Abstract

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

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. 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.

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.

Let denote the class of open connected intervals in . Let be the Lebesgue measure of .

Let . The fractional integral operator on is defined by

If , , and , then is bounded.

Definition 1. Let . The amalgam space consists of all Lebesgue measurable functions satisfying

Obviously, when , then . Moreover, when , is a Banach space ([7], Theorem 1). It is easy to see that if and , we have

In addition, when and , we have

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 of .

Proposition 2. Let . The dual space of is .

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 . For any locally integrable function , the Hardy-Littlewood maximal operator is defined as where the supremum is taken over all interval containing .

We now recall the boundedness of the Hardy-Littlewood maximal operator on the amalgam spaces from [2].

Theorem 3. Let . The Hardy-Littlewood maximal operator is bounded on .

The reader is referred to [2], Theorems 4.2 and 4.5 for the proof of the above result when . When , it follows from the well-known result that is bounded on the Lebesgue space .

We now recall the definition of the Muckenhoupt weight functions .

Definition 4. For , a locally integrable function is said to be an weight if where . A locally integrable function is said to be an weight if for some constants . The infimum of all such is denoted by . We define .

For any and , the weighted Lebesgue space consists of all Lebesgue measurable functions satisfying

Let . It is well known that is bounded on if and only if .

Let be any finite collection of semi-norms on and

For any , write where for any , write .

Definition 5. Let . The Hardy-amalgam space consists of all satisfying

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 and [6], Proposition 2, 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 . For any weighted function , the weighted Hardy space consists of all 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 and . Then, if and only if for some .

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. Let and . Suppose that We have a constant such that for any ,

Proof. Let . Define by As , is well defined.
In view of (15), we have . Consequently, Since , we have . Similarly, we have . Thus, and are well defined.
For any nonnegative function , define The Rubio de Francia operator possesses the following properties: The above properties are valid because , and Theorem 3 assures that is bounded on .
The identity in (18) shows that Similarly, the identity shows that Therefore, For any and , (5), (20), and (25) assert that Consequently, for any , we find that Moreover, (22) yields that . By applying Theorem 6 with , (20) and (27) yield for some .
Consequently, (26) gives Proposition 2 guarantees that Therefore, by taking the supremum over on both sides of (29), we have a constant such that for any

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, when and , 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.