β‰€β€‰π‘“βˆ—(𝑑/2)  +β€‰π‘”βˆ—(𝑑/2), for any π‘₯∈(0,2πœ‹], we have ξ€œπ‘₯0ξ€·(𝑓+𝑔)βˆ—(𝑑)𝑑1/π‘ξ€Έπ‘Ÿπ‘‘π‘‘π‘‘β‰€2π‘Ÿβˆ’1ξ‚΅ξ€œπ‘₯0ξ‚€π‘“βˆ—ξ‚€π‘‘2𝑑1/π‘ξ‚π‘Ÿπ‘‘π‘‘π‘‘+ξ€œπ‘₯0ξ‚€π‘”βˆ—ξ‚€π‘‘2𝑑1/π‘ξ‚π‘Ÿπ‘‘π‘‘π‘‘ξ‚Άπ‘Ÿβ‰€2𝑝+π‘Ÿβˆ’1ξ‚΅ξ€œ0(1/2)π‘₯ξ€·π‘“βˆ—(𝑑)𝑑1/π‘ξ€Έπ‘Ÿπ‘‘π‘‘π‘‘+ξ€œ0(1/2)π‘₯ξ€·π‘”βˆ—(𝑑)𝑑1/π‘ξ€Έπ‘Ÿπ‘‘π‘‘π‘‘ξ‚Ά.(2.4) Since (π‘Ž+𝑏)1/π‘Ÿβ‰€π‘Ž1/π‘Ÿ+𝑏1/π‘Ÿ for π‘Ž,𝑏>0, we have ‖𝑓+π‘”β€–π‘€π‘π‘Ÿβ‰€21/π‘βˆ’1/π‘Ÿ+1ξ€·β€–π‘“β€–π‘€π‘π‘Ÿ+β€–π‘”β€–π‘€π‘π‘Ÿξ€Έ,with21/π‘βˆ’1/π‘Ÿ+1>1forπ‘Ÿ,𝑝>1.(2.5)

Theorem 2.9. π‘€π‘π‘Ÿβ‰…πΏ(π‘π‘Ÿξ…ž,∞), where π‘Ÿ,π‘Ÿβ€²β‰₯1,1/π‘Ÿ+1/π‘Ÿξ…ž=1.

Proof. Suppose π‘”βˆˆπ‘€π‘π‘Ÿ. There is an absolute constant 𝐢 such that for all π‘₯>0, ξ‚΅π‘ŸπΆβ‰₯𝑝π‘₯1/π‘ξ€œπ‘₯0ξ€·π‘”βˆ—(𝑑)𝑑1/π‘ξ€Έπ‘Ÿπ‘‘π‘‘π‘‘ξ‚Ά1/π‘Ÿβ‰₯ξ‚΅π‘Ÿπ‘π‘₯1/π‘ξ€·π‘”βˆ—ξ€Έ(π‘₯)π‘Ÿξ€œπ‘₯0π‘‘π‘Ÿ/π‘βˆ’1𝑑𝑑1/π‘Ÿ=π‘₯1/π‘π‘Ÿβ€²π‘”βˆ—(π‘₯).(2.6) Thus, supπ‘₯>0π‘₯1/π‘π‘Ÿβ€²π‘”βˆ—(π‘₯)≀𝐢 implying that π‘”βˆˆπΏ(π‘π‘Ÿβ€²,∞).
Conversely, let π‘”βˆˆπΏ(π‘π‘Ÿξ…ž,∞). Then there is an absolute constant 𝐢 such that, π‘”βˆ—(𝑑)β‰€πΆπ‘‘βˆ’1/π‘π‘Ÿβ€². This implies that (π‘”βˆ—(𝑑)𝑑1/𝑝)π‘Ÿβ‰€πΆπ‘Ÿπ‘‘1/𝑝. Thus, supπ‘₯>0ξ‚΅π‘Ÿπ‘π‘₯1/π‘ξ€œπ‘₯0ξ€·π‘”βˆ—(𝑑)𝑑1/π‘ξ€Έπ‘Ÿπ‘‘π‘‘π‘‘ξ‚Ά1/π‘Ÿβ‰€supπ‘₯>0ξ‚΅πΆπ‘Ÿπ‘Ÿπ‘π‘₯1/π‘ξ€œπ‘₯0𝑑1/𝑝𝑑𝑑𝑑1/π‘Ÿ=πΆπ‘Ÿ1/π‘Ÿ.(2.7) This implies that π‘”βˆˆπ‘€π‘π‘Ÿ.

Remark 2.10. One can easily see from Theorem 2.9 that π‘€π‘βˆžβ‰…πΏ(𝑝,∞) and 𝑀𝑝1β‰…πΏβˆž. Moreover, ‖𝑔‖𝑀𝑝1=β€–π‘”β€–βˆž. To see this, note that ‖𝑔‖𝑀𝑝1=supπ‘₯>0ξ‚΅1𝑝π‘₯1/π‘ξ€œπ‘₯0π‘”βˆ—(𝑑)𝑑1/π‘π‘‘π‘‘π‘‘ξ‚Άβ‰€β€–π‘”β€–βˆžsupπ‘₯>0ξ‚΅1𝑝π‘₯1/π‘ξ€œπ‘₯0𝑑1/π‘βˆ’1ξ‚Ά=π‘‘π‘‘β€–π‘”β€–βˆž,‖𝑔‖𝑀𝑝1β‰₯π‘”βˆ—(π‘₯)𝑝π‘₯1/π‘ξ€œπ‘₯0𝑑1/π‘βˆ’1𝑑𝑑=π‘”βˆ—(π‘₯)(2.8) for all π‘₯ since π‘”βˆ— is decreasing. Taking the limit as π‘₯β†’0, we see that ‖𝑔‖𝑀𝑝1β‰₯π‘”βˆ—(0)=β€–π‘”β€–βˆž.

3. Main Results

3.1. Multiplication Operators

Theorem 3.1 (see multiplication operator on 𝐿(𝑝,1)). The multiplication operator π‘‡π‘”βˆΆπΏ(𝑝,1)→𝐿(π‘ξ…ž,1) for π‘ξ…žβ‰₯𝑝>1 is bounded if and only π‘”βˆˆπΏβˆž. Moreover, ‖𝑇𝑔‖=β€–π‘”β€–βˆž.

Proof. It is convenient to use 𝑀𝑝1 which is equivalent to 𝐿∞. Assume that =‖𝑇𝑔𝑓‖𝐿(π‘ξ…ž,1)≀𝐢‖𝑓‖𝐿(𝑝,1). Then for 𝑓=πœ’[0,π‘₯] where π‘₯∈(0,2πœ‹], ξ€œ02πœ‹ξ€·π‘‡π‘”πœ’[0,π‘₯]ξ€Έβˆ—(𝑑)𝑑1/π‘β€²βˆ’1ξ€œπ‘‘π‘‘=π‘₯0π‘”βˆ—(𝑑)𝑑1/π‘β€²βˆ’1ξ€œπ‘‘π‘‘β‰€πΆ02πœ‹πœ’βˆ—[]0,π‘₯(𝑑)𝑑1/π‘βˆ’1𝑑𝑑=𝐢𝑝π‘₯1/𝑝.(3.1) Multiplying and dividing the integrand on the left by 𝑑1/π‘βˆ’1, we get ξ€œπ‘₯0π‘”βˆ—(𝑑)𝑑1/π‘βˆ’1π‘‘π‘βˆ’π‘β€²/π‘π‘ξ…žπ‘‘π‘‘β‰€πΆπ‘π‘₯1/𝑝.(3.2) Since 𝑑↦𝑑(π‘βˆ’π‘ξ…ž)/π‘π‘ξ…ž is decreasing on [0,π‘₯] and 0<π‘₯≀2πœ‹, we have 1𝑝π‘₯1/π‘ξ€œπ‘₯0π‘”βˆ—(𝑑)𝑑1/π‘βˆ’1𝑑𝑑≀𝐢(2πœ‹)π‘βˆ’π‘β€²/𝑝𝑝′.(3.3) Taking the supremum over all π‘₯>0, we have that π‘”βˆˆπ‘€π‘1.
Assume that π‘”βˆˆπ‘€π‘1 and π‘₯>0. Since π‘ξ…ž>𝑝 we have β€–β€–π‘‡π‘”πœ’[0,π‘₯]‖‖𝐿(π‘ξ…ž,1)=ξ€œπ‘₯0π‘”βˆ—(𝑑)𝑑1/π‘β€²βˆ’1ξ€œπ‘‘π‘‘β‰€π‘₯0π‘”βˆ—(𝑑)𝑑1/π‘βˆ’1𝑑𝑑.(3.4) And so, β€–β€–π‘‡π‘”πœ’[0,π‘₯]‖‖𝐿(𝑝′,1)β€–β€–πœ’β‰€π‘€[0,π‘₯]‖‖𝐿(𝑝,1),where𝑀=supπ‘₯>01𝑝π‘₯1/π‘ξ€œπ‘₯0π‘”βˆ—(𝑑)𝑑1/π‘βˆ’1𝑑𝑑.(3.5) Using the atomic decomposition of 𝐿(𝑝,1), we get ‖‖𝑇𝑔𝑓‖‖𝐿(𝑝′,1)β‰€π‘€ξ…žβ€–π‘“β€–πΏ(𝑝,1)forsomepositiveconstantπ‘€ξ…ž.(3.6)
To prove the second part of the theorem, first note that the expression in (3.5) gives that β€–π‘‡π‘”β€–β‰€β€–π‘”β€–βˆž. Now take 𝑓=(1/π‘₯1/𝑝)πœ’[0,π‘₯]. We can easily see that ‖𝑓‖𝐿(𝑝,1)=1 and ‖𝑇𝑔𝑓‖𝐿(π‘ξ…ž,1)β‰₯π‘”βˆ—(π‘₯) for π‘₯∈[0,2πœ‹] since π‘”βˆ— is decreasing. Now taking the sup over ‖𝑓‖𝐿(𝑝,1)≀1 and the limit as π‘₯β†’0 gives ‖𝑇𝑔‖β‰₯β€–π‘”β€–βˆž. Thus, ‖𝑇𝑔‖=β€–π‘”β€–βˆž.

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,π‘ž)). The multiplication operator π‘‡π‘”βˆΆπΏ(𝑝,π‘ž)→𝐿(𝑝,π‘ž) is bounded if and only if π‘”βˆˆπΏβˆž for 1<π‘β‰€βˆž,1<π‘žβ‰€βˆž. Moreover, ‖𝑇𝑔‖=β€–π‘”β€–βˆž.

Remark 3.3. Since, by Theorem 2.9, 𝑀𝑝1βŠ†π‘€π‘π‘Ÿ for π‘Ÿ>1, the theorem implies that if the multiplication operator π‘‡π‘”βˆΆπΏ(𝑝,π‘ž)→𝐿(𝑝,π‘ž) defined by 𝑇𝑔𝑓=𝑔⋅𝑓 is bounded, then π‘”βˆˆπ‘€π‘π‘Ÿ for 𝑝,π‘ž>1.

Remark 3.4. It is worth observing that the norm ‖𝑓‖𝐿(𝑝,1)=supπ΄βŠ†π‘‹πœ‡(𝐴)β‰ 01πœ‡1/π‘ξ€œ(𝐴)0πœ‡(𝐴)π‘”βˆ—(𝑑)𝑑1/π‘βˆ’1𝑑𝑑(3.7) can also be used to prove the previous theorem, first on 𝐿(𝑝,1) and then on 𝐿(𝑝,π‘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 π‘€π‘π‘Ÿ. For sake of simplicity and without loss of generality, we modified it by replacing πœ‡(𝐴), π΄βŠ‚π‘‹ by π‘₯>0.

Theorem 3.5. If π‘“βˆˆπΏ(𝑝1,π‘ž1), and π‘”βˆˆπΏ(𝑝2,π‘ž2), where 1<𝑝1,𝑝2,π‘ž1,π‘ž2<∞, then π‘”β‹…π‘“βˆˆπΏ(π‘Ÿ,𝑠) where 1/π‘Ÿβ€‰= 1/𝑝1+1/𝑝2 and 1/𝑠=1/π‘ž1+1/π‘ž2.

Proof. Given 1<𝑝1,𝑝2,π‘ž1,π‘ž2<∞, assume π‘“βˆˆπΏ(𝑝1,π‘ž1) and π‘”βˆˆπΏ(𝑝2,π‘ž2). Let π‘Ÿ,𝑠 be such that 1/π‘Ÿ=1/𝑝1+1/𝑝2 and 1/𝑠=1/π‘ž1+1/π‘ž2. Since (𝑓⋅𝑔)βˆ—(𝑑)β‰€π‘“βˆ—(𝑑)π‘”βˆ—(𝑑), we have ξ€œ02πœ‹ξ€·(𝑓⋅𝑔)βˆ—(𝑑)𝑑1/π‘Ÿξ€Έπ‘ π‘‘π‘‘π‘‘β‰€ξ€œ02πœ‹ξ€·π‘“βˆ—(𝑑)𝑑1/𝑝1ξ€Έπ‘ β‹…ξ‚΅π‘”βˆ—(𝑑)𝑑1/𝑝2𝑠𝑑𝑑𝑑.(3.8) Using Holder’s inequality on the RHS with 𝑠/π‘ž1+𝑠/π‘ž2=1, we have ξ€œ02πœ‹ξ€·(𝑓⋅𝑔)βˆ—(𝑑)𝑑1/π‘Ÿξ€Έπ‘ π‘‘π‘‘π‘‘β‰€ξ‚΅ξ€œ02πœ‹ξ€·π‘“βˆ—(𝑑)𝑑1/𝑝1ξ€Έπ‘ž1𝑑𝑑𝑑𝑠/π‘ž1β‹…ξ‚΅ξ€œ02πœ‹ξ€·π‘”βˆ—(𝑑)𝑑1/π‘ž2ξ€Έπ‘ž2𝑑𝑑𝑑𝑠/π‘ž2.(3.9) Thus, we have ‖𝑔⋅𝑓‖𝐿(π‘Ÿ,𝑠)≀‖𝑓‖𝐿(𝑝1,π‘ž1)⋅‖𝑔‖𝐿(𝑝2,π‘ž2).(3.10)

Theorem 3.6. If π‘”βˆˆπ‘€π‘π‘Ÿ, then π‘‡π‘”βˆΆπΏ(π‘ž,𝑠)→𝐿(π‘π‘žπ‘Ÿξ…ž/(π‘π‘Ÿξ…ž+π‘ž),𝑠) is bounded, where 1/π‘Ÿ+1/π‘Ÿβ€²=1 and for 𝑠>0 and 𝑝,π‘ž>1.

Proof. Let π‘”βˆˆπ‘€π‘π‘Ÿβ‰…πΏ(π‘π‘Ÿξ…ž,∞)‖‖𝑇𝑔𝑓‖‖𝑠𝐿(π‘˜,𝑠)β‰€ξ€œ02πœ‹ξ€·π‘”βˆ—(𝑑)β‹…π‘“βˆ—(𝑑)𝑑1/π‘˜ξ€Έπ‘ π‘‘π‘‘π‘‘=ξ€œ02πœ‹ξ‚€π‘”βˆ—(𝑑)𝑑1/π‘π‘Ÿβ€²β‹…π‘“βˆ—(𝑑)𝑑1/π‘žξ‚π‘ π‘‘π‘‘π‘‘1ifπ‘˜=1π‘ž+1.π‘žπ‘Ÿβ€²(3.11) Therefore, ‖‖𝑇𝑔𝑓‖‖𝑠𝐿(π‘˜,𝑠)≀sup𝑑>0ξ‚€π‘”βˆ—(𝑑)𝑑1/π‘π‘Ÿβ€²ξ‚π‘ β‹…ξ€œ02πœ‹ξ€·π‘“βˆ—(𝑑)𝑑1/π‘žξ€Έπ‘ π‘‘π‘‘π‘‘.(3.12) That is, ‖‖𝑇𝑔𝑓‖‖𝐿(π‘˜,𝑠)β‰€β€–π‘”β€–π‘€π‘π‘Ÿβ‹…β€–π‘“β€–πΏ(π‘ž,𝑠),whereπ‘˜=π‘π‘Ÿβ€²π‘ž.π‘π‘Ÿβ€²+π‘ž(3.13)

Noting that π‘€π‘π‘Ÿβ‰…πΏ(π‘π‘Ÿβ€²,∞), π‘Ÿβ€²=π‘Ÿ/(π‘Ÿβˆ’1), it is easy to see that Theorem 3.6 shows that the result of Theorem 3.5 extends to the case where π‘ž2=∞.

3.2. Composition Operators

Theorem 3.7. The composition operator πΆπ‘”βˆΆπΏ(𝑝,π‘ž)→𝐿(𝑝,π‘ž) is bounded if and only if there is an absolute constant C such that πœ‡ξ€·π‘”βˆ’1ξ€Έ(𝐴)β‰€πΆπœ‡(𝐴),(3.14) for all πœ‡-measurable sets π΄βŠ†[0,2πœ‹] and for 1<π‘β‰€βˆž,1β‰€π‘žβ‰€βˆž. Moreover, ‖𝐢𝑔‖=‖𝑔‖1/𝑝.

Proof. We will prove this theorem for 𝐿(𝑝,1) and use the interpolation theorem to conclude for 𝐿(𝑝,π‘ž).
First assume that πΆπ‘”βˆΆπΏ(𝑝,1)→𝐿(𝑝,1) is bounded that is, there is an absolute constant 𝐢 such that ‖‖𝐢𝑔𝑓‖‖𝐿(𝑝,1)≀𝐢‖𝑓‖𝐿(𝑝,1).(3.15) Let 𝐴 be a πœ‡-measurable set in [0,2πœ‹] and let 𝑓=πœ’π΄. Then, (3.15) is equivalent to β€–β€–πΆπ‘”πœ’π΄β€–β€–πΏ(𝑝,1)β€–β€–πœ’β‰€πΆπ΄β€–β€–πΏ(𝑝,1)⟺1π‘ξ€œ02πœ‹ξ€·πΆπ‘”πœ’π΄ξ€Έβˆ—(𝑑)𝑑1/π‘βˆ’1β‰€πΆπ‘‘π‘‘π‘ξ€œ02πœ‹πœ’βˆ—π΄(𝑑)𝑑(1/𝑝)βˆ’1𝑑𝑑;(3.16) that is, 1π‘ξ€œ02πœ‹ξ€·πœ’π΄ξ€Έβˆ˜π‘”βˆ—(𝑑)𝑑1/π‘βˆ’1πΆπ‘‘π‘‘β‰€π‘ξ€œ02πœ‹πœ’[0,πœ‡(𝐴)](𝑑)𝑑(1/𝑝)βˆ’1𝑑𝑑.(3.17) Since (πœ’π΄βˆ˜π‘”)=πœ’π‘”βˆ’1(𝐴), then (πœ’π΄βˆ˜π‘”)βˆ—=πœ’[0,πœ‡(π‘”βˆ’1(𝐴))]. Therefore, the previous inequality gives 1π‘ξ€œπœ‡(π‘”βˆ’10(𝐴))𝑑1/π‘βˆ’1πΆπ‘‘π‘‘β‰€π‘ξ€œ0πœ‡(𝐴)𝑑1/π‘βˆ’1𝑑𝑑.(3.18) And hence, πœ‡ξ€·π‘”βˆ’1ξ€Έ(𝐴)β‰€πΆπ‘πœ‡(𝐴).(3.19)
On the other hand, assume that there is some constant 𝐢>0 such that πœ‡(π‘”βˆ’1(𝐴))β‰€πΆπœ‡(𝐴). Then, β€–β€–πΆπ‘”πœ’π΄β€–β€–πΏ(𝑝,1)=1π‘ξ€œ02πœ‹ξ€·πœ’π΄ξ€Έβˆ˜π‘”βˆ—(𝑑)𝑑1/π‘βˆ’1=1π‘‘π‘‘π‘ξ€œ02πœ‹πœ’[0,πœ‡(π‘”βˆ’1(𝐴))](𝑑)𝑑1/π‘βˆ’1=ξ€·πœ‡ξ€·π‘”π‘‘π‘‘βˆ’1(𝐴)ξ€Έξ€Έ1/𝑝≀𝐢1/𝑝(πœ‡(𝐴))1/𝑝.(3.20) Consequently, β€–β€–πΆπ‘”πœ’π΄β€–β€–πΏ(𝑝,1)≀𝐢1/𝑝(πœ‡(𝐴))1/𝑝.(3.21) As a consequence of Theorem 2.5 or the result by Weiss and Stein in [9], we have ‖‖𝐢𝑔𝑓‖‖𝐿(𝑝,1)≀𝐢1/𝑝‖𝑓‖𝐿(𝑝,1).(3.22)
To prove the second part of the theorem, note that from the above, we have ‖‖𝐢𝑔‖‖=sup‖𝑓‖𝐿(𝑝,1)≀1‖‖𝐢𝑔𝑓‖‖𝐿(𝑝,1)‖𝑓‖𝐿(𝑝,1)≀𝐢1/𝑝.(3.23) But inf{πΆβˆΆπœ‡(π‘”βˆ’1(𝐴))β‰€πΆπœ‡(𝐴)}=‖𝑔‖. Thus, ‖𝐢𝑔‖≀‖𝑔‖1/𝑝. To obtain the other inequality, let 𝑓=(1/[πœ‡(𝐴)]1/𝑝)πœ’π΄. This gives ‖𝑓‖𝐿(𝑝,1)=1 and ‖‖𝐢𝑔𝑓‖‖𝐿(𝑝,1)=ξƒ―πœ‡ξ€·π‘”βˆ’1ξ€Έ(𝐴)ξƒ°πœ‡(𝐴)1/𝑝.(3.24) Thus, ‖‖𝐢𝑔‖‖=sup‖𝑓‖𝐿(𝑝,1)≀1‖‖𝐢𝑔𝑓‖‖𝐿(𝑝,1)β‰₯supπœ‡(𝐴)β‰ 0ξ‚»πœ‡(π‘”βˆ’1(𝐴))ξ‚Όπœ‡(𝐴)1/𝑝=‖𝑔‖1/𝑝.(3.25) Now to show the result for 𝐿(𝑝,π‘ž), note that the operator 𝐢𝑔 is linear on 𝐿(𝑝,π‘ž) and that for 𝑝0 and 𝑝1 such that 𝑝0<𝑝<𝑝1, we have β€–πΆπ‘”πœ’π΄β€–πΏ(𝑝0,1)≀𝑀0(πœ‡(𝐴))1/𝑝0 and β€–πΆπ‘”πœ’π΄β€–πΏ(𝑝1,1)≀𝑀1(πœ‡(𝐴))1/𝑝1. Since 𝐿(𝑝𝑖,1)βŠ†πΏ(𝑝𝑖,∞),𝑖=0,1, then for some absolute constants 𝐢0 and 𝐢1 we have β€–β€–πΆπ‘”πœ’π΄β€–β€–πΏ(𝑝0,∞)≀𝐢0(πœ‡(𝐴))1/𝑝0,β€–β€–πΆπ‘”πœ’π΄β€–β€–πΏ(𝑝1,∞)≀𝐢0(πœ‡(𝐴))1/𝑝1.(3.26)
Hence, by the interpolation theorem we conclude that there is a constant 𝐢>0 such that ‖‖𝐢𝑔𝑓‖‖𝐿(𝑝,π‘ž)≀𝐢‖𝑓‖𝐿(𝑝,π‘ž)for𝑝0<𝑝<𝑝1,βˆ€π‘žandβˆ€π‘“βˆˆπΏ(𝑝,π‘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 πœ‡ 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 π‘Ž to achieve (3.14).(2)If instead πœ‡ 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 π΄βŠ†π‘‹, we have βˆ«πœ‡(𝐴)=𝐴|𝑑|βˆ’1𝑑𝑑. Hence (3.14) is achieved for a measurable function 𝑔 such that π‘”βˆ’1(𝐴)βŠ†π΄. The left multiplication by the reciprocal of an absolute constant π‘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 β€–π‘‡πœ’π΄β€–π‘‹β‰€π‘πœ‡(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 π‘“βˆˆπΏ(𝑝,1) if and only if βˆ‘π‘“(𝑑)=βˆžπ‘›=0π‘π‘›πœ’π΄π‘›(𝑑) with βˆ‘βˆžπ‘›=0|𝑐𝑛|πœ‡(𝐴𝑛)1/𝑝<∞. This β€œatomic decomposition of 𝐿(𝑝,1)” provides us with a technique to study operators on 𝐿(𝑝,π‘ž) and in particular 𝐿𝑝. In other words, to study operators on 𝐿(𝑝,π‘ž), all we need is to study the actions of the operator on characteristic functions which can then be lifted to 𝐿(𝑝,π‘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,π‘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 Ξ”2 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,π‘ž),1<𝑝,π‘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.


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