#### Abstract

Let be a class of sublinear operators satisfying certain size conditions introduced by Soria and Weiss, and let be the commutators generated by functions and . This paper is concerned with two-weight, weak-type norm estimates for these sublinear operators and their commutators on the weighted Morrey and amalgam spaces. Some boundedness criteria for such operators are given, under the assumptions that weak-type norm inequalities on weighted Lebesgue spaces are satisfied. As applications of our main results, we can obtain the weak-type norm inequalities for several integral operators as well as the corresponding commutators in the framework of weighted Morrey and amalgam spaces.

#### 1. Introduction and Main Results

##### 1.1. Sublinear Operators

Let be the -dimensional Euclidean space equipped with the Euclidean norm and the Lebesgue measure . Suppose that represents a linear or a sublinear operator, which satisfies that for any with compact support and , where is a universal constant independent of and . The size condition (1) was first introduced by Soria and Weiss in [1]. It can be proved that (1) is satisfied by many integral operators in harmonic analysis, such as the Hardy–Littlewood maximal operator, Calderón–Zygmund singular integral operators, Ricci–Stein’s oscillatory singular integrals, and Bochner–Riesz operators at the critical index and so on.

Similarly, for any given , we assume that represents a linear or a sublinear operator with order , which satisfies that for any with compact support and , where is also a universal constant independent of and . It can be easily checked that (2) is satisfied by some important operators such as the fractional maximal operator, Riesz potential operators, and fractional oscillatory singular integrals.

Let be a locally integrable function on ; suppose that the commutator operator stands for a linear or a sublinear operator, which satisfies that for any with compact support and , where is an absolute constant independent of and .

Similarly, for any given , we assume that the commutator operator stands for a linear or a sublinear operator, which satisfies that for any with compact support and , where is also an absolute constant independent of and . Clearly, based on the above assumptions,

##### 1.2. Weighted Morrey Spaces

The classical Morrey space was introduced by Morrey [2] in connection with elliptic partial differential equations. Let and . We recall that a real-valued function is said to belong to the space on the -dimensional Euclidean space , if the following norm is finite: where is the Euclidean ball with center and radius . In particular, one has

In [3], Komori and Shirai considered the weighted case and gave the definitions of the weighted Morrey spaces as follows.

*Definition 1. *Let and . For two weights and on , the weighted Morrey space is defined by
where the norm is given by
and the supremum is taken over all cubes in .

*Definition 2. *Let , , and be a weight on . We define the weighted weak Morrey space as the set of all measurable functions satisfying
By definition, it is clear that

##### 1.3. Weighted Amalgam Spaces

Let ; a function is said to be in the Wiener amalgam space of and , if the function belongs to , where is the open ball in centered at with radius , is the characteristic function of the ball , and is the usual Lebesgue norm in . In [4], Fofana introduced a new class of function spaces which turn out to be the subspaces of . More precisely, for , 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 . It was shown in [4] that the space is nontrivial if and only if . Throughout this paper, we will always assume that the condition is satisfied. Let us consider the following special cases: (1)If we take , then . It is easy to check that

Hence, the amalgam space is equal to the Lebesgue space with the same norms provided that (2)If , then we can see that the amalgam space is equal to the Morrey space with equivalent norms, where

In this paper, we will consider the weighted version of .

*Definition 3. *Let , and let , , and be three weights on . We denote by the weighted amalgam space, the space of all locally integrable functions such that
with and the usual modification when .

*Definition 4. *Let , and let and be two weights on . We denote by the weighted weak amalgam space consisting of all measurable functions such that
with and the usual modification when .

Note that when , this kind of weighted (weak) amalgam space was introduced by Feuto in [5] (see also [6]). We remark that Feuto considered ball instead of cube in his definition, but these two definitions are apparently equivalent. Also, note that when and , then is just the weighted Morrey space with , and is just the weighted weak Morrey space with .

The two-weight problem for classical integral operators has been extensively studied. In [7–9] and [10], the authors gave some -type conditions which are sufficient for the two-weight, weak-type inequalities for Calderón–Zygmund operators, fractional integral operators, and their commutators on the weighted Lebesgue spaces. In [11], the authors established the two-weight, weak-type estimates for the maximal Bochner–Riesz operators and their commutators. Inspired by the above results, it is natural and interesting to study the weak-type estimates for sublinear operators (1) and (2), as well as the corresponding commutators (3) and (4).

Let be the conjugate index of whenever ; that is, . The main purpose of this paper is to investigate the two-weight, weak-type norm inequalities in the setting of weighted Morrey and amalgam spaces. Our main results can be stated as follows. On the boundedness properties of the sublinear operators and their commutators on weighted Morrey spaces, we will prove the following.

Theorem 5. *Let , , and satisfy condition (1). Given a pair of weights , suppose that for some and for all cubes in ,
**Furthermore, we suppose that satisfies the weak-type inequality
where does not depend on and . If , then the operator is bounded from into .*

Theorem 6. *Let , , and satisfy condition (2) with . Given a pair of weights , suppose that for some and for all cubes in ,
**Furthermore, we suppose that satisfies the weak-type inequality
where does not depend on and . If , then the operator is bounded from into .*

Theorem 7. *Let , , , and satisfy condition (3). Given a pair of weights , suppose that for some and for all cubes in ,
where . Furthermore, we suppose that satisfies the weak-type inequality
where does not depend on and . If , then the commutator operator is bounded from into .*

Theorem 8. *Let , , , and satisfy condition (4) with . Given a pair of weights , suppose that for some and for all cubes in ,
where . Furthermore, we suppose that satisfies the weak-type inequality
where does not depend on and . If , then the commutator operator is bounded from into .*

Concerning the boundedness properties on weighted amalgam spaces for these operators, we have the following results.

Theorem 9. *Let and . Given a pair of weights , assume that for some and for all cubes in ,
**Furthermore, we assume that satisfies the weak-type inequality (17). If , then the operator is bounded from into .*

Theorem 10. *Let , , and . Given a pair of weights , assume that for some and for all cubes in ,
**Furthermore, we assume that satisfies the weak-type inequality (19). If , then the operator is bounded from into .*

Theorem 11. *Let , and . Given a pair of weights , assume that for some and for all cubes in ,
where . Furthermore, we assume that satisfies the weak-type inequality (21). If , then the commutator operator is bounded from into .*

Theorem 12. *Let , , , and . Given a pair of weights , assume that for some and for all cubes in ,
where . Furthermore, we assume that satisfies the weak-type inequality (23). If , then the commutator operator is bounded from into .*

*Remark 13. *It should be pointed out that the conclusions of our main theorems are natural generalizations of the corresponding weak-type estimates on the weighted Lebesgue spaces. The operators satisfying the assumptions of the above theorems include Calderón–Zygmund operators, Bochner–Riesz operators, and fractional integral operators. Hence, we are able to apply our main theorems to these classical integral operators.

#### 2. Notations and Definitions

##### 2.1. Weights

A nonnegative function defined on will be called a weight if it is locally integrable. will denote the cube centered at and has side length ; all cubes are assumed to have their sides parallel to the coordinate axes. Given a cube and , stands for the cube concentric with having side length times as long, i.e., . For any given weight and any Lebesgue measurable set of , we denote the characteristic function of by , the Lebesgue measure of by , and the weighted measure of by , where . We also denote the complement of . Given a weight , we say that satisfies the *doubling condition*, if there exists a universal constant such that for any cube , we have

When satisfies condition (28), we denote for brevity. A weight is said to belong to Muckenhoupt’s class for , if there exists a constant such that holds for every cube . The class is defined as the union of the classes for , i.e., . If is an weight, then we have (see [12]). Moreover, this class is characterized as the class of all weights satisfying the following property: there exists a number and a finite constant such that (see [12]) holds for every cube and all measurable subsets of . Given a weight on and for , the weighted Lebesgue space is defined as the set of all measurable functions such that

We also define the weighted weak Lebesgue space () as the set of all measurable functions satisfying

##### 2.2. Orlicz Spaces and BMO

We next recall some basic facts about Orlicz spaces needed for the proofs of the main results. For further details, we refer the reader to [13]. A function is said to be a Young function if it is continuous, convex, and strictly increasing satisfying and as . Given a Young function , we define the -average of a function over a cube by means of the following Luxemburg norm:

In particular, when , , it is easy to check that that is, the Luxemburg norm coincides with the normalized norm. The main examples that we are going to consider are with .

Let us now recall the definition of the space of (see [14]). A locally integrable function is said to be in , if where denotes the mean value of over , namely, and the supremum is taken over all cubes in . After identifying functions that differ by a constant, the space becomes a Banach space.

Throughout this paper, always denotes a positive constant independent of the main parameters involved, but it may be different from line to line. We will use appearing in the first section of this paper to denote certain constants. We also use to denote the equivalence of and ; that is, there exist two positive constants and independent of and such that .

#### 3. Proofs of Theorems 5 and 6

*Proof of Theorem 5. *Let with and . For an arbitrary fixed cube , we set . Decompose as
where denotes the characteristic function of the set . Then, for any given , we write
We first consider the term . Using assumption (17) and the condition , we get
This is just our desired estimate. Let us estimate the second term . To this end, we observe that when and , one has . We then decompose into a geometrically increasing sequence of concentric cubes and obtain the following pointwise estimate by condition (1).
This pointwise estimate (40) together with Chebyshev’s inequality implies that
It follows directly from Hölder’s inequality with exponent that
Moreover, for any positive integer , we apply Hölder’s inequality again with exponent to get
Thus, in view of (43), we conclude that
The last inequality is obtained by the -type condition (16) on . Since , we can easily check that there exists a *reverse doubling constant* independent of such that (see [3], Lemma 4.1)
which implies that for any positive integer ,
by iteration. Hence,
where the last series is convergent since the *reverse doubling constant* and . This yields our desired estimate
Summing up the above estimates for and , and then taking the supremum over all cubes and all , we finish the proof of Theorem 5.

*Proof of Theorem 6. *Let with and . For an arbitrary fixed cube in , we decompose as
where . For any given , we then write
Let us consider the first term . Using assumption (19) and the condition , we have
This is exactly what we want. We now deal with the second term . Note that , whenever and . For and all , using the standard technique and condition (2), we can see that
This pointwise estimate (52) together with Chebyshev’s inequality yields
By using Hölder’s inequality with exponent , we can deduce that
Moreover, we apply estimate (43) to get