Research Article | Open Access
Two-Weight, Weak-Type Norm Inequalities for Fractional Integral Operators and Commutators on Weighted Morrey and Amalgam Spaces
Let and be the fractional integral operator of order γ, and let be the linear commutator generated by a symbol function b and , . This paper is concerned with two-weight, weak-type norm estimates for such operators on the weighted Morrey and amalgam spaces. Based on weak-type norm inequalities on weighted Lebesgue spaces and certain -type conditions on pairs of weights, we can establish the weak-type norm inequalities for fractional integral operator as well as the corresponding commutator in the framework of weighted Morrey and amalgam spaces. Furthermore, some estimates for the extreme case are also obtained on these weighted spaces.
1.1. Fractional Integral Operators
Let be the n-dimensional Euclidean space equipped with the Euclidean norm and the Lebesgue measure . For given γ, , the fractional integral operator (or Riesz potential) with order γ is defined by (see  for the basic properties of )
Weighted norm inequalities for fractional integral operators arise naturally in harmonic analysis and have been extensively studied by several authors. The study of two-weight problem for was initiated by Sawyer in his pioneer paper . By a weight , we mean that is a nonnegative and locally integrable function. In , Sawyer concerned the following question. Suppose that . For which pairs of weights on is the fractional integral operator bounded from into weak-? A necessary and sufficient condition for the weak-type inequality was given by Sawyer. More specifically, he showed the following.
Theorem 1 (see ). Let and . Given a pair of weights on , the weak-type inequalityholds for any if and only iffor all cubes Q in . Here, denotes the characteristic function of the cube Q, denotes the conjugate index of p, and C is a universal constant.
Sawyer’s result is interesting and important, and it promotes a series of research works on this subject (see, e.g., [3, 4, 5–9]), but it has the defect that condition (3) involves the fractional integral operator itself. In , Cruz-Uribe and Pérez considered the case when and found a sufficient -type condition on a pair of weights which ensures the boundedness of the operator from into weak-, where . The condition (4) given below is simpler than (3) in the sense that it does not involve the operator itself, and hence it can be more easily verified.
Theorem 2 (see ). Let and . Given a pair of weights on , suppose that for some and for all cubes Q in ,
Then, the fractional integral operator satisfies the weak-type inequalitywhere C does not depend on f nor on .
The proof of Theorem 2 is quite complicated. It depends on an inequality relating the Hardy–Littlewood maximal function and the sharp maximal function which is strongly reminiscent of the good-λ inequality of Fefferman and Stein. For another, more elementary proof, see also . This solves a problem posed by Sawyer and Wheeden in . Moreover, in , Li improved this result by replacing the “power bump” in (4) by a smaller “Orlicz bump” (see also ). On the other hand, for given , the linear commutator generated by a suitable function b and is defined by
This commutator was first introduced by Chanillo in . In , Liu and Lu obtained a sufficient -type condition for the commutator to satisfy a two-weight weak-type inequality, where . That condition is an -type condition in the scale of Orlicz spaces (see (7) given below).
Theorem 3 (see ). Let , and . Given a pair of weights on , suppose that for some and for all cubes Q in ,where and , that is,
Then, the linear commutator satisfies the weak-type inequalitywhere C does not depend on f nor on .
In , Martell considered the case when and gave a verifiable condition which is sufficient for the two-weight, weak-type inequality for fractional integral operator . The condition (10) given below (in the Euclidean setting of ) is also simpler than the one in Theorem 1.
Theorem 4 (see ). Let and . Given a pair of weights on , suppose that for some and for all cubes Q in ,
Then, the fractional integral operator satisfies the weak-type inequalitywhere C does not depend on f nor on .
Furthermore, in , Zhang sharpened Martell’s result by replacing the local norm on the left-hand side of (10) by the smaller Orlicz space norm. On the other hand, by comparing Theorem 4 with Theorems 2 and 3, it is natural to conjecture that when , there is a two-weight, weak-type inequality for the commutator of fractional integral operator. By using the same method as in the proof of Theorem 1 in  and certain Orlicz norm, we are able to obtain the following sufficient condition on a pair of weights to ensure the boundedness of , whenever b belongs to . More specifically, the following statement is true.
Theorem 5. Let , and . Given a pair of weights on , suppose that for some and for all cubes Q in ,where . Then, the linear commutator satisfies the weak-type inequalitywhere C does not depend on f nor on .
Question. In view of Theorems 2–5, it is a natural and interesting problem to find some sufficient conditions for which the two-weight, weak-type norm inequalities hold for the operators and , in the endpoint case .
In this paper, we are mainly interested in the weighted Morrey spaces and weighted amalgam spaces. Let us recall their definitions.
1.2. Weighted Morrey Spaces
The classical Morrey space was introduced by Morrey  in connection with elliptic partial differential equations. Let and . We recall that a real-valued function f is said to belong to the space on the n-dimensional Euclidean space , if the following norm is finite:where is the Euclidean ball with center and radius as well as the Lebesgue measure . Here, is the volume of the unit ball of . In particular, one has
In , Komori and Shirai considered the weighted case and introduced a version of weighted Morrey space, which is a natural generalization of weighted Lebesgue space.
Definition 1. Let and . For two weights and ν on , the weighted Morrey space is defined bywhere the norm is given byand the supremum is taken over all cubes Q in .
Definition 2. Let , , and w be a weight on . We define the weighted weak Morrey space as the set of all measurable functions f satisfyingBy 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 an open ball in centered at y with radius 1, is the characteristic function of the ball , and is the usual Lebesgue norm in . In , Fofana introduced a new class of function spaces which turned out to be the subspaces of . More precisely, for , we define the amalgam space of and as the set of all measurable functions f satisfying and , wherewith the usual modification when or , and is the Lebesgue measure of the ball . As it was shown in  that the space is nontrivial if and only if , in the remaining of this paper, we will always assume that this condition is satisfied. Let us consider the following two special cases:(1)If we take , then . By Fubini’s theorem, it is easy to check that where the last equality holds since . Hence, the amalgam space is equal to the Lebesgue space with the same norms provided that .(2)If , then we can see that in such a situation, the amalgam space is equal to the classical Morrey space with equivalent norms, where .
In this paper, we will consider the weighted version of .
Definition 3. Let and let be three weights on . We denote by the weighted amalgam space, the space of all locally integrable functions f, such thatwith and the usual modification when .
Definition 4. Let , and let be two weights on . We denote by the weighted weak amalgam space consisting of all measurable functions f such thatwith and the usual modification when .
Note that in the particular case when , this kind of weighted (weak) amalgam space was introduced by Feuto in  (see also ). We remark that Feuto  considered ball B instead of cube Q in his definition, but these two definitions are evidently equivalent. Also, note that when and , then is just the weighted Morrey space with , and is just the weighted weak Morrey space with .
Recently, in [17–19], the author studied the two-weight, weak-type inequalities for fractional integral operator, as well as its commutators on weighted Morrey and amalgam spaces, under some -type conditions (4) and (7) on the pair . As a continuation of the works mentioned above, in this paper, we consider related problems about two-weight, weak-type inequalities for and , under some other -type conditions (10) and (12) on and .
2. Statement of Our Main Results
We are now in a position to state our main results. Let be the conjugate index of whenever ; that is, . First, we give the two-weight, weak-type norm inequalities for the fractional integral operator in the setting of weighted Morrey and amalgam spaces.
Theorem 6. Let . Given a pair of weights on , suppose that for some and for all cubes Q in ,
If , then the fractional integral operator is bounded from into .
Theorem 7. Let , and . Given a pair of weights on , assume that for some and for all cubes Q in ,
If and , then the fractional integral operator is bounded from into with .
Next, we introduce the definition of the space of (see ). Suppose that and letwhere denotes the mean value of b on Q, namely,and the supremum is taken over all cubes Q in . Define
If we regard two functions whose difference is a constant as one, then the space is a Banach space with respect to the norm . Concerning the two-weight weak-type estimates for the linear commutator in the context of weighted Morrey and amalgam spaces, we have the following results.
Theorem 8. Let , , , and . Given a pair of weights on , suppose that for some and for all cubes Q in ,where . If , then the linear commutator is bounded from into .
Theorem 9. Let , , , and . Given a pair of weights on , assume that for some and for all cubes Q in ,where . If and , then the linear commutator is bounded from into with .
Theorem 10. Let and . Given a pair of weights on , suppose that for some and for all cubes Q in ,
If and , then the fractional integral operator is bounded from into BMO.
In addition, we will also discuss the extreme case of Theorem 7. In order to do so, we need to introduce the following new -type space.
Definition 5. Let and . The space is defined as the set of all locally integrable functions f satisfying , whereHere, the -norm is taken with respect to the variable y. We also use the notation to denote the mean value of f on .
Observe that if , then is just the classical space given above.
Now, we can show that is bounded from into our new -type space defined above. This new result may be viewed as a supplement of Theorem 7.
Theorem 11. Let , , and . Given a pair of weights on , assume that for some and for all cubes Q in ,
If , , and , then the fractional integral operator is bounded from into .
3. Notation and Definitions
In this section, we recall some standard definitions and notation.
For given and , we denote by the cube centered at y and has side length , and all cubes are assumed to have their sides parallel to the coordinate axes. Given a cube and , stands for the cube concentric with Q and having side length times as long, i.e., . A nonnegative function defined on will be called a weight if it is locally integrable. For any given weight and any Lebesgue measurable set E of , we denote the characteristic function of E by , the Lebesgue measure of E by , and the weighted measure of E by , where . We also denote the complement of E. Given a weight , we say that satisfies the doubling condition, if there exists a finite constant such that for any cube Q in , we have
When satisfies this condition (34), we denote for brevity. A weight is said to belong to Muckenhoupt’s class for , if there exists a constant such thatholds for every cube Q in . The class is defined as the union of the classes for , i.e., . If is an weight, then we have (see ). 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 )holds for every cube and all measurable subsets E of Q. Given a weight on and for , the weighted Lebesgue space is defined to be the collection of all measurable functions f satisfying
For a weight and , define the distribution function of f with bywhere λ is a positive number. We say that f is in the weighted weak Lebesgue space , if there exists a constant such that
3.2. Orlicz Spaces
We next recall some basic facts from the theory of Orlicz spaces needed for the proofs of the main results. For more information about these spaces the reader may consult the book . Let be a Young function. That is, a continuous, convex, and strictly increasing function satisfying and as . For a Young function and a cube Q in , we will consider the -average of a function f given by the following Luxemburg norm:
In particular, when with , it is easy to see thatthat is, the Luxemburg norm in such a situation coincides with the normalized norm. The main examples that we are going to consider are with .
Throughout the paper, C always denotes a positive constant independent of the main parameters involved, but it may be different from line to line. We will use to denote the equivalence of A and B; that is, there exist two positive constants and independent of A and B such that .
4. Proofs of Theorems 6 and 7
Proof of Theorem 6. Let with and . For an arbitrary fixed cube in , we decompose f aswhere and denotes the characteristic function of . For any given , we then writeLet us consider the first term . Using Theorem 4 and the condition , we haveThis is exactly what we want. We now deal with the second term . Note that , whenever and . For and all , using the standard technique, we can see that