# Bessel-Riesz Operators on Lebesgue Spaces and Morrey Spaces Defined in Measure Metric Spaces

Authors: Saba Mehmood; Eridani; Fatmawati; Wasim Raza
Journal: International Journal of Differential Equations
 (2023)

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## 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.


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## Body

## 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 Table1 contains a list of notations used in this article.Table 1
The list of notations used in this paper.
ℝThe real numbersℝ+The positive real numbersℝnThe Euclidean spacesΣSigma-algebraμA measure defined on sigma-algebraδQuasimetric functionLpμThe class of Lebesgue measurable set1≤p<∞ϕFunctionℝ+⟶ℝ+Lϕpμ(Classical) Morrey space1≤p≤qXQuasimetric measure space with a complete measureμ and quasimetric function δBBa,r Centered at a∈X with radius r>0B˜Ba,2r Centered at a∈X with radius 2r>0Gα,γBessel–Riesz kernel0<α<γ,γ≥0Gα,γfxBessel–Riesz operators0<α<γ,γ≥0MqfxHardy–Littlewood maximal operator


## 2. Preliminaries

### 2.1. Measurable Spaces

Definition 1.
A distinguished collectionΣ of subsets of X is called sigma algebra if it satisfies the following axioms:(1)
IfA∈Σ, then Ac∈Σ,(2)
IfA1,A2,… are countable family of sets in Σ then ∪i=1∞Ai∈Σ,(3)
The intersection of countable sets also belongs toΣ i.e., ∩i=1∞Ai∈Σ.Definition 2.
A measureμ defined on sigma-algebraΣ, is a function μ:Σ⟶ℝ+ such that(1)
μ∅=0, where ∅ is an empty set.(2)
IfA1,A2,… is a sequence of disjoint sets in Σ, then(1)μ∪i=1∞Ai=∑n=1∞μAi.
IfΣ is σ-algebra of subsets of X, then the pair X,Σ is said to be a measurable space and members of Σ are called Σ-measurable sets. Moreover, if X,Σ is a measurable space, and if μ is a measure on Σ, then the triplet X,Σ,μ is said to be a measure space.Definition 3 (see [8]).
We assume thatX≔X,δ,μ is a topological space, endowed with a complete measure μ such that the space of compactly supported continuous functions is dense in L1X,μ and there exists a function (quasimetric) δ:X×X⟶0,∞ satisfying the conditions:(i)
δx,y>0 for all x≠y, and δx,x=0 for all x∈X;(ii)
There exists a constanta0≥1, such that δx,y≤a0δy,x for all x,y∈X;(iii)
There exists a constanta1≤1, such that δx,y≤a1δx,z+δz,y for all x,y,z∈X.
We assume that the ballsBa,r≔x∈X:δa,x<r are μ-measurable and 0<μBa,r<∞ for a∈X,r>0. For every neighborhood V of x∈X, there exists r>0 such that Bx,r⊂V. We also assume that μX=∞,μa=0 and Ba,r2/Ba,r1≠∅ for all a∈X,0<r1<r2<∞. The triple X,δ,μ will be called quasimetric measure space.Definition 4 (see [20]).
In measure metric spacesX,δ,μ, we will have the properties μB∼rβ for every balls B≔Ba,r. For 0<α<β,γ>0, we define(2)Gα,γ:=δx,yα−β1+δx,yγ.
For anyx in X, the function Gα,γ may be regarded as the product of two kernels, Jγ be a Bessel kernel, and Gα be a Riesz kernel. We define Lpμ as any measurable functions f such that 1≤p<∞.(3)f:Lpμ=∫Xftpdμt1/p<∞.
Ifγ=0, then we have Gα 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 Gα boundedness results is correct. Spanne [2] has demonstrated the boundedness of Iα on Morrey spaces. Additionally, Adams [6] and Chiarenza [12] derived a stronger conclusion about the boundedness of Riesz operators. Using the multiplication operator W, Kurata et al. [1] demonstrated that the Gα,γfx function is confined on generalized Morrey spaces. The boundedness of Gα,γfx 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 a∈X, and r>0, then(4)∫XGα,γdμx=∑k∈ℤ∫2kr≤δa,x<2k+1rGα,γdμx∼∑k∈ℤ2krα1+2krγ.
We know thatGα,γ∈L1μ, provided that 0<α<γ.
Now, we introduce the following operator:(5)Gα,γfx:=∫Xδx,yα−1fy1+δx,yγdμy,x∈X.


### 2.2. Hardy–Littlewood Maximal Operator

TheMq operator which includes Hardy–Littlewood maximal operator defined as follows [11]:(6)Mqfx:=supx∈B1μB∫Bfyqdμy1/q.A typical outcome forMq is that it is constrained by Lpμ for 1≤q<p≤∞.Definition 5.
Morrey Spaces [8] For 1≤p<∞ and an appropriate ϕ:ℝ+⟶ℝ+. In order to define the Morrey space, we use the formula Lϕpμ=LϕpX,μ is the set of all functions, where f∈LlocpX for(7)f:Lϕpμ:=supB1ϕB1μB∫Bfypdμy1/p<∞.
Here, we defineϕB≔ϕμB.


## 2.1. Measurable Spaces

Definition 1.
A distinguished collectionΣ of subsets of X is called sigma algebra if it satisfies the following axioms:(1)
IfA∈Σ, then Ac∈Σ,(2)
IfA1,A2,… are countable family of sets in Σ then ∪i=1∞Ai∈Σ,(3)
The intersection of countable sets also belongs toΣ i.e., ∩i=1∞Ai∈Σ.Definition 2.
A measureμ defined on sigma-algebraΣ, is a function μ:Σ⟶ℝ+ such that(1)
μ∅=0, where ∅ is an empty set.(2)
IfA1,A2,… is a sequence of disjoint sets in Σ, then(1)μ∪i=1∞Ai=∑n=1∞μAi.
IfΣ is σ-algebra of subsets of X, then the pair X,Σ is said to be a measurable space and members of Σ are called Σ-measurable sets. Moreover, if X,Σ is a measurable space, and if μ is a measure on Σ, then the triplet X,Σ,μ is said to be a measure space.Definition 3 (see [8]).
We assume thatX≔X,δ,μ is a topological space, endowed with a complete measure μ such that the space of compactly supported continuous functions is dense in L1X,μ and there exists a function (quasimetric) δ:X×X⟶0,∞ satisfying the conditions:(i)
δx,y>0 for all x≠y, and δx,x=0 for all x∈X;(ii)
There exists a constanta0≥1, such that δx,y≤a0δy,x for all x,y∈X;(iii)
There exists a constanta1≤1, such that δx,y≤a1δx,z+δz,y for all x,y,z∈X.
We assume that the ballsBa,r≔x∈X:δa,x<r are μ-measurable and 0<μBa,r<∞ for a∈X,r>0. For every neighborhood V of x∈X, there exists r>0 such that Bx,r⊂V. We also assume that μX=∞,μa=0 and Ba,r2/Ba,r1≠∅ for all a∈X,0<r1<r2<∞. The triple X,δ,μ will be called quasimetric measure space.Definition 4 (see [20]).
In measure metric spacesX,δ,μ, we will have the properties μB∼rβ for every balls B≔Ba,r. For 0<α<β,γ>0, we define(2)Gα,γ:=δx,yα−β1+δx,yγ.
For anyx in X, the function Gα,γ may be regarded as the product of two kernels, Jγ be a Bessel kernel, and Gα be a Riesz kernel. We define Lpμ as any measurable functions f such that 1≤p<∞.(3)f:Lpμ=∫Xftpdμt1/p<∞.
Ifγ=0, then we have Gα 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 Gα boundedness results is correct. Spanne [2] has demonstrated the boundedness of Iα on Morrey spaces. Additionally, Adams [6] and Chiarenza [12] derived a stronger conclusion about the boundedness of Riesz operators. Using the multiplication operator W, Kurata et al. [1] demonstrated that the Gα,γfx function is confined on generalized Morrey spaces. The boundedness of Gα,γfx 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 a∈X, and r>0, then(4)∫XGα,γdμx=∑k∈ℤ∫2kr≤δa,x<2k+1rGα,γdμx∼∑k∈ℤ2krα1+2krγ.
We know thatGα,γ∈L1μ, provided that 0<α<γ.
Now, we introduce the following operator:(5)Gα,γfx:=∫Xδx,yα−1fy1+δx,yγdμy,x∈X.


## 2.2. Hardy–Littlewood Maximal Operator

TheMq operator which includes Hardy–Littlewood maximal operator defined as follows [11]:(6)Mqfx:=supx∈B1μB∫Bfyqdμy1/q.A typical outcome forMq is that it is constrained by Lpμ for 1≤q<p≤∞.Definition 5.
Morrey Spaces [8] For 1≤p<∞ and an appropriate ϕ:ℝ+⟶ℝ+. In order to define the Morrey space, we use the formula Lϕpμ=LϕpX,μ is the set of all functions, where f∈LlocpX for(7)f:Lϕpμ:=supB1ϕB1μB∫Bfypdμy1/p<∞.
Here, we defineϕB≔ϕμB.


## 3. Main Results

Theorem 1.
For1<p<∞,0<α<min1,γ, there exists C>0 such that(8)Gα,γfx:Lpμ≤CGα,γ:L1μ⋅f:Lpμ.Proof.
Supposef≥0, and for every r>0,x∈X, we have(9)Gα,γfx=∫δx,y<r+∫δx,y≥rδx,yα−1fy1+δx,yγdμy=:I1x,r+I2x,r.
It is easy to see that the following estimates is holds. Then we have(10)I1x,r=∫δx,y<rδx,yα−1fy1+δx,yγdμy=∑k=−∞−1∫2kr≤δx,y<2k+1rδx,yα−1fy1+δx,yγdμy∼∑k=−∞−12krα−11+2krγ∫Bx,2k+1rfydμy≤CM1fx⋅Gα,γ:L1μ,I2x,r=∑k=0∞∫2kr≤δx,y<2k+1rδx,yα−1fy1+δx,yγdμy∼∑k=0∞2krα−11+2krγ∫Bx,2k+1rfydμy≤Cf:Lp∑k=0∞2krα2k+1r−1/p1+2krγ≤Cr−1/pGα,γ:L1μ⋅f:Lpμ.
So, for everyx∈X, and r>0, we have(11)Gα,γfx≤CM1fx+r−1/pf:LpμGα,γ:L1μ.
If we chooser>0 such that M1fx=r−1/pf:Lpμ, then(12)Gα,γfx≤CM1fx⋅Gα,γ:L1μ,and this finish the proof. (QED).
Next, by using Definition5 of Morrey spaces, we haveTheorem 2.
Suppose1<p<∞,0<α<min1,γ, and ϕ is a decreasing functions. Then there exists C>0 such that(13)Gα,γfx:Lϕpμ≤CGα,γ:L1μ⋅f:Lϕpμ.Proof.
Suppose forB≔Ba,r, and B˜≔Ba,2r, we define(14)f=f1+f2≔fχB˜+fχB˜c,f∈Lϕpμ.
Sincef1:Lpμ≤ϕB˜μB˜1/pf:Lϕpμ<∞, then f1∈Lpμ. By the previous result, for every B, we have(15)∫BGα,γf1xypdμy1/p≤Gα,γf1x:Lpμ≤CGα,γ:L1μ⋅f:Lpμ≤CϕBμB1/pGα,γ:L1μ⋅f:Lϕpμ.
It is easy to see that for everyx∈Ba,r, we have the following estimate:(16)Gα,γf2x≤∑k=1∞∫2kr≤δx,y<2k+1rδx,yα−1fy1+δx,yγdμy∼∑k=1∞2krα−11+2krγ∫Bx,2k+1rfydμy≤Cf:Lϕpμ∑k=1∞2krαϕ2k+1r1+2krγ≤Cϕrf:Lϕpμ⋅Gα,γ:L1μ,and this complete the proof. (QED).
Suppose0<α<1, and 1<p<∞, for every r>0, we have(17)∫XGα,γxpdμx∼∑k∈ℤ2kr1+pα−11+2krpγ,and Gα,γ∈Lpμ, provided that 1<p<1/1−α.Theorem 3.
For1<p<∞,0<α<min1,γ, there exists C>0 such that(18)Gα,γfx:Lpμ≤CGα,γ:Lpμ⋅f:Lpμ.Proof.
Supposef≥0, and for every r>0,x∈X, Gα,γ∈Lpμ we have(19)Gα,γfx=∫δx,y<r+∫δx,y≥rδx,y1+pα−1fy1+δx,yγpdμy=:I1x,r+I2x,r.
It is easy to see that the following estimates is holds. That is, we have(20)I1x,r=∫δx,y<rδx,y1+pα−1fy1+δx,yγpdμy=∑k=−∞−1∫2kr≤δx,y<2k+1rδx,y1+pα−1fy1+δx,yγpdμy∼∑k=−∞−12kr1+pα−11+2krpγ∫Bx,2k+1rfydμy≤CM1fx⋅Gα,γ:Lpμ,I2x,r=∑k=0∞∫2kr≤δx,y<2k+1rδx,y1+pα−1fy1+δx,yγpdμy∼∑k=0∞2kr1+pα−11+2krpγ∫Bx,2k+1rfydμy≤Cf:Lp∑k=0∞2krα1+p2k+1r−1/p1+2krpγ≤Cr−1/pGα,γ:Lpμ⋅f:Lpμ.
So, for everyx∈X, and r>0, we have(21)Gα,γfx≤CM1fx+r−1/pf:LpμGα,γ:Lpμ.
If we chooser>0 such that M1fx=r−1/pf:Lpμ, then(22)Gα,γfx≤CM1fx⋅Gα,γ:Lpμ,and this finish the proof. (QED).
Next, by using Definition5, the following theorem exists:Theorem 4.
Suppose1<p<∞,0<α<min1,γ, and ϕ is a decreasing functions. Then there exists C>0 such that(23)Gα,γfx:Lϕpμ≤CGα,γ:Lpμ⋅f:Lϕpμ.Proof.
Suppose forB≔Ba,r, and B˜≔Ba,2r, we define(24)f=f1+f2≔fχB˜+fχB˜c,f∈Lϕpμ.
Sincef1:Lpμ≤ϕB˜μB˜1/pf:Lϕpμ<∞, then f1∈Lpμ and Gα,γ∈Lpμ By the previous result, for every B, we have(25)∫BGα,γf1xpdμy1/p≤Gα,γf1x:Lpμ≤CGα,γ:Lpμ⋅f1:Lpμ≤CϕBμB1/pGα,γ:Lpμ⋅f:Lϕpμ.
It is easy to see that for everyx∈Ba,r, we have the following estimate:(26)Gα,γf2x≤∑k=1∞∫2kr≤δx,y<2k+1rδx,y1+pα−1fy1+δx,yγpdμy∼∑k=1∞2kr1+pα−11+2krpγ∫Bx,2k+1rfydμy≤Cf:Lϕpμ∑k=1∞2krα1+pϕ2k+1r1+2krpγ≤Cϕrf:Lϕpμ⋅Gα,γ:Lpμ,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 thatGα,γ belongs to some Lebesgue spaces.Lemma 1.
IfμBa,r∼rβ, then(27)∫XGα,γx,ydμy<∞,where(28)Gα,γx,y=δx,yα−β1+δx,yγ.Let(29)∫XGα,γx,ydμy=∫δx,y<rGα,γx,ydμy+∫δx,y≥RGα,γx,ydμy,by simplifying(30)∫δx,y<rGα,γx,ydμy=∑k=−∞−1∫2kR≤δx,y<2k+1RGα,γx,ydμy∼∑k=−∞−12kRα−β∫δx,y<2k+1RGα,γx,ydμy≤C1∑k=−∞−12kRα−βμβ2k+1Rx=C1∑k=−∞−12kRα≤C1Rα<∞,∫δx,y≥RGα,γx,ydμy=∑k=0∞∫2kR≤δx,y<2k+1RGα,γx,ydμy=C2∑k=0∞2k+1Rα−β1+2k+1Rγ≤C2∑k=0∞2kRα−β2kRγ=C2∑k=0∞2kRα−β−γ≤C2Rα−β−γ<∞.For everyx∈X, and R>0, (3) implies(31)∫XGα,γx,ydμy=CRα+Rα−β−γ.Hence, we obtain(32)∫XGα,γx,ydμy<∞∀x∈X.To prove thatGα,γ is the member of some Lebesgue spaces, we have the following theorem:Theorem 5.
If0<α<β and γ∈0,∞, then we have for every R>0,(33)Gα,γ:Lpp∼∑k∈Z2kRα−βp+β2kRγp,1≤ββ+γ−α<p<ββ−α.Proof.
For everyR>0, we have(34)∫XGα,γx,ypdμy=∫0≤δx,yGα,γx,ypdμy=∑k∈Z∫2kR≤δx,y<2k+1Rδx,yα−βp1+δx,yγpdμy∼∑k∈Z2kRα−βp1+2kRγp∫2kR≤δx,y<2k+1Rdμy∼∑k∈Z2kRα−βp+β1+2kRγp.
Using Lemma1, for 1≤p<∞, we define Gα,γ∈Lpμ, if and only if(35)Gα,γ:Lp=∫ℝnGα,γx,ypdμy1/p<∞.
From the above definition, it easy to see that(36)Gα,γ:Lp<∞⇔1≤ββ+γ−α<p<ββ−α.
With the result of the membership ofGα,γ on Lebesgue spaces, we can prove the boundedness of Gα,γfx on Lebesgue spaces by using Young inequality.Theorem 6 (see [18]).
Assumep,q,r∈1,∞ with(37)1r=1p+1q−1.Iff∈Lpℝn and g∈Lqℝn, then f∗g exists and belongs to Lrℝn. Moreover,(38)f∗gr≤fpgq.


### 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 haveGα,γ∈LpX, where β/β+γ−α<p<β/β−α and the inclusion property of Morrey spaces, so Gα,γ∈Lϕpμ. Accordingly, we have the following theorem.Theorem 7.
Let0<α<β, 0<γ, then we have(39)Gα,γfxLp2,q2≤Cp1,q1Gα,γLϕpμfLp1,q1,for every f∈Lp1,q1X, where 1<p1<q1<p, 1≤s≤p, β/β+γ−α<p<β/β−α, 1/p2=1/p1−q1/p1p´ and 1/q2=1/q1−1/p´.Suppose0<α<β, 0<γ, and take β/β+γ−α<p<β/β−α,1≤s≤p. Let f∈Lp1,q1X, 1<p1<q1<p´. For every x∈X , Gα,γfx≔I1x+I2x, where I1x≔∫δx,y<rδx,yα−βfy/1+δx,yγdμy and I2x≔∫δx,y≥Rδx,yα−βfy/1+δx,yγdμy. To estimate I1 and I2, we use dyadic decomposition. Now, estimate I1:(40)I1x≤C1∑k=−1−∞2kRα−β1+2kRγ∫2kR≤δx,y<2k+1Rfydμy≤C2Mqfx∑k=−1−∞2kRα−β+β/s2kRβ/s´1+2kRγ.By using Hölder’s inequality, we get(41)I1x≤C3Mqfx∑k=−1−∞2kRα−βs+β1+2kRγs1/s∑k=−1−∞2kRβ1/s´≤C4Mqfx∫δx,y<rGα,γsδx,ydμy1/sRβ1/s−1/t≤C4Gα,γLϕpμMqfxRβ/p´.Hölder’s inequality is used again to estimateI2:(42)I2x≤C5∑k=0∞2kRα−β1+2kRγ∫2kR≤δx,y<2k+1Rfydμy≤C5∑k=0∞2kRα−β1+2kRγ∫2kR≤δx,y<2k+1Rfyp1dμy1/p12kRβ/p1´.Next, we write(43)I2x≤C6fLp1,q1∑k=0∞2kRα−β+β−β/q11+2kRγ∫2kR≤δx,y<2k+1Rdμy1/s2kRβ/s≤C6fLp1,q1∑k=0∞∫2kR≤δx,y<2k+1Rδx,yα−βfy/1+δx,yγdμy1/s2kRβ/q1−β2kRβ/s,and we obtain(44)I2x≤C6fLp1,q1Gα,γLϕpμ∑k=0∞2kRβ/p´−β/q1≤C7fLp1,q1Gα,γLϕpμRβ1/p´−1/q1.Summing the two estimates, we get(45)Gα,γfx≤CGα,γLϕpμMqfxRβ/p´+fLp1,q1Rβ/p´−β/q1,for each x∈X. Assume that Mpf is finite everywhere and f is not a unique instance of 0. Make sure that Rβ/q1=fLp1,q1/Mqfx. We get(46)Gα,γfx≤CGα,γLϕpμfLp1,q1q1/p´Mqfx1−q1/p´.Define1/p2=1/p1−q1/p1p´ and 1/q2=1/q1−1/p´. For arbitrary r>0, we have(47)∫δx,y<rGα,γfxp2dμx1/p2≤CGα,γLϕpμfLp1,q11−p1/p2∫δx,y<r|Mqfxp1dμx1/p2.Divide byrβ/p2−β/q2 and take supremum to get(48)Gα,γfxLp2,q2=supr>0∫δx,y<rGα,γfxp2dμx1/p2srβ/p2−β/q2≤CGα,γLϕpμfLp1,q11−p1/p2∫δx,y<r|Mqfxp1dμx1/p2rβ/p2−β/q2=CGα,γLϕpμfLp1,q11−p1/p2MpfLp1,q1p1/p2.Using the boundedness ofM on Morrey spaces (Chiarenza–Frascas theorem [2]), we obtain an inequality(49)Gα,γfxLp2,q2≤CGα,γLϕpμfLp1,q11−p1/p2.Corollary 1.
Suppose we havep1=q1,p2=q2, and s=p1≤β/β+γ−α<p<β/β−α. If for some C1>0, μBa,r≤C1rβ, f∈Lp1μ, and Gα,γ∈Lsμ, then there exist C2>0 such that(50)Gα,γfxf:Lq1μ≤C2Gα,γx,y:Lsμ.f:Lp1μs.By the above corollary, we can say thatGα,γfx is bounded from Lp1μ to Lq1μ. Moreover, norm of Gα,γfx is dominated by norm of kernel Bessel-Riesz. For the boundedness of Gα,γfx.


## 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 haveGα,γ∈LpX, where β/β+γ−α<p<β/β−α and the inclusion property of Morrey spaces, so Gα,γ∈Lϕpμ. Accordingly, we have the following theorem.Theorem 7.
Let0<α<β, 0<γ, then we have(39)Gα,γfxLp2,q2≤Cp1,q1Gα,γLϕpμfLp1,q1,for every f∈Lp1,q1X, where 1<p1<q1<p, 1≤s≤p, β/β+γ−α<p<β/β−α, 1/p2=1/p1−q1/p1p´ and 1/q2=1/q1−1/p´.Suppose0<α<β, 0<γ, and take β/β+γ−α<p<β/β−α,1≤s≤p. Let f∈Lp1,q1X, 1<p1<q1<p´. For every x∈X , Gα,γfx≔I1x+I2x, where I1x≔∫δx,y<rδx,yα−βfy/1+δx,yγdμy and I2x≔∫δx,y≥Rδx,yα−βfy/1+δx,yγdμy. To estimate I1 and I2, we use dyadic decomposition. Now, estimate I1:(40)I1x≤C1∑k=−1−∞2kRα−β1+2kRγ∫2kR≤δx,y<2k+1Rfydμy≤C2Mqfx∑k=−1−∞2kRα−β+β/s2kRβ/s´1+2kRγ.By using Hölder’s inequality, we get(41)I1x≤C3Mqfx∑k=−1−∞2kRα−βs+β1+2kRγs1/s∑k=−1−∞2kRβ1/s´≤C4Mqfx∫δx,y<rGα,γsδx,ydμy1/sRβ1/s−1/t≤C4Gα,γLϕpμMqfxRβ/p´.Hölder’s inequality is used again to estimateI2:(42)I2x≤C5∑k=0∞2kRα−β1+2kRγ∫2kR≤δx,y<2k+1Rfydμy≤C5∑k=0∞2kRα−β1+2kRγ∫2kR≤δx,y<2k+1Rfyp1dμy1/p12kRβ/p1´.Next, we write(43)I2x≤C6fLp1,q1∑k=0∞2kRα−β+β−β/q11+2kRγ∫2kR≤δx,y<2k+1Rdμy1/s2kRβ/s≤C6fLp1,q1∑k=0∞∫2kR≤δx,y<2k+1Rδx,yα−βfy/1+δx,yγdμy1/s2kRβ/q1−β2kRβ/s,and we obtain(44)I2x≤C6fLp1,q1Gα,γLϕpμ∑k=0∞2kRβ/p´−β/q1≤C7fLp1,q1Gα,γLϕpμRβ1/p´−1/q1.Summing the two estimates, we get(45)Gα,γfx≤CGα,γLϕpμMqfxRβ/p´+fLp1,q1Rβ/p´−β/q1,for each x∈X. Assume that Mpf is finite everywhere and f is not a unique instance of 0. Make sure that Rβ/q1=fLp1,q1/Mqfx. We get(46)Gα,γfx≤CGα,γLϕpμfLp1,q1q1/p´Mqfx1−q1/p´.Define1/p2=1/p1−q1/p1p´ and 1/q2=1/q1−1/p´. For arbitrary r>0, we have(47)∫δx,y<rGα,γfxp2dμx1/p2≤CGα,γLϕpμfLp1,q11−p1/p2∫δx,y<r|Mqfxp1dμx1/p2.Divide byrβ/p2−β/q2 and take supremum to get(48)Gα,γfxLp2,q2=supr>0∫δx,y<rGα,γfxp2dμx1/p2srβ/p2−β/q2≤CGα,γLϕpμfLp1,q11−p1/p2∫δx,y<r|Mqfxp1dμx1/p2rβ/p2−β/q2=CGα,γLϕpμfLp1,q11−p1/p2MpfLp1,q1p1/p2.Using the boundedness ofM on Morrey spaces (Chiarenza–Frascas theorem [2]), we obtain an inequality(49)Gα,γfxLp2,q2≤CGα,γLϕpμfLp1,q11−p1/p2.Corollary 1.
Suppose we havep1=q1,p2=q2, and s=p1≤β/β+γ−α<p<β/β−α. If for some C1>0, μBa,r≤C1rβ, f∈Lp1μ, and Gα,γ∈Lsμ, then there exist C2>0 such that(50)Gα,γfxf:Lq1μ≤C2Gα,γx,y:Lsμ.f:Lp1μs.By the above corollary, we can say thatGα,γfx is bounded from Lp1μ to Lq1μ. Moreover, norm of Gα,γfx is dominated by norm of kernel Bessel-Riesz. For the boundedness of Gα,γfx.


## 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.



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