Abstract
In this paper, we establish the boundedness of the modified fractional integral operator from mixed Morrey spaces to the bounded mean oscillation space and Lipschitz spaces, respectively.
1. Introduction
Let , and the fractional integral operator and the modified fractional integral operator are defined bywhere is a locally integrable function on .
The well-known Hardy–Littlewood–Sobolev inequality yields the boundedness of from to , where and . For the endpoint situation, we know is bounded from to (see [1]). Moreover, Peetre [2] proved that is bounded from to . Here, is the bounded mean oscillation space, which consists of all locally integrable functions on such thatwhere is the ball centered at with radius , denotes its Lebesgue measure, and . If one regards two functions whose difference is a constant as one, the space is a Banach space with respect to the norm .
Other than Lebesgue spaces, Morrey spaces are also important function spaces to study the boundedness of integral operators in harmonic analysis. The classical Morrey space was introduced by Morrey [3] to study the regularity of elliptic partial differential equations. Now, we recall the definition of the Morrey space .
For , the classical Morrey space consists of all functions with
One can see that Morrey spaces are natural generalizations of Lebesgue spaces. The mapping properties of on Morrey spaces were first studied by Peetre [2] and further generalized by Adams [4]. We refer readers to [5–19] and the references therein for more studies about boundedness of the fractional integral operator on Morrey-type and anisotropic spaces. Recently, the mapping properties of from Morrey spaces to and Lipschitz spaces were also obtained in [20–22]. Here, we review the definition of Lipschitz spaces briefly. Let ; we say a locally integrable function belongs to the Lipschitz space if
If one regards two functions whose difference is a constant as one, the space is a Banach space with respect to the norm . As we know, a locally integrable function which belongs to also means that there exists a constant such that for all . If is the smallest constant satisfying the inequality, then .
With the development of the theory of function spaces, Morrey spaces have been extended to different settings. One of the extensions is the mixed Morrey space, which was recently defined by Nogayama et al. [23–25] to uniform mixed Lebesgue spaces and Morrey spaces. Note that there exists another mixed Morrey space by using the iteration of Morrey norm introduced by Ragusa and Scapellato [26]. We refer the readers to [21, 27, 28] for the boundedness of various operators on these mixed Morrey spaces of iteration type.
To give the definition of mixed Morrey spaces , we first recall the definition of mixed Lebesgue spaces introduced in [29]. Let . Then, the mixed Lebesgue norm is defined bywhere is a measurable function. If for some , then we have to make appropriate modifications. We define the mixed Lebesgue space to be the set of all measurable functions with .
Now, we can give the definition of mixed Morrey spaces introduced by Nogayama [24].
Definition 1. Let , and . A measurable function belongs to the mixed Morrey space if and only ifIn [24], the author proved the boundedness of from one mixed Morrey space to another, which inspires us to consider the mapping of at or beyond the endpoint situation, i.e., the boundedness of from to or .
Throughout the paper, we use the following notations.
Let be the collection of all locally integrable functions on . We use and to denote the characteristic function and the Lebesgue measure of a measurable set .
The letter denotes -tuples of the numbers in , i.e., , where and . By definition, the inequality, for example, , means for all . For , we denote , where satisfies .
By , we mean that for some constant , and means that and .
2. Main Results
We first recall the boundedness of on mixed Lebesgue spaces, which will be used to prove our main results. Here and in the following, we denote and .
Lemma 1. Let and . Then,if and only if
The proof of Lemma 1 can be found in Lemma 3.1 of [30].
In [24], Nogayama established the following result on the boundedness of from one mixed Morrey space to another.
Lemma 2. Let , , and . Assume that and . Also, assume that and . Then, for ,
For more studies on the boundedness of operators on mixed Morrey spaces, we refer the readers to [31, 32]. One can see that in Lemma 2 satisfies . It is natural to ask what happens if or . In this section, we give an affirmative answer. More precisely, we will establish the boundedness of modified fractional integral operator from to or .
When in Lemma 2, we have the following theorem.
Theorem 1. Let , , and . Then, for all , we have
Proof. By the definition of , we only need to show for any and , there holdsWrite , and letThen, we haveDefine such that and . For the term , by using Lemma 1 and Hölder’s inequality on mixed Lebesgue spaces, we obtainNow, we turn to estimate . By a direct computation, we haveFrom the estimates of and , we get (11), which finishes the proof.
When in Lemma 2, we have the following mapping property of from to .
Theorem 2. Let , , , and . Then, for , we have
Proof. The proof is similar to that of Theorem 1.
By the definition of Lipschitz spaces, we only need to show for any and , there holdsWe also write and letSimilar to (13), we haveChoose such that and . For the term , by using Lemma 1 and Hölder’s inequality on mixed Lebesgue spaces, we getsince .
For , by a similar method as in the proof of Theorem 1, we havesince .
From the estimates of and , we get (17). The proof is complete.
It is worth mentioning that our results in Theorems 1 and 2 extend the corresponding results of classical Morrey spaces to mixed Morrey spaces.
Data Availability
Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by the Natural Science Foundation of Henan Province (no. 202300410338), the Science and Technology Project of Henan Province (no. 212102210394), the Program for Science Technology Innovation Talents in Universities of Henan Province (no. 22HASTIT021), and the Nanhu Scholar Program for Young Scholars of Xinyang Normal University.