# Atomic Decomposition of Weighted Lorentz Spaces and Operators

Authors: Eddy Kwessi; Geraldo de Souza; Fidele Ngwane; Asheber Abebe
Journal: Journal of Function Spaces
 (2014)

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## Abstract

We obtain an atomic decomposition of weighted Lorentz spaces for a
class of weights satisfying the Δ2 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.


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## Body

## 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 Λp(w)={f:RN→R:∥f∥Λp(w)=(∫0∞(f*(x))pw(x)dx)1/p<∞}, where f* is the decreasing rearrangement of f and w is a weight function. He proved that, for p≥1, ∥·∥Λp(w) is a norm if and only if the weight w is decreasing. Carro and Soria in [3] proved that ∥·∥Λp(w) is a quasi-norm in general provided that W(t)=∫0s‍w(s)ds satisfies the Δ2 condition, that is, W(2t)≤CW(t), for some constant C>1. In this paper, we study Λ1(w) that we denote by LΦ, with Φ(t)=tw(t), in the sense that f:[0,2π]→R belongs to LΦ if and only if ∥f∥LΦ=∫02π‍f*(t)(Φ(t)/t)dt<∞. Our interest in this special space stems from the fact that, as demonstrated in [4] with w(t)=tp/q, this space has some interesting properties that allow an easy study of operators on L(p,q) 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 L2[0,2π] can be thought of as an atomic decomposition of the space L2[0,2π]. Coifman in [5] gave the unifying definition of an atom and showed that Hardy’s spaces H1(D), the spaces of holomorphic functions on the unit disc D, 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 L(p,1) 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.
DefineCΦ as the space of weights Φ:[0,∞)→[0,∞) so that,(1)
Φ(0)=0,(2)
Φ is increasing,(3)
Φ(t)/t is decreasing,(4)
there is a positive constantC such that, ∫0x(Φ(t)/t)≤CΦ(x), (Dini’s Condition),(5)
Φ satisfies the Δ2 condition; that is, there is a an constant K>1 such that Φ(2x)≤KΦ(x).Note that the spaceCΦ is nonempty since Φ(t)=tα, for α∈(0,1), belongs to CΦ. Hereafter, μ will denote a nonatomic measure defined on [0,2π].Definition 2.
One defines the weighted Lorentz spaceLΦ as
(1)LΦ={f:[0,2π]⟶R;∥f∥LΦ=∫02π‍f*(t)Φ(t)tdt<∞},
where f* is the decreasing rearrangement of f defined as f*(t)=inf⁡⁡{y>0:μ({x:|f(x)|>y})≤t}, and μ is a nonatomic measure defined on [0,2π].Remark 3.
ForΦ(t)=t1/p, LΦ is identical to the classical Lorentz space L(p,1).Definition 4.
One will also consider the following space:(2)AΦ(μ)={f:[0,2π]⟶R,f(t)=∑n=1∞cnχAn;∑n=1∞|cn|Φ(μ(An))<∞},
where the An’s are μ-measurable sets in [0,2π] and χA represents the characteristic function on the set A.We will show in Theorem14 that this space is an atomic decomposition of the space LΦ.Put∥f∥AΦ(μ)=inf⁡∑n=1∞|cn|Φ(μ(An)) where the infimum is taken over all possible representations of f. The next result is proved in the appendix.Proposition 5.
If one endowsAΦ(μ) with ∥·∥AΦ(μ), then(1)
∥·∥AΦ(μ)is a norm,(2)
(AΦ(μ),∥·∥AΦ(μ))is a Banach space.Definition 6.
For1≤r≤∞, define for a measurable function g:[0,2π]→R the quantity
(3)∥g∥MΦ,r={sup⁡x>0(1Φ(x)∫0x‍(g*(t)Φ(t))rdtt)1/rif1≤r<∞,sup⁡x>0g*(x)Φ(x)ifr=∞.
The spaceMΦ,r is the set of measurable functions g for which ∥g∥MΦ,r<∞. This space generalizes the space Mrp introduced in [4]. In the next theorem and remark, we give further properties of theses spaces in the weighted case.Theorem 7.
ForΦ∈CΦ, one has that(1)
if1≤r≤∞, then ∥·∥MΦ,r is a quasi-norm on MΦ,r,(2)
L∞≅MΦ,1 with ∥g∥∞≅∥g∥MΦ,1.The remark below is stated only for completeness and the proof can be found in [4].Remark 8.
ForΦ∈CΦ and r>1, we have MΦ,r≅MΦ1/r′,∞, where 1/r+1/r′=1.Definition 9.
ForΦ∈CΦ, define Ψ(t)=t/Φ(t), t>0. For a measure μ defined on [0,2π], consider the following spaces
(4)ΣΦ1(μ)={g:[0,2π]⟶Rmeasurable:1Φ(μ(A))∑ddd∥g∥ΣΦ1(μ)=sup⁡μ(A)≠0⁡1Φ(μ(A))|∫Ag(t)dμ(t)|<∞},LΨ∞={g:[0,2π]⟶Rmeasurable:sup⁡t>0∥g∥LΨ∞=sup⁡t>0⁡Ψ(t)g*(t)<∞}.The first space was basically introduced by Lorentz in [2] for Φ(t)=t1/p. We will prove in Theorem 10 that these spaces are norm-equivalent.Theorem 10.
LetΦ∈CΦ and Ψ(t)=t/Φ(t), t>0. For a measurable function g:[0,2π]→R, one has
(5)∥g∥ΣΦ1(μ)≅∥g∥LΨ∞.Theorem 11 (Hölder’s type inequalities).
LetΦ∈CΦ and Ψ(t)=t/Φ(t), t>0.(1)
Forf∈LΦ and g∈LΨ∞, one has
(6)|∫02πf(t)g(t)dμ(t)|≤∥f∥LΦ∥g∥LΨ∞.(2)
Forf∈AΦ(μ) and g∈ΣΦ1(μ), one has
(7)|∫02πf(t)g(t)dμ(t)|≤∥f∥AΦ(μ)∥g∥ΣΦ1(μ).


## 3. Atomic Decomposition

We start with this important result on the dual of the spacesLΦ and AΦ(μ).Theorem 12.
LetΦ∈CΦ and Ψ(t)=t/Φ(t), t>0. Then one has the following.(1)
(LΦ)*≅LΨ∞; that is, ϕ∈(LΦ)* if and only if there is a unique g∈LΨ∞ so that for all f∈LΦ(8)ϕ(f)=∫02πf(t)g(t)dμ(t)with∥ϕ∥≅∥g∥LΨ∞.(2)
Likewise, one has(AΦ(μ))*≅ΣΦ1(μ); that is, ψ∈(AΦ(μ))* if and only if there is a unique g∈ΣΦ1(μ) so that for all f∈AΦ(μ)(9)ψ(f)=∫02πf(t)g(t)dμ,∥ψ∥=∥g∥ΣΦ1(μ).Proof.
Letg∈LΨ∞. Define ϕ(f)=∫02π‍f(t)g(t)dμ(t). By Theorem 11, we have
(10)|ϕ(f)|≤∥f∥LΦ∥g∥LΨ∞.
Thus, using the linearity of the integral, we conclude thatϕ∈(LΦ)*. On the other hand, let ϕ∈(LΦ)*. For a μ-measurable set A⊆[0,2π], define λ(A)=ϕ(χA). Then there is a constant M>0 such that
(11)|λ(A)|=|ϕ(χA)|≤M∥χA∥LΦ.
Since(12)∥χA∥LΦ=∫02πχ[0,μ(A)](t)Φ(t)tdt=∫0μ(A)Φ(t)t,
then using Dini’s condition (4) in Definition 1, we have
(13)∥χA∥LΦ≤CΦ(μ(A)).
It follows from (11) and (13) that |λ(A)|<MCΦ(μ(A)) and condition 1 in Definition 1 yield λ≪μ. By the Radon-Nikodym Theorem and the definition of functions in LΦ, there is an integrable function g on [0,2π] such that, for all f∈LΦ,
(14)ϕ(f)=∫02πf(t)g(t)dμ(t).
To prove thatg∈LΨ∞, observe that
(15)|∫A‍g(t)dμ(t)|=|ϕ(χA)|≤M∥χA∥LΦ<MCΦ(μ(A)).
Thus taking the supremum overμ-measurable sets A such that μ(A)≠0, we have
(16)sup⁡μ(A)≠01Φ(μ(A))|∫A‍g(t)dμ(t)|≤MC,thatis,g∈∑Φ1(μ).
The proof is complete using the equivalence in Theorem10. 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.
LetX and Y be two vector normed spaces and let T∈L(X,Y), the space of bounded linear operators from X onto Y. Let T* be the adjoint operator of T defined by T*f=f∘T for all f∈Y*, the dual space of Y. Then(a)
T*∈L(Y*,X*) and ∥T*∥=∥T∥;(b)
T* is injective if and only if the range of T is dense in Y.The next result is the most important of the present paper and gives an equivalent representation of functions inLΦ as “linear” combinations of simple functions.Theorem 14 (atomic decomposition ofLΦ).
ForΦ∈CΦ, one has
(17)LΦ≅AΦ(μ),with∥f∥LΦ≅∥f∥AΦ(μ).Proof.
Let us show first thatAΦ(μ)⊆LΦ. Take f(t)=χA(t). Then using Dini’s condition (4) in Definition 1, we have
(18)∥χA∥LΦ=∫02πχA*(t)Φ(t)tdt=∫0μ(A)Φ(t)tdt≤CΦ(μ(A)).
Thus iff(t)=∑n=1∞cnχAn(t) for cn∈R, then
(19)∥f∥LΦ≤∑n=1∞|cn|∥χAn∥LΦ.
And (18) implies
(20)∥f∥LΦ≤C∑n=1∞|cn|Φ(μ(A)).
Taking the infimum over all representations off, we have
(21)∥f∥LΦ≤C∥f∥AΦ(μ),thatis,AΦ(μ)⊆LΦ.
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:A1:
AΦ(μ)⊆LΦ and ∥f∥LΦ≤M∥f∥AΦ(μ), by inequality (21).A2:
AΦ(μ) is dense in LΦ, see [11].A3:
(AΦ(μ))*=(LΦ)* since by Theorems 10 and 12  (LΦ)*≅LΨ∞≅ΣΦ1(μ)≅(AΦ(μ))*.
Using Theorem13, we conclude from A1 that the inclusion map i:AΦ(μ)→LΦ is a bounded linear map and that ∥i*∥=∥i∥ where i* is the duality map i*:(LΦ)*→(AΦ(μ))*. From A2, it follows that duality map i* is injective and from A3 that i* is the identity map. Therefore, we have that i is an isomorphism and thus ∥f∥LΦ≅∥f∥AΦ(μ).Remark 15.
The spaceAΦ(μ) is called an atomic decomposition of LΦ in the sense that each function of LΦ coincides with a function of AΦ(μ) and thus can be written as a “linear” combination of atoms, where the atoms are the “simple” functions χAn.


## 4. Operators on Weighted Lorentz Spaces

In this section, we study two types of operators: the multiplication and composition operators of weighted Lorentz spacesLΦ.Theorem 16 (multiplication operator).
ForΦ∈CΦ and for f∈LΦ, define the multiplication operator Tg as Tgf=f·g. Then Tg:LΦ→LΦ is bounded if and only if g∈MΦ,1. Moreover, ∥Tg∥≅∥g∥MΦ,1.Proof.
IfTg is bounded, then there is an absolute constant M such that
(22)∥Tgf∥LΦ≤M∥f∥LΦ,∀f∈LΦ.
Takef=χA for A⊆[0,2π]. Then from (22), it follows that
(23)∫02π(χA(t)g(t))*Φ(t)tdt≤M∫02πχA*(t)Φ(t)tdt,
which after simplification is equivalent to
(24)∫0μ(A)g*(t)Φ(t)tdt≤M∫0μ(A)Φ(t)tdt.
Using Dini’s condition (4) in Definition1, we have
(25)∫0μ(A)g*(t)Φ(t)tdt≤MCΦ(μ(A)),
and hence
(26)1Φ(μ(A))∫0μ(A)g*(t)Φ(t)tdt≤K.
Thus,(27)sup⁡x>0,μ(A)=x1Φ(x)∫0xg*(t)Φ(t)tdt≤K.
This completes the proof thatg∈MΦ,1.
On the other hand, ifg∈MΦ,1, then for A⊆[0,2π], we have
(28)∥TgχA∥LΦ=∫02π(χA·g)*(t)Φ(t)tdt=∫0μ(A)g*(t)Φ(t)tdt=Φ(μ(A))(1Φ(μ(A))∫0μ(A)g*(t)Φ(t)tdt).
Therefore,(29)∥TgχA∥LΦ≤∥g∥MΦ,1·Φ(μ(A)).
Since from Theorem14, LΦ≅AΦ(μ); then for f∈LΦ, we have f(t)=∑n=1∞cnΦ(μ(An)), for some cn’s ∈R. And so,
(30)∥Tgf∥LΦ≤∑n=1∞|cn|∥TgχAn∥LΦ≤∥g∥MΦ,1∑n=1∞|cn|Φ(μ(An)).
Taking the infimum over all representations off and using the equivalence between LΦ and AΦ(μ), we have
(31)∥Tgf∥LΦ≤∥g∥MΦ,1∥f∥LΦ.
To prove the second statement of the theorem, observe that (31) implies that
(32)∥Tg∥≤∥g∥MΦ,1.
If we takef(t)=(1/Φ(x))χ[0,x](t) for some x>0, then ∥f∥LΦ=1 and since g*(t) and Φ(t)/t are decreasing, we have
(33)∥Tgf∥LΦ=1Φ(x)∫02π(χ[0,x])*(t)Φ(t)tdt=1Φ(x)∫0xg*(t)Φ(t)tdt≥1Φ(x)g*(x)∫0xΦ(t)tdt≥g*(x).
Consequently,∥Tg∥LΦ≥g*(x) and so sup⁡∥f∥LΦ=1∥Tgf∥LΦ≥g*(x). Thus
(34)∥Tg∥≥g*(x),∀x>0.
Taking the limit asx→0, we have
(35)∥Tg∥≥∥g∥∞.
From the inequality(B.4) in the proof of Theorem 7 (Appendix B), it follows that
(36)∥Tg∥≥C∥g∥MΦ,1.
The result then follows by combining (32) and (36).Definition 17.
Leth be a nonsingular measurable transformation on [0,2π], and let Φ∈CΦ. We define XΦ(μ) as
(37)XΦ(μ)={h:[0,2π]⟶[0,2π]:(μ(h-1(A)))Φ(μ(h-1(A)))≤MΦ(μ(A))},
where A is a μ-measurable set in [0,2π]. M is an absolute real constant and h-1 is the inverse image of the μ-measurable subset A of [0,2π]. Put ∥h∥XΦ(μ)=sup⁡μ(A)≠0(Φ(μ(h-1(A)))/Φ(μ(A))).Theorem 18 (composition operator).
ForΦ∈CΦ and for f∈LΦ, define the composition operator Ch as Chf=f∘h. Then Ch:LΦ→LΦ is bounded if and only if h∈XΦ(μ). Moreover, ∥Ch∥≅∥h∥XΦ(μ).Proof.
The technique of this proof mirrors that of Theorem16. For, assume that the operator Ch is bounded; that is, there is some absolute constant M such that
(38)∥Chf∥LΦ≤M∥f∥LΦ.
Takingf=χA for some μ-measurable set A⊆[0,2π], the inequality (38) implies
(39)∫02π(χA∘h)*(t)Φ(t)tdt≤M∫02πχ[0,μ(A)](t)Φ(t)tdt.
Since(χA∘h)(t)=χh-1(A)(t), (39) entails
(40)∫0μ(h-1(A))Φ(t)tdt≤M∫0μ(A)Φ(t)tdt.
Using the fact thatΦ(t)/t is decreasing and Dini’s condition (4) in Definition 1, respectively, on the LHS and the RHS of (40), we have
(41)Φ(μ(h-1(A)))≤MΦ(μ(A)).
This proves thath∈XΦ(μ).
On the other hand, suppose thath∈XΦ(μ). Then for a μ-measurable subset A of [0,2π] we have
(42)∥ChχA∥LΦ=∫0μ(h-1(A))‍Φ(t)tdt.
Using Dini’s condition (4) and the fact thath∈XΦ(μ), there is an absolute constant M such that
(43)∥ChχA∥LΦ≤CΦ(μ(h-1(A)))≤CMΦ(μ(A)).
Now letf∈LΦ. Then f(t)=∑n=1∞cnχAn(t) since LΦ≅AΦ(μ). Then, using (43), it follows that
(44)∥Chf∥LΦ≤MC∑n=1∞|cn|Φ(μ(An)).
Taking the infimum over all representations off and using the equivalence LΦ≅AΦ(μ) we have
(45)∥Chf∥LΦ≤∥f∥LΦ,
showing that the operator Ch is bounded on LΦ.
Note that, from (45), we get ∥Ch∥≤MC. Without loss of generality, consider the constant M to be such that M=inf⁡⁡{K>0:K≥Φ(μ(h-1(A)))/Φ(μ(A))}. Then
(46)∥Ch∥≤CM≤C(sup⁡μ(A)≠0Φ(μ(h-1(A)))Φ(μ(A)))=C∥h∥XΦ(μ).
Moreover, if we takef(t)=χA(t)/Φ(μ(A)) for a μ-measurable subset A in [0,2π], we have ∥f∥LΦ=1 and using the fact that Φ(t)/t is decreasing
(47)∥Chf∥LΦ=1Φ(μ(A))∫0μ(g-1(A))Φ(t)tdt≥Φ(μ(h-1(A)))Φ(μ(A)).
Hence(48)∥Ch∥=sup⁡∥f∥LΦ=1∥Chf∥LΦ≥Φ(μ(h-1(A)))Φ(μ(A)).
Taking the supremum overA⊆[0,2π] such that μ(A)≠0, we have that
(49)∥Ch∥≥∥h∥XΦ(μ).
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 spacesLΦ(μ) 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 LΦ.


## 5. Discussion

The special atoms spacesAΦ(μ) originally introduced by de Souza in [12] for Φ(t)=t1/p 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 f defined on [0,2π] such that
(50)∫02π∫02π|f(x)-f(y)||x-y|2-1/pdxdy<∞,p>1.Another interesting use of the special atoms space is the real characterization of some spaces of analytic functionsF in the unit disc such that
(51)∫01∫02π|F′(reiθ)|(1-r)(1/p)-1dθdt<∞,p>1,
where F′ represents the derivative of F (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 L(p,1), p>1 and the weak-Lp spaces also known as L(p,∞), p>1. Indeed in [8], we show that f∈L(p,1) for p>1 if and only if
(52)f(t)=∑n=1∞cnχAn(t),
where ∑n=1∞|cn|μ(A)1/p<∞, the An’s are μ-measurable sets in [0,2π]. It was also shown in [8] that (52) is equivalent to
(53)f(t)=∑n=1∞cn[χAn(t)-χBn(t)],
where ∑n=1∞|cn|μ(An∪Bn)1/p<∞ and An∩Bn=∅ for μ-measurable set An, Bn in [0,2π].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 T:L(p,1)→B is bounded, where B is a Banach space closed under absolute value and satisfying ∥f∥B=∥|f|∥B if ∥TχA∥B≤Cμ(A)1/p, for an absolute constant C.Another interesting observation is that the dual ofL(p,1) can be identified as the set of measurable functions g:[0,2π]→R such that either of the following is satisfied, for μ-measurable subsets A,B of [0,2π],
(54)sup⁡μ(A)≠01μ(A)1/p|∫Ag(t)dμ(t)|<∞,(55)sup⁡μ(X)≠0X=A∪BA∩B=∅μ(A)=μ(B)1μ(X)1/p|∫Ag(t)dμ(t)-∫Bg(t)dμ(t)|.In fact, (54) and (55) provide a natural generalization of Lipschitz spaces. Indeed in (54), letting g(t)=f′(t) for a differentiable function f on [0,2π], A=[x,x+h], and μ be the Lebesgue measure yields
(56)sup⁡h>0|f(x+h)-f(x)|h1/p<∞.Also in (55), letting g(t)=f′(t), A=[x-h,x], B=[x,x+h], and μ be the Lebesgue measure yields
(57)sup⁡h>0|f(x+h)+f(x-h)-2f(x)|(2h)1/p<∞.In [4], Kwessi et al. use this new representation of L(p,1) to study operators such as the multiplication and composition operators on L(p,q) via interpolation. The key part is to show that the study of the boundedness of such operators on L(p,q) and in particular on L(p,p)=Lp 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 LΦ.



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