Atomic Decomposition of Weighted Lorentz Spaces and Operators

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 Holder’s-type and duality-Riesz type inequalities on these weighted Lorentz spaces.


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 Λ  () = { : R  → R : ‖‖ Λ  () = (∫ ∞ 0 ( * ()) ()) 1/ < ∞}, where  * is the decreasing rearrangement of  and  is a weight function.He proved that, for  ≥ 1, ‖ ⋅ ‖ Λ  () is a norm if and only if the weight  is decreasing.Carro and Soria in [3] proved that ‖ ⋅ ‖ Λ  () is a quasi-norm in general provided that () = ∫  0 () satisfies the Δ 2 condition, that is, (2) ≤ (), for some constant  > 1.In this paper, we study Λ 1 () that we denote by  Φ , with Φ() = (), in the sense that  : [0, 2] → R belongs to  Φ if and only if ‖‖  Φ = ∫ 2 0  * ()(Φ()/) < ∞.Our interest in this special space stems from the fact that, as demonstrated in [4] with () =  / , this space has some interesting properties that allow an easy study of operators on (, ) 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  2 [0, 2] can be thought of as an atomic decomposition of the space  2 [0, 2].Coifman in [5] gave the unifying definition of an atom and showed that Hardy's spaces  1 (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 (, 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.

Preliminaries
We begin with some preliminary definitions and results (proofs can be found in the appendices) that will be helpful throughout the paper.
Definition 4. One will also consider the following space: where the   's are -measurable sets in [0, 2] and   represents the characteristic function on the set .
We will show in Theorem 14 that this space is an atomic decomposition of the space  Φ . Put where the infimum is taken over all possible representations of .The next result is proved in the appendix.

Proposition 5. If one endows 𝐴
The space  Φ, is the set of measurable functions  for which ‖‖  Φ, < ∞.This space generalizes the space    introduced in [4].In the next theorem and remark, we give further properties of theses spaces in the weighted case.
Theorem 7.For Φ ∈  Φ , one has that The remark below is stated only for completeness and the proof can be found in [4].

Atomic Decomposition
We start with this important result on the dual of the spaces  Φ and  Φ ().
Theorem 12. Let Φ ∈  Φ and Ψ() = /Φ(),  > 0. Then one has the following. ( Thus, using the linearity of the integral, we conclude that  ∈ ( Φ ) * .On the other hand, let  ∈ ( Φ ) * .For a measurable set  ⊆ [0, 2], define () = (  ).Then there is a constant  > 0 such that Since then using Dini's condition (4) in Definition 1, we have It follows from ( 11) and ( 13) that |()| < Φ((A)) and condition 1 in Definition 1 yield  ≪ .By the Radon-Nikodym Theorem and the definition of functions in  Φ , there is an integrable function  on [0, 2] such that, for all  ∈  Φ , To prove that  ∈  ∞ Ψ , observe that Thus taking the supremum over -measurable sets  such that () ̸ = 0, we have sup The proof is complete using the equivalence in Theorem 10.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.Let  and  be two vector normed spaces and let  ∈ (, ), the space of bounded linear operators from  onto .Let  * be the adjoint operator of  defined by  *  =  ∘  for all  ∈  * , the dual space of .Then The next result is the most important of the present paper and gives an equivalent representation of functions in  Φ as "linear" combinations of simple functions.
Then using Dini's condition (4) in Definition 1, we have And (18) implies Taking the infimum over all representations of , we have 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: A 2 :  Φ () is dense in  Φ , see [11].
Remark 15.The space  Φ () is called an atomic decomposition of  Φ in the sense that each function of  Φ coincides with a function of  Φ () and thus can be written as a "linear" combination of atoms, where the atoms are the "simple" functions    .

Operators on Weighted Lorentz Spaces
In this section, we study two types of operators: the multiplication and composition operators of weighted Lorentz spaces  Φ .
which after simplification is equivalent to Using Dini's condition (4) in Definition 1, we have and hence Thus, sup This completes the proof that  ∈  Φ,1 .
Remark 19.The previous result in part shows that boundedness of operators other than the aforementioned ones on weighted Lorentz spaces  Φ () 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  Φ .

Discussion
The special atoms spaces  Φ () originally introduced by de Souza in [12] for Φ() =  1/ 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  defined on [0, 2] such that Another interesting use of the special atoms space is the real characterization of some spaces of analytic functions  in the unit disc such that where   represents the derivative of  (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 (, 1),  > 1 and the weak-  spaces also known as (, ∞),  > 1. Indeed in [8], we show that  ∈ (, 1) for  > 1 if and only if where It was also shown in [8] that ( 52) is equivalent to where 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  : (,1) →  is bounded, where  is a Banach space closed under absolute value and satisfying ‖‖  = ‖||‖  if ‖  ‖  ≤ () 1/ , for an absolute constant .
Another interesting observation is that the dual of (, 1) can be identified as the set of measurable functions  : [0, 2] → R such that either of the following is satisfied, for -measurable subsets ,  of [0, 2], sup In fact, (54) and (55) provide a natural generalization of Lipschitz spaces.Indeed in (54), letting () =   () for a differentiable function  on [0, 2],  = [,  + ℎ], and  be the Lebesgue measure yields In [4], Kwessi et al. use this new representation of (, 1) to study operators such as the multiplication and composition operators on (, ) via interpolation.The key part is to show that the study of the boundedness of such operators on (, ) and in particular on (, ) =   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  Φ .
Proof.The technique of this proof mirrors that of Theorem 16.For, assume that the operator  ℎ is bounded; that is, there is some absolute constant  such that      ℎ      Φ ≤           Φ .