Abstract
The boundedness of Bessel–Riesz operators defined on Lebesgue spaces and Morrey spaces in measure metric spaces is discussed in this research study. The maximal operator and traditional dyadic decomposition are used to study the Bessel-Riesz operators. We investigate the interaction between the kernel and space parameters to get the results and see how this affects kernel-bound operators.
1. Introduction
The Bessel–Riesz operator is a type of singular integral operator which acts on a function by convolving it with a Bessel–Riesz kernel. These operators are defined on metric spaces and are particularly useful in the study of measure spaces. They possess many important properties such as being bounded, compact, and having a certain degree of regularity. The study of Bessel-Riesz operators is a fundamental tool in harmonic analysis and has applications in other areas of mathematics such as partial differential equations and complex analysis.
The Schrödinger’s equation has been used to derive the Bessel–Riesz operators. The wave function or state function of a quantum mechanical system is described by Schrödinger’s equation has been used to derive the Bessel–Riesz operators. Schrödinger’s equation is a linear partial differential equation. In quantum physics, Schrödinger’s equation is analogous to Newton’s law. Kurata et al. [1] investigated Schrödinger’s operator and the boundedness of integral operators on generalized Morrey spaces in 1999.
According to Garca-Cuerva and Eduardo Gatto [2], Riesz potential is limited between Lebesgue spaces in terms of Euclidean settings. In Kokilashvili [3] (see also Edmunds, [4], Chapter 6), the nondoubling measure is covered in great detail, including conclusions addressing the boundedness of the fractional integral operator on Lebesgue spaces with the nondoubling measure. Theorems of the Sobolev and Adams types for fractional integrals built on quasimetric measure spaces were used to show the result for potentials in Euclidean settings [3]. Garcia-Cuerva and Jos Mara [5] demonstrate that fractional integrals on measure metric spaces are bounded. Adams [6] and Peetre [7] studied the Riesz potential in Euclidean situations using Morrey spaces.
According to Eridani’s proof [8], fractional integral operators are constrained in weighted Morrey spaces with quasimetric measure spaces. In [9, 10], it was proven that maximum and fractional integral operators in classical Morrey spaces are bounded. The boundedness of these operators in Morrey spaces was recently established by Idris et al. ([11], Theorem 6, pp.3). They found similar results to the Chiarenza result [12] about the boundedness of fractional integral operators on Morrey spaces. Additionally, results for weighted Morrey spaces and generalized Morrey spaces can be found in [13–17]. The examination by Eridani et al. [8] of the boundedness of fractional integral operators in Morrey spaces based on quasimetric measure spaces is novel in this study. The paper by Idris et al. [11] also includes some discoveries on Young with weight and demonstrates that these operators on Young are bound in Euclidean contexts. Euclidean spaces are the most straightforward way to represent metric spaces.
A separate approach will be used to demonstrate the boundedness of these operators on Young in measure metric spaces. We shall also discover that the norm of the kernels constrains the norm of the operators. In [18], Young inequality was utilized to demonstrate that Bessel-Riesz operators are bounded. With measure metric spaces, we will look into the boundedness of these integral operators on Lebesgue and Morrey spaces. The Young inequality cannot be used indefinitely to produce the same results. To demonstrate that Bessel–Riesz operators are bound on Young, we will employ the maximal operator. The boundedness of Bessel-Riesz operators on quasimetric spaces, a recent scientific innovation, is demonstrated in this article as an extension of the findings [19] from earlier work on Morrey spaces. The kernels of the operators hold some parameters. The usual dyadic decomposition and the Hardy–Littlewood maximal operator will all be applied in the proofs as before. We will find that the norm of the kernels dominates the norm of these integral operators. This paper will discuss an integral operator’s boundedness in measure metric spaces.
The following Table 1 contains a list of notations used in this article.
2. Preliminaries
2.1. Measurable Spaces
Definition 1. A distinguished collection of subsets of is called sigma algebra if it satisfies the following axioms:(1)If , then ,(2)If are countable family of sets in then ,(3)The intersection of countable sets also belongs to i.e., .
Definition 2. A measure defined on sigma-algebra, is a function such that(1), where is an empty set.(2)If is a sequence of disjoint sets in , thenIf is -algebra of subsets of , then the pair is said to be a measurable space and members of are called -measurable sets. Moreover, if is a measurable space, and if is a measure on , then the triplet is said to be a measure space.
Definition 3 (see [8]). We assume that is a topological space, endowed with a complete measure such that the space of compactly supported continuous functions is dense in and there exists a function (quasimetric) satisfying the conditions:(i) for all , and for all ;(ii)There exists a constant , such that for all ;(iii)There exists a constant , such that for all .We assume that the balls are -measurable and for . For every neighborhood of , there exists such that . We also assume that and for all . The triple will be called quasimetric measure space.
Definition 4 (see [20]). In measure metric spaces , we will have the properties for every balls . For , we defineFor any in X, the function may be regarded as the product of two kernels, be a Bessel kernel, and be a Riesz kernel. We define as any measurable functions such that .If , then we have which also referred to as Riesz Potentials or fractional integrals. Since the 1920s, research on Riesz potentials has been conducted. These operators’ boundedness has been studied by Hardy [21, 22] and Sobolev [23]. See [1], for example, of using the aforementioned operators in settings involving Euclidean spaces. We have evidence that Eridani [8] citation of the boundedness results is correct. Spanne [2] has demonstrated the boundedness of on Morrey spaces. Additionally, Adams [6] and Chiarenza [12] derived a stronger conclusion about the boundedness of Riesz operators. Using the multiplication operator , Kurata et al. [1] demonstrated that the function is confined on generalized Morrey spaces. The boundedness of on Young in Euclidean settings was discussed by Idris et al. [11] in the cited article. In this section, we will talk about Bessel–Riesz operators with boundedness in measure metric spaces. For every , and , thenWe know that , provided that .
Now, we introduce the following operator:
2.2. Hardy–Littlewood Maximal Operator
The operator which includes Hardy–Littlewood maximal operator defined as follows [11]:
A typical outcome for is that it is constrained by for .
Definition 5. Morrey Spaces [8] For and an appropriate . In order to define the Morrey space, we use the formula is the set of all functions, where forHere, we define .
3. Main Results
Theorem 1. For , there exists such that
Proof. Suppose , and for every , we haveIt is easy to see that the following estimates is holds. Then we haveSo, for every , and , we haveIf we choose such that , thenand this finish the proof. (QED).
Next, by using Definition 5 of Morrey spaces, we have
Theorem 2. Suppose , and is a decreasing functions. Then there exists such that
Proof. Suppose for , and , we defineSince , then . By the previous result, for every , we haveIt is easy to see that for every , we have the following estimate:and this complete the proof. (QED).
Suppose , and , for every , we haveand , provided that .
Theorem 3. For , there exists such that
Proof. Suppose , and for every , we haveIt is easy to see that the following estimates is holds. That is, we haveSo, for every , and , we haveIf we choose such that , thenand this finish the proof. (QED).
Next, by using Definition 5, the following theorem exists:
Theorem 4. Suppose , and is a decreasing functions. Then there exists such that
Proof. Suppose for , and , we defineSince , then and By the previous result, for every , we haveIt is easy to see that for every , we have the following estimate:and this complete the proof (QED).
At infinity, the Bessel–Riesz kernel disappears more quickly than the fractional integral operator does. By accomplishing this, we can demonstrate that the Bessel–Riesz kernel is a member of some Lebesgue spaces. First, the following lemma, which is helpful to demonstrate that belongs to some Lebesgue spaces.
Lemma 1. If , thenwhere
Letby simplifying
For every , and , (3) implies
Hence, we obtain
To prove that is the member of some Lebesgue spaces, we have the following theorem:
Theorem 5. If and , then we have for every ,
Proof. For every , we haveUsing Lemma 1, for , we define , if and only ifFrom the above definition, it easy to see thatWith the result of the membership of on Lebesgue spaces, we can prove the boundedness of on Lebesgue spaces by using Young inequality.
Theorem 6 (see [18]). Assume with
If and , then exists and belongs to . Moreover,
3.1. Boundedness of the Bessel–Riesz Operators on Morrey Spaces
This section will cover Bessel–Riesz operators on Morrey spaces in measure metric spaces and their boundedness. In the previous section, we have , where and the inclusion property of Morrey spaces, so . Accordingly, we have the following theorem.
Theorem 7. Let , , then we havefor every , where , , , and .
Suppose , , and take . Let , . For every , , where and . To estimate and , we use dyadic decomposition. Now, estimate :
By using Hölder’s inequality, we get
Hölder’s inequality is used again to estimate :
Next, we writeand we obtain
Summing the two estimates, we getfor each . Assume that is finite everywhere and is not a unique instance of 0. Make sure that . We get
Define and . For arbitrary , we have
Divide by and take supremum to get
Using the boundedness of on Morrey spaces (Chiarenza–Frascas theorem [2]), we obtain an inequality
Corollary 1. Suppose we have , and . If for some , , , and , then there exist such that
By the above corollary, we can say that is bounded from to . Moreover, norm of is dominated by norm of kernel Bessel-Riesz. For the boundedness of .
4. Concluding Remark
This paper has investigated the existence of Morrey space for the Bessel–Riesz operators that are dominated by the Bessel–Riesz kernel. The standard dyadic decomposition and maximal operators were used to further establish the boundedness of Bessel–Riesz operators. Additionally, Saba et al. [24], have shown that the generalized Bessel–Riesz operator is bounded. Future consideration will be an ongoing research direction that is to extend the study of Bessel–Riesz operators to more general settings such as weighted spaces, variable exponent spaces, and other metric measure spaces.
Data Availability
While the results of the research are being commercialized, the data that were used to support the findings of this study are currently under embargo. After the publication of this article, requests for data will be taken into consideration by the relevant author.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
As part of the Staff Exchange Program, Universitas Airlangga, a component of this study was finished in 2019 by the second author. Department of Mathematics, Faculty of Science and Technology, Universitas Airlangga, is funding this research. The research findings have been presented and published in ICOMCOS 2020 [19] and ICOWOBAS 2021 [20].