#### Abstract

In this paper, we first introduce some new classes of weighted amalgam spaces. Then, we give the weighted strong-type and weak-type estimates for fractional integral operators on these new function spaces. Furthermore, the weighted strong-type estimate and endpoint estimate of linear commutators generated by and are established as well. In addition, we are going to study related problems about two-weight, weak-type inequalities for and on the weighted amalgam spaces and give some results. Based on these results and pointwise domination, we can prove norm inequalities involving fractional maximal operator and generalized fractional integrals in the context of weighted amalgam spaces, where and is the infinitesimal generator of an analytic semigroup on with Gaussian kernel bounds.

#### 1. Introduction

One of the most significant operators in harmonic analysis is the fractional integral operator. Let be a positive integer. The -dimensional Euclidean space is endowed with the Lebesgue measure and the Euclidean norm . For given , , the fractional integral operator (or Riesz potential) of order is defined by

The boundedness properties of between various function spaces have been studied extensively. It is well-known that the Hardy-Littlewood-Sobolev theorem states that the fractional integral operator is bounded from to for , , and . Also, we know that is bounded from to for and (see ). In 1974, Muckenhoupt and Wheeden  studied the weighted boundedness of and obtained the following two results (for sharp weighted norm inequalities, see ).

Theorem 1 (see ). Let , , , and . Then, the fractional integral operator is bounded from to .

Theorem 2 (see ). Let , , , and . Then, the fractional integral operator is bounded from to.

For , the linear commutator generated by a suitable function and is defined by

This commutator was first introduced by Chanillo in . In 1991, Segovia and Torrea  showed that is bounded from () to whenever (see  for sharp weighted bounds; see also  for the unweighted case). This corresponds to the norm inequalities satisfied by . Let us recall the definition of the space of (see ). is the Banach function space modulo constants with the norm defined by where the supremum is taken over all balls in and stands for the mean value of over ; that is, .

Theorem 3 (see ). Let , , , and . Suppose that , then the linear commutator is bounded from to .

In the endpoint case and , since linear commutator has a greater degree of singularity than itself, a straightforward computation shows that fails to be of weak type () when (see  for some counterexamples). However, if we restrict ourselves to a bounded domain in , then the following weighted endpoint estimate for commutator of the fractional integral operator is valid, which was established by Cruz-Uribe and Fiorenza  in 2007 (see also  for the unweighted case).

Theorem 4 (see ). Let , , , and . Suppose that , then for any given and any bounded domain in , there exists a constant , which does not depend on , Ω, and , such that where and .

Let , a function is said to be in the Wiener amalgam space of and , if the function belongs to , where is the open ball centered at and with radius , is the characteristic function of the ball , and is the usual Lebesgue norm in . Define

Then, we know that becomes a Banach function space with respect to the norm . This amalgam space was first introduced by Wiener in the 1920’s, but its systematic study goes back to the works of Holland  and Fournier and Stewart . Let . We define the amalgam space of and as the set of all measurable functions satisfying and , where with the usual modification when or and is the Lebesgue measure of the ball . This generalization of amalgam space was originally introduced by Fofana in . As proved in , the space is nontrivial if and only if ; thus, in the remaining of the paper, we will always assume that this condition is fulfilled. Note that

(i) for , one can easily see that , where is the Wiener amalgam space defined by (5)

(ii)if and , then is just the classical Morrey space defined by (with , see )

(iii) if and , then reduces to the usual Lebesgue space

In  (see also [15, 16]), Feuto considered a weighted version of the amalgam space . A nonnegative measurable function defined on is called a weight if it is locally integrable. Let and be a weight on . We denote by the weighted amalgam space, the space of all locally integrable functions satisfying , where with the usual modification when and is the weighted measure of . Similarly, for , we can see that becomes a Banach function space with respect to the norm . Furthermore, we denote by the weighted weak amalgam space consisting of all measurable functions such that (see )

Notice that

(i) if and , then is just the weighted Morrey space defined by (with , see ) and is just the weighted weak Morrey space defined by (with , see )

(ii) if and , then reduces to the weighted Lebesgue space and reduces to the weighted weak Lebesgue space

Recently, many works in classical harmonic analysis have been devoted to norm inequalities involving several integral operators in the setting of weighted amalgam spaces (see [1416, 19] and ). These results obtained are extensions of well-known analogues in the weighted Lebesgue spaces.

Let be the fractional integral operator, and let be its linear commutator. The aim of this paper is twofold. We first define some new classes of weighted amalgam spaces. As the weighted amalgam space may be considered as an extension of the weighted Lebesgue space, it is natural and important to study the weighted boundedness of and in these new spaces. We will prove that as well as its commutator , which are known to be bounded on weighted Lebesgue spaces, are bounded on weighted amalgam spaces under appropriate conditions. In addition, we will discuss two-weight, weak-type norm inequalities for and in the context of weighted amalgam spaces and give some partial results. Using these results and pointwise domination, we will establish the corresponding strong-type and weak-type estimates for fractional maximal operator and generalized fractional integrals , where and is the infinitesimal generator of an analytic semigroup on with Gaussian kernel bounds.

The present paper is organized as follows. In Section 2, we first state some preliminary definitions and results about weights, Orlicz spaces, and weighted amalgam spaces, and the main results of the present paper are also given in Section 2. The following Sections 3, 4, and 5 are devoted to their proofs. Finally, in Section 6, we discuss some related two-weight problems.

#### 2. Statement of Our Main Results

##### 2.1. Notations and Preliminaries

Let us first recall the definitions of two weight classes: and .

Definition 5 ( weights ). A weight is said to belong to the class for , if there exists a positive constant such that for any ball in , where we denote the conjugate exponent of by . The class is defined replacing the above inequality by for any ball in . We also define .

Definition 6 ( weights ). A weight is said to belong to the class for , if there exists a positive constant such that for any ball in , The class () is defined replacing the above inequality by for any ball in .
There is a close connection between weights and weights (see ).

Lemma 7. Suppose that , , and . Then, the following statements are true:
(i) If , then implies and
(ii) if , then if and only if

Given a ball and , we write for the ball with the same center as whose radius is times that of . For any and , we denote by the complement of in ; that is, . Given a weight , we say that satisfies the doubling condition if there exists a universal constant such that for any ball in , we have

When satisfies this doubling condition (16), we denote for brevity. An important fact here is that if is in , then (see ). Moreover, if , then there exists a number such that (see ) holds for any measurable subset of a ball .

Given a weight on , for , the weighted Lebesgue space is defined as the set of all functions such that

We also denote by () the weighted weak Lebesgue space consisting of all measurable functions such that

We next recall some definitions and basic facts about Orlicz spaces needed for the proofs of our main results. For further information on this subject, we refer to . A function is said to be a Young function if it is continuous, convex, and strictly increasing satisfying and as . An important example of Young function is with some . Given a Young function , we define the -average of a function over a ball by means of the following Luxemburg norm:

In particular, when , , it is easy to see that is a Young function and that is, the Luxemburg norm coincides with the normalized norm. Recall that the following generalization of Hölder’s inequality holds where is the complementary Young function associated with , which is given by , . Obviously, is a Young function, and its complementary Young function is . In the present situation, we denote and by and , respectively. Now, the above generalized Hölder’s inequality reads

There is a further generalization of Hölder’s inequality that turns out to be useful for our purpose (see ): Let , , and be Young functions such that for all , where is the inverse function of . Then, for all functions and and for all balls in ,

##### 2.2. Weighted Amalgam Spaces

Let us begin with the definitions of the weighted amalgam spaces with Lebesgue measure in (8) and (9) replaced by weighted measure.

Definition 8. Let , and let be three weights on . We denote by the weighted amalgam space, the space of all locally integrable functions such that If , then we denote for brevity, i.e., . Furthermore, we denote by the weighted weak amalgam space consisting of all measurable functions for which with the usual modification when .
The aim of this paper is to extend Theorems 14 to the corresponding weighted amalgam spaces. We are going to prove that the fractional integral operator , which is bounded on weighted Lebesgue spaces, is also bounded on our new weighted spaces under appropriate conditions. Our first two results in this paper are stated as follows.

Theorem 9. Let , , , and . Assume that and , then the fractional integral operator is bounded from to with .

Theorem 10. Let , , , and . Assume that and , then the fractional integral operator is bounded from to with .

Let be the linear commutator generated by and function . For the strong-type estimate of on the weighted amalgam spaces, we have the following result.

Theorem 11. Let , , , and . Assume that , , and , then the linear commutator is bounded from to with .

To obtain endpoint estimate for the linear commutator , we first need to define the weighted -average of a function over a ball by means of the weighted Luxemburg norm; that is, given a Young function and , we define (see , for instance)

When , this norm is denoted by , and when , this norm is also denoted by . The complementary Young function is given by with corresponding mean Luxemburg norm denoted by . For and for every ball in , we can also show the weighted version of (23). Namely, the following generalized Hölder’s inequality in the weighted setting is true (see , for instance). Now, we introduce the new amalgam spaces of type as follows.

Definition 12. Let , , and let be three weights on . We denote by the weighted amalgam space of type, the space of all locally integrable functions defined on with finite norm : where Note that for all . Then, for any ball in and , it is immediate that by definition, i.e., the inequality holds for any ball in . From this, we can further see the following inclusion: when and are some other weights.
In the endpoint case , we will prove the following weak-type estimate of linear commutator in the setting of weighted amalgam spaces.

Theorem 13. Let ,, , and . Assume that , , and , then for any given and any ball in , there exists a constant independent of , , and such that where and . From the above definitions, we can roughly say that the linear commutator is bounded from to .

Moreover, we will discuss the extreme case of Theorem 9. In order to do so, we need to introduce a new -type space given below.

Definition 14. Let and . We define the space as the set of all locally integrable functions satisfying , where Here, the -norm is taken with respect to the variable . We also use the notation to denote the mean value of over .
Observe that if , then is the classical space.
Now, we can show that is bounded from to our new -type space defined above. This result can be regarded as a supplement of Theorem 9.

Theorem 15. Let , , , and let and . If and , then the fractional integral operator is bounded from to .

Throughout this paper, the letter always denotes a positive constant that is independent of the essential variables but whose value may vary at each occurrence. We also use to denote the equivalence of and ; that is, there exist two positive constants and independent of quantities and such that . Equivalently, we could define the above notions of this section with cubes in place of balls and we will use whichever is more appropriate, depending on the circumstances.

#### 3. Proofs of Theorems 9 and 10

In this section, we will prove the conclusions of Theorems 9 and 10.

Proof of Theorem 9. The proof is inspired by [14, 15]. Let and with and . For an arbitrary point , set for the ball centered at and of radius , . We represent as where is the characteristic function of . By the linearity of the fractional integral operator , one can write Here and in what follows, for any positive number , we use the convention . Below, we will give the estimates of and , respectively. By the weighted -boundedness of (see Theorem 1), we have Observe that when . This fact implies that Since , we get by Lemma 7(i). Moreover, since , then by doubling inequality (16), we obtain Substituting the above inequality (40) into (39), we can see that Let us now turn to the estimate of . First, it is clear that when and , we get . We then decompose into a geometrically increasing sequence of concentric balls and obtain the following pointwise estimate: