# Boundedness of Fractional Integral Operators on Hardy-Amalgam Spaces

Authors: Ka Luen Cheung; Kwok-Pun Ho; Tat-Leung Yee
Journal: Journal of Function Spaces
 (2021)

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## Abstract

We establish the boundedness of the fractional integral operators on the Hardy-amalgam spaces.


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## Body

## 1. Introduction

In this paper, we establish the boundedness of the fractional integral operators on Hardy-amalgam spaces.The amalgam spaces were introduced by Wiener [1]. The amalgam spaces are important function spaces for the Fourier analysis and the mapping properties of operators [2–8]. The amalgam spaces also provide the foundation for the time-frequency analysis [9], especially, the introduction of the Wiener amalgam space and the modulation spaces.The study of the amalgam spaces has been further extended to the Hardy-amalgam space [10, 11] and the slice spaces [12–14]. The reader is referred to [6, 12] for the boundedness of the Riesz transform, the Calderón-Zygmund operators and the intrinsic square function on the Hardy-amalgam spaces.The above results on Hardy-amalgam spaces motivate us to investigate the mapping properties of the fractional integral operators on the Hardy-amalgam spaces.The studies of the mapping properties of the fractional integral operators on Hardy type spaces begun from the classical Hardy spaces introduced by Stein and Weiss [15]. The mapping properties of the fractional type integrals on Hardy spaces and the weighted norm inequalities of the fractional integral operators on Hardy spaces were established in [16, 17], respectively.In this paper, we use the extrapolation theory for amalgam spaces developed in [6] to obtain our main result. The extrapolation theory was introduced by Rubio de Francia in [18–20]. By using the extrapolation theory, we do not need to develop the atomic decomposition of the Hardy-amalgam spaces. In addition, we use the idea from [21] to refine the extrapolation theory so that we do not need to use the density argument in [6].This paper is organized as follows. Section2 presents the definitions of the amalgam spaces and the Hardy-amalgam spaces. It also includes the mapping properties of the fractional integral operators on weighted Hardy spaces which is an essential component for applying the extrapolation theory. The mapping properties of the fractional integral operators on the Hardy-amalgam spaces are established in Section 3.


## 2. Preliminaries and Definitions

In this section, we present the definitions and the preliminary results used to obtain the main result. In particular, this section contains the duality of the amalgam spaces, the boundedness of the Hardy-Littlewood maximal operator on the amalgam spaces, and the mapping properties of the fractional integral operators on the weighted Hardy spaces.LetI denote the class of open connected intervals in ℝ. Let I be the Lebesgue measure of I∈I.Let0<α<1. The fractional integral operator on ℝ is defined by
(1)Iαfx=∫ℝfyx−y1−αdy.If0<α<1, 1<p<1/α, and 1/p=1/q+α, then Iα:Lp⟶Lq is bounded.Definition 1.
Let0<p,q<∞. The amalgam space Lp,lq consists of all Lebesgue measurable functions fsatisfying
(2)fLp,lq=∑n∈ℤ∫nn+1fxpdxq/p1/q<∞.Obviously, when1≤p=q<∞, then Lp,lq=Lp. Moreover, when 1≤p,q<∞, Lp,lq is a Banach space ([7], Theorem 1). It is easy to see that if p∈0,∞ and 0<q1≤q2<∞, we have
(3)Lp,lq1⊂Lp,lq2.In addition, when0<p1≤p2<∞ and q∈0,∞, we have
(4)Lp2,lq⊂Lp1,lq.The amalgam spaces had been extended to the slice spaces introduced and studied in [13, 22] and Orlicz slice spaces in [14]. For the mapping properties of operators on slice spaces and Orlicz-slice spaces, the reader is referred to [14, 23].The following result present the dual space ofLp,lq.Proposition 2.
Let1<p,q<∞. The dual space of Lp,lq is Lp′,lq′.The reader is referred to [7] for the proof of the above result. Particularly, we have the Hölder inequality for the amalgam space
(5)∫ℝfxgxdx≤fLp,lqgLp′,lq′.We now study the boundedness of the Hardy-Littlewood maximal operator onLp,lq. For any locally integrable function f, the Hardy-Littlewood maximal operator is defined as
(6)Mfx=supI∋x1I∫Ifydy,where the supremum is taken over all interval I∈I containing x.We now recall the boundedness of the Hardy-Littlewood maximal operator on the amalgam spaces from [2].Theorem 3.
Let1<p,q<∞. The Hardy-Littlewood maximal operator M is bounded on Lp,lq.The reader is referred to [2], Theorems 4.2 and 4.5 for the proof of the above result when p≠q. When p=q, it follows from the well-known result that M is bounded on the Lebesgue space Lp.We now recall the definition of the Muckenhoupt weight functionsAp.Definition 4.
For1<p<∞, a locally integrable function ω:ℝ⟶0,∞ is said to be an Ap weight if
(7)ωAp=supI∈I1I∫Iωxdx1I∫Iωx−p′/pdxp/p′<∞,where p′=p/p−1. A locally integrable function ω:ℝ⟶0,∞ is said to be an A1 weight if
(8)1I∫Iωydy≤Cωx,a.e.x∈I,for some constants C>0. The infimum of all such C is denoted by ωA1. We define A∞=∪p≥1Ap.For anyp∈0,∞ and u:ℝ⟶0,∞, the weighted Lebesgue space Lup consists of all Lebesgue measurable functions f satisfying
(9)fLup=∫ℝfxpuxdx1/p<∞.Letp∈1,∞. It is well known that M is bounded on Lpu if and only if u∈Ap.LetF=·αi,βi be any finite collection of semi-norms on S and
(10)SF=ψ∈S:ψαi,βi≤1,forall·αi,βi∈F.For anyf∈S′, write
(11)MFfx=supψ∈sFsupt>0f∗ψtx,where for any t>0, write ψtx=t−1ψx/t.Definition 5.
Letp,q∈0,∞. The Hardy-amalgam space Hp,lq consists of all f∈S′ satisfying
(12)fHp,lq=MFfLp,lq<∞.We use the grand maximal function to define the Hardy-amalgam spaces while in [6], we use the Littlewood-Paley function. In view of the definition of Lp,lq and [6], Proposition 2, Lp,lq is a ball quasi-Banach function space. Thus, [24], Section 5 shows that they are equivalent definition for the Hardy-amalgam spaces. For simplicity, we refer the reader to [25, 26] for the definitions of ball quasi-Banach function spaces and the Hardy spaces built on the ball quasi-Banach function spaces. In particular, the Orlicz-slice Hardy spaces were introduced in [14]. The intrinsic square function characterization of the Orlicz slice Hardy spaces was established in [27]. The mapping properties of the maximal Bochner-Riesz means, the parametric Marcinkiewicz integrals, and the multiplier operators on the Orlicz-slice Hardy spaces were given in [23].For the atomic decomposition and the dual space of the Hardy-amalgam spaces, see [10, 11].We need to use the weighted Hardy spaces to establish our main result. Thus, we also recall the definition of the weighted Hardy spaces.Letp∈0,∞. For any weighted function ω:ℝ⟶0,∞, the weighted Hardy space Hpω consists of all f∈S′ satisfying
(13)fHpω=MFfLpω<∞.For the studies of the weighted Hardy spaces, the reader is referred to [28, 29].We now present the boundedness of the fractional integral operators on weighted Hardy spaces.Theorem 6.
Let0<p<1/α and 1/q=1/p−α. Then, ω∈A∞ if and only if
(14)IαfHqω≤CfHpωp/q,for some C>0.For the proof of the above theorem, see [17], Corollary 6.2 and Theorem 8.1.


## 3. Main Result

The main result of this paper, the boundedness of the fractional integral operators on the Hardy-amalgam spaces is established in this section.Theorem 7.
Let0<α<1 and 0<p,q<1/α. Suppose that
(15)1p−1r=1q−1s=α.
We have a constantC>0 such that for any f∈Hp,lq,
(16)IαfHr,ls≤CfHp,lq.Proof.
Letβ∈0,minp,q. Define θ by
(17)1β−1θ=α.
As0<β<minp,q<1/α, θ is well defined.
In view of (15), we have 1/β−1/θ=α=1/p−1/r. Consequently,
(18)1r−1θ=1p−1β<0.
Sinceβ<p, we have r>θ. Similarly, we have s>θ. Thus, r/θ′ and s/θ′ are well defined.
For any nonnegative functionh∈Lr/θ′,ls/θ′, define
(19)Rhx=∑k=0∞Mkhx2kMLr/θ′,ls/θ′k.
The Rubio de Francia operatorR possesses the following properties:
(20)hx≤Rhx,(21)RhLr/θ′,ls/θ′≤2hLr/θ′,ls/θ′,(22)RhA1≤2MLr/θ′,ls/θ′.
The above properties are valid becauser/θ′,s/θ′>1, and Theorem 3 assures that M is bounded on Lr/θ′,ls/θ′.
The identity in (18) shows that
(23)βθp/β′=βθpp−β=rr−θ=r/θ′.
Similarly, the identity1/s−1/θ=1/q−1/β shows that
(24)βθq/β′=βθqq−β=ss−θ=sθ′.
Therefore,(25)gLr/θ′,ls/θ′β/θ=gβ/θLp/β′,lq/β′.
For anyf∈Hp,lq and h∈Lr/θ′,ls/θ′, (5), (20), and (25) assert that
(26)∫ℝMFfxβRhxβ/θdx≤MFfβLp/β,lq/βRhβ/θLp/β′,lq/β′=MFfLp,lqβRhLr/θ′,ls/θ′β/θ=fHp,lqβhLr/θ′,ls/θ′β/θ.
Consequently, for anyh∈Lr/θ′,ls/θ′, we find that
(27)Hp,lq↪HβRhβ/θ.
Moreover, (22) yields that Rh∈A1. By applying Theorem 6 with ω=Rh, (20) and (27) yield
(28)∫ℝMFIαfxθ∣hx∣dx≤∫ℝMFIαfxθRhxdx=IαfHθRhθ≤CfHβRhβ/θθ=C∫ℝMFfxβRhxβ/θdxθ/β,for some C>0.
Consequently, (26) gives
(29)∫ℝMFIαfxθhxdx≤CfHp,lqθhLr/θ′,ls/θ′.
Proposition2 guarantees that
(30)sup∫ℝMFIαfxθhxdx:hLr/θ′,ls/θ′≤1=MFIαfθLr/θ,ls/θ=MFIαfLr,lsθ=IαfHr,lsθ.
Therefore, by taking the supremum overhLr/θ′,ls/θ′≤1 on both sides of (29), we have a constant C>0 such that for any f∈Hp,lq(31)IαfHr,ls≤CfHp,lq.By using the idea from [21], we can get rid of the density argument used in [6]. Moreover, the above method have been applied in [23, 30–32] to study the mapping properties of the Calderón-Zygmund operators and some sublinear operators on the Hardy local Morrey spaces with variable exponents, the Orlicz-slice Hardy spaces, the Herz-Hardy spaces with variable exponents, and the Hardy-Morrey spaces with variable exponents, respectively.Particularly, whenp=q and r=s, the above result becomes the well-known result of the boundedness of the fractional integral operators on Hardy spaces established in [15].



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