Boundedness of Fractional Integral Operators on Hardy- Amalgam Spaces

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 CalderónZygmund 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 Hardyamalgam 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. Section 2 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.


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][3][4][5][6][7][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][13][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][19][20]. By using the extrapolation theory, we do not need to develop the atomic decomposition of the Hardyamalgam 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. Section 2 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.

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.

Let I denote the class of open connected intervals in ℝ.
Let jIj be the Lebesgue measure of I ∈ I.
The following result present the dual space of ðL p , l q Þ.
The reader is referred to [7] for the proof of the above result. Particularly, we have the Hölder inequality for the amalgam space We now study the boundedness of the Hardy-Littlewood maximal operator on ðL p , l q Þ. For any locally integrable function f , the Hardy-Littlewood maximal operator is defined as 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].
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 L p .
We now recall the definition of the Muckenhoupt weight functions A p .
where p ′ = p/ðp − 1Þ. A locally integrable function ω : ℝ ⟶ ½0,∞Þ is said to be an A 1 weight if for some constants C > 0. The infimum of all such C is denoted by ½ω A 1 . We define A ∞ = ∪ p≥1 A p .
For any p ∈ ð0,∞Þ and u : ℝ ⟶ ½0,∞Þ, the weighted Lebesgue space L p u consists of all Lebesgue measurable functions f satisfying Let p ∈ ð1,∞Þ. It is well known that M is bounded on L p ðuÞ if and only if u ∈ A p .
Let F = fk·k α i ,β i g be any finite collection of semi-norms on S and For any f ∈ S ′, write where for any t > 0, write ψ t ðxÞ = t −1 ψðx/tÞ.
Definition 5. Let p, q ∈ ð0,∞Þ. The Hardy-amalgam space ð H p , l q Þ consists of all f ∈ S ′ satisfying We use the grand maximal function to define the Hardy-2 Journal of Function Spaces amalgam spaces while in [6], we use the Littlewood-Paley function. In view of the definition of ðL p , l q Þ and [6], Proposition 2, ðL p , l q Þ 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.
Let p ∈ ð0,∞Þ. For any weighted function ω : ℝ ⟶ ð0, ∞Þ, the weighted Hardy space H p ðωÞ consists of all f ∈ S ′ satisfying 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. Let 0 < p < 1/α and 1/q = ð1/pÞ − α. Then, ω ∈ A ∞ if and only if for some C > 0.
For the proof of the above theorem, see [17], Corollary 6.2 and Theorem 8.1.

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. Let 0 < α < 1 and 0 < p, q < 1/α. Suppose that We have a constant C > 0 such that for any f ∈ ðH p , l q Þ, Proof. Let β ∈ ð0, min ðp, qÞÞ. Define θ by As 0 < β < min ðp, qÞ < 1/α, θ is well defined.
Consequently, (26) gives Proposition 2 guarantees that Therefore, by taking the supremum over khk ðL ðr/θÞ′ ,l ðs/θÞ′ Þ ≤ 1 on both sides of (29), we have a constant C > 0 such that for any f ∈ ðH p , l q Þ 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][31][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, when p = 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].

Data Availability
No data were used to support this study.

Conflicts of Interest
The authors declare that they have no conflicts of interest.