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Abstract and Applied Analysis
VolumeΒ 2011Β (2011), Article IDΒ 908491, 16 pages
http://dx.doi.org/10.1155/2011/908491
Research Article

On the Convolution Equation Related to the Diamond Klein-Gordon Operator

1Department of Mathematics, Khon Kaen University, Khon Kaen 40002, Thailand
2Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok 10400, Thailand

Received 11 July 2011; Revised 3 August 2011; Accepted 31 August 2011

Academic Editor: ElenaΒ Braverman

Copyright Β© 2011 Amphon Liangprom and Kamsing Nonlaopon. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We study the distribution 𝑒𝛼π‘₯(β‹„+π‘š2)π‘˜π›Ώ for π‘šβ‰₯0, where (β‹„+π‘š2)π‘˜ is the diamond Klein-Gordon operator iterated π‘˜ times, 𝛿 is the Dirac delta distribution, π‘₯=(π‘₯1,π‘₯2,…,π‘₯𝑛) is a variable in ℝ𝑛, and 𝛼=(𝛼1,𝛼2,…,𝛼𝑛) is a constant. In particular, we study the application of 𝑒𝛼π‘₯(β‹„+π‘š2)π‘˜π›Ώ for solving the solution of some convolution equation. We find that the types of solution of such convolution equation, such as the ordinary function and the singular distribution, depend on the relationship between π‘˜ and 𝑀.

1. Introduction

The 𝑛-dimensional ultrahyperbolic operator β–‘π‘˜ iterated π‘˜ times is defined byβ–‘π‘˜=ξƒ©πœ•2πœ•π‘₯21+πœ•2πœ•π‘₯22πœ•+β‹―+2πœ•π‘₯2π‘βˆ’πœ•2πœ•π‘₯2𝑝+1βˆ’πœ•2πœ•π‘₯2𝑝+2πœ•βˆ’β‹―βˆ’2πœ•π‘₯2𝑝+π‘žξƒͺπ‘˜,(1.1) where 𝑝+π‘ž=𝑛 is the dimension of ℝ𝑛, and π‘˜ is a nonnegative integer. We consider the linear differential equation of the formβ–‘π‘˜π‘’(π‘₯)=𝑓(π‘₯),(1.2) where 𝑒(π‘₯) and 𝑓(π‘₯) are generalized functions, and π‘₯=(π‘₯1,π‘₯2,…,π‘₯𝑛)βˆˆβ„π‘›.

Gelfand and Shilov [1] have first introduced the fundamental solution of (1.2), which was initially complicated. Later, Trione [2] has shown that the generalized function 𝑅𝐻2π‘˜(π‘₯) defined by (2.2) with 𝛾=2π‘˜ is the unique fundamental solution of (1.2). Tellez [3] has also proved that 𝑅𝐻2π‘˜(π‘₯) exists only when 𝑛=𝑝+π‘ž with odd 𝑝.

Kananthai [4] has first introduced the operator β‹„π‘˜ called the diamond operator iterated π‘˜ times, which is defined byβ‹„π‘˜=βŽ‘βŽ’βŽ’βŽ£ξƒ©π‘ξ“π‘–=1πœ•2πœ•π‘₯2𝑖ξƒͺ2βˆ’ξƒ©π‘+π‘žξ“π‘—=𝑝+1πœ•2πœ•π‘₯2𝑗ξƒͺ2⎀βŽ₯βŽ₯βŽ¦π‘˜,(1.3) where 𝑛=𝑝+π‘ž is the dimension of ℝ𝑛, for all π‘₯=(π‘₯1,π‘₯2,…,π‘₯𝑛)βˆˆβ„π‘› and nonnegative integers π‘˜. The operator β‹„π‘˜ can be expressed in the formβ‹„π‘˜=β–΅π‘˜β–‘π‘˜=β–‘π‘˜β–΅π‘˜,(1.4) where β–‘π‘˜ is defined by (1.1), and β–΅π‘˜ is the Laplace operator iterated π‘˜ times defined byβ–΅π‘˜=ξƒ©πœ•2πœ•π‘₯21+πœ•2πœ•π‘₯22πœ•+β‹―+2πœ•π‘₯2𝑛ξƒͺπ‘˜.(1.5) Note that in case π‘˜=1, the generalized form of (1.5) is called the local fractional Laplace operator; see [5] for more details. On finding the fundamental solution of this product, he uses the convolution of functions which are fundamental solutions of the operators β–‘π‘˜ and β–΅π‘˜. He found that the convolution (βˆ’1)π‘˜π‘…π‘’2π‘˜(π‘₯)βˆ—π‘…π»2π‘˜(π‘₯) is the fundamental solution of the operator β‹„π‘˜, that is,β‹„π‘˜ξ€·(βˆ’1)π‘˜π‘…π‘’2π‘˜(π‘₯)βˆ—π‘…π»2π‘˜ξ€Έ(π‘₯)=𝛿,(1.6) where 𝑅𝐻2π‘˜(π‘₯) and 𝑅𝑒2π‘˜(π‘₯) are defined by (2.2) and (2.7), respectively (with 𝛾=2π‘˜), and 𝛿 is the Dirac-delta distribution. The fundamental solution (βˆ’1)π‘˜π‘…π‘’2π‘˜(π‘₯)βˆ—π‘…π»2π‘˜(π‘₯) is called the diamond kernel of Marcel Riesz. A number of effective results on the diamond kernel of Marcel Riesz have been presented by Kananthai [6–12].

In 1997, Kananthai [13] has studied the properties of the distribution 𝑒𝛼π‘₯β–‘π‘˜π›Ώ and the application of the distribution 𝑒𝛼π‘₯β–‘π‘˜π›Ώ for finding the fundamental solution of the ultrahyperbolic equation by using the convolution method. Later in 1998, he has also studied the properties of the distribution 𝑒𝛼π‘₯β‹„π‘˜π›Ώ and its application for solving the convolution equation𝑒𝛼π‘₯β‹„π‘˜π›Ώβˆ—π‘’(π‘₯)=π‘’π‘šπ›Όπ‘₯ξ“π‘Ÿ=0πΆπ‘Ÿβ‹„π‘Ÿπ›Ώ.(1.7) Recently, Nonlaopon gave some generalizations of his paper [6]; see [14] for more details.

In 2000, Kananthai [15] has studied the application of the distribution 𝑒𝛼π‘₯β–‘π‘˜π›Ώ for solving the convolution equation𝑒𝛼π‘₯β–‘π‘˜π›Ώβˆ—π‘’(π‘₯)=π‘’π‘šπ›Όπ‘₯ξ“π‘Ÿ=0πΆπ‘Ÿβ–‘π‘Ÿπ›Ώ,(1.8) which is related to the ultrahyperbolic equation.

In 2009, Sasopa and Nonlaopon [16] have studied the properties of the distribution 𝑒𝛼π‘₯β–‘π‘˜π‘π›Ώ and its application to solve the convolution equation𝑒𝛼π‘₯β–‘π‘˜π‘π›Ώβˆ—π‘’(π‘₯)=π‘’π‘šπ›Όπ‘₯ξ“π‘Ÿ=0πΆπ‘Ÿβ–‘π‘Ÿπ‘π›Ώ.(1.9) Here, β–‘π‘˜π‘ is the operator related to the ultrahyperbolic type operator iterated π‘˜ times, which is defined byβ–‘π‘˜π‘=1𝑐2𝑝𝑖=1πœ•2πœ•π‘₯2π‘–βˆ’π‘+π‘žξ“π‘—=𝑝+1πœ•2πœ•π‘₯2𝑗ξƒͺπ‘˜,(1.10) where 𝑝+π‘ž=𝑛 is the dimension of ℝ𝑛.

In 1988, Trione [17] has studied the fundamental solution of the ultrahyperbolic Klein-Gordon operator iterated π‘˜ times, which is defined byξ€·β–‘+π‘š2ξ€Έπ‘˜=𝑝𝑖=1πœ•2πœ•π‘₯2π‘–βˆ’π‘+π‘žξ“π‘—=𝑝+1πœ•2πœ•π‘₯2𝑗+π‘š2ξƒͺπ‘˜.(1.11) The fundamental solution of the operator (β–‘+π‘š2)π‘˜ is given byπ‘Š2π‘˜(π‘₯,π‘š)=βˆžξ“π‘Ÿ=0(βˆ’1)π‘ŸΞ“(π‘˜+π‘Ÿ)ξ€·π‘šπ‘Ÿ!Ξ“(π‘˜)2ξ€Έπ‘Ÿ(βˆ’1)π‘Ÿπ‘…π»2π‘˜+2π‘Ÿ(π‘₯),(1.12) where 𝑅𝐻2π‘˜+2π‘Ÿ(π‘₯) is defined by (2.2) with 𝛾=2π‘˜+2π‘Ÿ. Next, Tellez [18] has studied the convolution product of π‘Šπ›Ό(π‘₯,π‘š)βˆ—π‘Šπ›½(π‘₯,π‘š), where 𝛼 and 𝛽 are any complex parameter. In addition, Trione [19] has studied the fundamental (𝑃±𝑖0)πœ†-ultrahyperbolic solution of the Klein-Gordon operator iterated π‘˜ times and the convolution of such fundamental solution.

Liangprom and Nonlaopon [20] have studied the properties of the distribution 𝑒𝛼π‘₯(β–‘+π‘š2)π‘˜π›Ώ and its application for solving the convolution equation𝑒𝛼π‘₯ξ€·β–‘+π‘š2ξ€Έπ‘˜π›Ώβˆ—π‘’(π‘₯)=𝑒𝑀𝛼π‘₯ξ“π‘Ÿ=0πΆπ‘Ÿξ€·β–‘+π‘š2ξ€Έπ‘Ÿπ›Ώ,(1.13) where (β–‘+π‘š2)π‘˜ is defined by (1.11).

In 2007, Tariboon and Kananthai [21] have introduced the operator (β‹„+π‘š2)π‘˜ called diamond Klein-Gordon operator iterated π‘˜ times, which is defined byξ€·β‹„+π‘š2ξ€Έπ‘˜=βŽ‘βŽ’βŽ’βŽ£ξƒ©π‘ξ“π‘–=1πœ•2πœ•π‘₯2𝑖ξƒͺ2βˆ’ξƒ©π‘+π‘žξ“π‘—=𝑝+1πœ•2πœ•π‘₯2𝑗ξƒͺ2+π‘š2⎀βŽ₯βŽ₯βŽ¦π‘˜,(1.14) where 𝑝+π‘ž=𝑛 is the dimension of ℝ𝑛, for all π‘₯=(π‘₯1,π‘₯2,…,π‘₯𝑛)βˆˆβ„π‘›,π‘šβ‰₯0 and nonnegative integers π‘˜. Later, Lunnaree and Nonlaopon [22, 23] have studied the fundamental solution of operator (β‹„+π‘š2)π‘˜, and this fundamental solution is called the diamond Klein-Gordon kernel. They have also studied the Fourier transform of the diamond Klein-Gordon kernel and its convolution.

In this paper, we aim to study the properties of the distribution 𝑒𝛼π‘₯(β‹„+π‘š2)π‘˜π›Ώ and the application of 𝑒𝛼π‘₯(β‹„+π‘š2)π‘˜π›Ώ for solving the convolution equation𝑒𝛼π‘₯ξ€·β‹„+π‘š2ξ€Έπ‘˜π›Ώβˆ—π‘’(π‘₯)=𝑒𝑀𝛼π‘₯ξ“π‘Ÿ=0πΆπ‘Ÿξ€·β‹„+π‘š2ξ€Έπ‘Ÿπ›Ώ,(1.15) where (β‹„+π‘š2)π‘˜ is defined by (1.14), 𝑒(π‘₯) is the generalized function, and πΆπ‘Ÿ is a constant. On finding the type of solution 𝑒(π‘₯) of (1.15), we use the method of convolution of the generalized functions.

Before we proceed to that point, the following definitions and concepts require clarifications.

2. Preliminaries

Definition 2.1. Let π‘₯=(π‘₯1,π‘₯2,…,π‘₯𝑛) be a point of the 𝑛-dimensional Euclidean space ℝ𝑛. Let 𝑒=π‘₯21+π‘₯22+β‹―+π‘₯2π‘βˆ’π‘₯2𝑝+1βˆ’π‘₯2𝑝+2βˆ’β‹―βˆ’π‘₯2𝑝+π‘ž(2.1) be the nondegenerated quadratic form, where 𝑝+π‘ž=𝑛 is the dimension of ℝ𝑛. Let Ξ“+={π‘₯βˆˆβ„π‘›βˆΆπ‘₯1>0and𝑒>0} be the interior of a forward cone, and let Ξ“+ denote its closure. For any complex number 𝛾, we define the function π‘…π»π›ΎβŽ§βŽͺ⎨βŽͺβŽ©π‘’(π‘₯)=(π›Ύβˆ’π‘›)/2𝐾𝑛(𝛾),forπ‘₯βˆˆΞ“+,0,forπ‘₯βˆ‰Ξ“+,(2.2) where the constant 𝐾𝑛(𝛾) is given by πΎπ‘›πœ‹(𝛾)=(π‘›βˆ’1)/2Ξ“((2+π›Ύβˆ’π‘›)/2)Ξ“((1βˆ’π›Ύ)/2)Ξ“(𝛾).Ξ“((2+π›Ύβˆ’π‘)/2)Ξ“((π‘βˆ’π›Ύ)/2)(2.3)

The function 𝑅𝐻𝛾(π‘₯) is called the ultrahyperbolic kernel of Marcel Riesz, which was introduced by Nozaki [24]. It is well known that 𝑅𝐻𝛾(π‘₯) is an ordinary function if Re(𝛾)β‰₯𝑛 and is a distribution of 𝛾 if Re(𝛾)<𝑛. Let supp 𝑅𝐻𝛾(π‘₯) denote the support of 𝑅𝐻𝛾(π‘₯) and suppose that supp 𝑅𝐻𝛾(π‘₯)βŠ‚Ξ“+, that is, supp 𝑅𝐻𝛾(π‘₯) is compact.

By putting 𝑝=1 in 𝑅𝐻2π‘˜(π‘₯) and taking into account Legendre's duplication formulaΞ“(2𝑧)=22π‘§βˆ’1πœ‹βˆ’1/2ξ‚€1Ξ“(𝑧)Γ𝑧+2,(2.4) we obtain𝐼𝐻𝛾𝑣(π‘₯)=(π›Ύβˆ’π‘›)/2𝐻𝑛(𝛾),(2.5)𝑣=π‘₯21βˆ’π‘₯22βˆ’π‘₯23βˆ’β‹―βˆ’π‘₯2𝑛, where𝐻𝑛(𝛾)=πœ‹(π‘›βˆ’2)/22π›Ύβˆ’1Γ𝛾+2βˆ’π‘›2Γ𝛾2.(2.6) The function 𝐼𝐻𝛾(π‘₯) is called the hyperbolic kernel of Marcel Riesz.

Definition 2.2. Let π‘₯=(π‘₯1,π‘₯2,…,π‘₯𝑛) be a point of ℝ𝑛 and πœ”=π‘₯21+π‘₯22+β‹―+π‘₯2𝑛. The elliptic kernel of Marcel Riesz is defined by π‘…π‘’π›Ύπœ”(π‘₯)=(π›Ύβˆ’π‘›)/2π‘Šπ‘›(𝛾),(2.7) where 𝑛 is the dimension of ℝ𝑛, π›Ύβˆˆβ„‚, and π‘Šπ‘›πœ‹(𝛾)=𝑛/22𝛾Γ(𝛾/2)Ξ“((π‘›βˆ’π›Ύ)/2).(2.8)

Note that 𝑛=𝑝+π‘ž. By putting π‘ž=0 (i.e., 𝑛=𝑝) in (2.2) and (2.3), we can reduce 𝑒(π›Ύβˆ’π‘›)/2 to πœ”π‘(π›Ύβˆ’π‘)/2, where πœ”π‘=π‘₯21+π‘₯22+β‹―+π‘₯2𝑝, and reduce 𝐾𝑛(𝛾) to πΎπ‘πœ‹(𝛾)=(π‘βˆ’1)/2Ξ“((1βˆ’π›Ύ)/2)Ξ“(𝛾)Ξ“((π‘βˆ’π›Ύ)/2).(2.9)

Using the Legendre’s duplication formula Ξ“(2𝑧)=22π‘§βˆ’1πœ‹βˆ’1/2ξ‚€1Ξ“(𝑧)Γ𝑧+2,Ξ“ξ‚€1(2.10)2Γ1+𝑧2ξ‚βˆ’π‘§=πœ‹sec(πœ‹π‘§),(2.11) we obtain 𝐾𝑝1(𝛾)=2ξ‚€secπ›Ύπœ‹2ξ‚π‘Šπ‘(𝛾).(2.12) Thus, in case π‘ž=0, we have 𝑅𝐻𝛾𝑒(π‘₯)=(π›Ύβˆ’π‘)/2𝐾𝑝(𝛾)=2cosπ›Ύπœ‹2𝑒(π›Ύβˆ’π‘)/2π‘Šπ‘ξ‚€(𝛾)=2cosπ›Ύπœ‹2𝑅𝑒𝛾(π‘₯).(2.13) In addition, if 𝛾=2π‘˜ for some nonnegative integer π‘˜, then 𝑅𝐻2π‘˜(π‘₯)=2(βˆ’1)π‘˜π‘…π‘’2π‘˜(π‘₯).(2.14)

Lemma 2.3. The convolution (βˆ’1)π‘˜π‘…π‘’2π‘˜(π‘₯)βˆ—π‘…π»2π‘˜(π‘₯) is the fundamental solution of the diamond operator iterated π‘˜ times, that is, β‹„π‘˜ξ€·(βˆ’1)π‘˜π‘…π‘’2π‘˜(π‘₯)βˆ—π‘…π»2π‘˜ξ€Έ(π‘₯)=𝛿.(2.15)

For the proof of this Lemma, see [4, 12].

It can be shown that π‘…π‘’βˆ’2π‘˜(π‘₯)βˆ—π‘…π»βˆ’2π‘˜(π‘₯)=(βˆ’1)π‘˜β‹„π‘˜π›Ώ, for all nonnegative integers π‘˜.

Definition 2.4. Let π‘₯=(π‘₯1,π‘₯2,…,π‘₯𝑛) be a point of ℝ𝑛. The function 𝑇𝛾(π‘₯,π‘š) is defined by 𝑇𝛾(π‘₯,π‘š)=βˆžξ“π‘Ÿ=0βŽ›βŽœβŽœβŽβˆ’π›Ύ2π‘ŸβŽžβŽŸβŽŸβŽ ξ€·π‘š2ξ€Έπ‘Ÿ(βˆ’1)𝛾/2+π‘Ÿπ‘…π‘’π›Ύ+2π‘Ÿ(π‘₯)βˆ—π‘…π»π›Ύ+2π‘Ÿ(π‘₯),(2.16) where 𝛾 is a complex parameter, and π‘š is a nonnegative real number. Here, 𝑅𝐻𝛾+2π‘Ÿ(π‘₯) and 𝑅𝑒𝛾+2π‘Ÿ(π‘₯) are defined by (2.2) and (2.7), respectively.

From the definition of 𝑇𝛾(π‘₯,π‘š), by putting 𝛾=βˆ’2π‘˜, we have π‘‡βˆ’2π‘˜(π‘₯,π‘š)=βˆžξ“π‘Ÿ=0βŽ›βŽœβŽœβŽπ‘˜π‘ŸβŽžβŽŸβŽŸβŽ ξ€·π‘š2ξ€Έπ‘Ÿ(βˆ’1)βˆ’π‘˜+π‘Ÿπ‘…π‘’2(βˆ’π‘˜+π‘Ÿ)(π‘₯)βˆ—π‘…π»2(βˆ’π‘˜+π‘Ÿ)(π‘₯).(2.17) Since the operator (β‹„+π‘š2)π‘˜ defined by (1.14) is linearly continuous and has 1-1 mapping, this possesses its own inverses. From Lemma 2.3, we obtainπ‘‡βˆ’2π‘˜(π‘₯,π‘š)=βˆžξ“π‘Ÿ=0βŽ›βŽœβŽœβŽπ‘˜π‘ŸβŽžβŽŸβŽŸβŽ ξ€·π‘š2ξ€Έπ‘Ÿβ‹„π‘˜βˆ’π‘Ÿξ€·π›Ώ=β‹„+π‘š2ξ€Έπ‘˜π›Ώ.(2.18)

Substituting π‘˜=0 in (2.18) yields that we have 𝑇0(π‘₯,π‘š)=𝛿. On the other hand, putting 𝛾=2π‘˜ in (2.16) yields𝑇2π‘˜βŽ›βŽœβŽœβŽ0βŽžβŽŸβŽŸβŽ ξ€·π‘š(π‘₯,π‘š)=βˆ’π‘˜2ξ€Έ0(βˆ’1)π‘˜+0𝑅𝑒2π‘˜+0(π‘₯)βˆ—π‘…π»2π‘˜+0+(π‘₯)βˆžξ“π‘Ÿ=1βŽ›βŽœβŽœβŽπ‘ŸβŽžβŽŸβŽŸβŽ ξ€·π‘šβˆ’π‘˜2ξ€Έπ‘Ÿ(βˆ’1)π‘˜+π‘Ÿπ‘…π‘’2π‘˜+2π‘Ÿ(π‘₯)βˆ—π‘…π»2π‘˜+2π‘Ÿ(π‘₯).(2.19) The second summand of the right-hand side of (2.19) vanishes when π‘š=0. Hence, we obtain 𝑇2π‘˜(π‘₯,π‘š=0)=(βˆ’1)π‘˜π‘…π‘’2π‘˜(π‘₯)βˆ—π‘…π»2π‘˜(π‘₯),(2.20) which is the fundamental solution of the diamond operator.

For the proofs of Lemmas 2.5 and 2.6, see [23].

Lemma 2.5. Given the equation ξ€·β‹„+π‘š2ξ€Έπ‘˜π‘’(π‘₯)=𝛿,(2.21) where (β‹„+π‘š2)π‘˜ is the diamond Klein-Gordon operator iterated π‘˜ times, defined by ξ€·β‹„+π‘š2ξ€Έπ‘˜=βŽ‘βŽ’βŽ’βŽ£ξƒ©π‘ξ“π‘–=1πœ•2πœ•π‘₯2𝑖ξƒͺ2βˆ’ξƒ©π‘+π‘žξ“π‘—=𝑝+1πœ•2πœ•π‘₯2𝑗ξƒͺ2+π‘š2⎀βŽ₯βŽ₯βŽ¦π‘˜(2.22) with a nonnegative integer π‘˜ and the Dirac-delta distribution 𝛿, then 𝑒(π‘₯)=𝑇2π‘˜(π‘₯,π‘š) is the fundamental solution of the diamond Klein-Gordon operator iterated π‘˜ times (β‹„+π‘š2)π‘˜, where 𝑇2π‘˜(π‘₯,π‘š) is defined by (2.16) with 𝛾=2π‘˜.

Lemma 2.6. Let 𝑇2π‘˜(π‘₯,π‘š) be the diamond Klein-Gordon kernel defined by (2.16), then 𝑇2π‘˜(π‘₯,π‘š) is a tempered distribution and can be expressed by 𝑇2π‘˜(π‘₯,π‘š)=𝑇2π‘˜βˆ’2𝑣(π‘₯,π‘š)βˆ—π‘‡2𝑣(π‘₯,π‘š),(2.23) where 𝑣 is a nonnegative integer and 𝑣<π‘˜. Moreover, if one puts 𝑙=π‘˜βˆ’π‘£ and β„Ž=𝑣, then one obtains 𝑇2𝑙(π‘₯,π‘š)βˆ—π‘‡2β„Ž(π‘₯,π‘š)=𝑇2𝑙+2β„Ž(π‘₯,π‘š)(2.24) for 𝑙+β„Ž=π‘˜.

3. Properties of the Distribution 𝑒𝛼π‘₯(β‹„+π‘š2)π‘˜π›Ώ

Lemma 3.1. The following equality holds: 𝑒𝛼π‘₯ξ€·β‹„+π‘š2ξ€Έπ‘˜π›Ώ=πΏπ‘˜π›Ώ,(3.1) and 𝑒𝛼π‘₯(β‹„+π‘š2)π‘˜π›Ώ is the tempered distribution of order 4π‘˜ with support {0}, where 𝐿 is the partial differential operator and is defined by 𝐿≑⋄+π‘š2ξ€Έ+π‘›ξ“π‘Ÿ=1𝛼2π‘Ÿβ–‘βˆ’2π‘›ξ“π‘π‘Ÿ=1𝑖=1ξƒ©π›Όπ‘Ÿπœ•3πœ•π‘₯2π‘–πœ•π‘₯π‘Ÿ+π›Όπ‘–πœ•3πœ•π‘₯π‘–πœ•π‘₯2π‘Ÿξƒͺ+2π‘›ξ“π‘Ÿ=1𝑝+π‘žξ“π‘—=𝑝+1ξƒ©π›Όπ‘Ÿπœ•3πœ•π‘₯2π‘—πœ•π‘₯π‘Ÿ+π›Όπ‘—πœ•3πœ•π‘₯π‘—πœ•π‘₯2π‘Ÿξƒͺ+4π‘›ξ“π‘Ÿ=1π›Όπ‘Ÿξƒ©π‘ξ“π‘–=1π›Όπ‘–πœ•2πœ•π‘₯π‘–πœ•π‘₯π‘Ÿβˆ’π‘+π‘žξ“π‘—=𝑝+1π›Όπ‘—πœ•2πœ•π‘₯π‘—πœ•π‘₯π‘Ÿξƒͺβˆ’2π‘›ξ“π‘Ÿ=1𝛼2π‘Ÿξƒ©π‘ξ“π‘–=1π›Όπ‘–πœ•πœ•π‘₯π‘–βˆ’π‘+π‘žξ“π‘—=𝑝+1π›Όπ‘—πœ•πœ•π‘₯𝑗ξƒͺ+𝑝𝑖=1𝛼2π‘–βˆ’π‘+π‘žξ“π‘—=𝑝+1𝛼2𝑗ξƒͺβ–΅ξƒ©βˆ’2𝑝𝑖=1𝛼2π‘–βˆ’π‘+π‘žξ“π‘—=𝑝+1𝛼2jξƒͺπ‘›ξ“π‘Ÿ=1π›Όπ‘Ÿπœ•πœ•π‘₯π‘Ÿ+𝑝𝑖=1𝛼2π‘–βˆ’π‘+π‘žξ“π‘—=𝑝+1𝛼2𝑗ξƒͺπ‘›ξ“π‘Ÿ=1𝛼2π‘Ÿ.(3.2) As before, β–‘ is the ultrahyperbolic operator defined by (1.1) (with π‘˜=1), and β–΅ is the Laplace operator defined by πœ•β–΅=2πœ•π‘₯1+πœ•2πœ•π‘₯2πœ•+β‹―+2πœ•π‘₯𝑛.(3.3)

Proof. Let πœ‘βˆˆπ’Ÿ be the space of testing functions which are infinitely differentiable with compact supports, and let π’Ÿβ€² be the space of distributions. Now, 𝑒𝛼π‘₯ξ€·β‹„+π‘š2=𝛿,πœ‘(π‘₯)𝛿,β‹„+π‘š2𝑒𝛼π‘₯ξ¬πœ‘(π‘₯),(3.4) for 𝑒𝛼π‘₯(β‹„+π‘š2)π›Ώβˆˆπ’Ÿβ€². A direct computation shows that ξ€·β‹„+π‘š2𝑒𝛼π‘₯πœ‘(π‘₯)=𝑒𝛼π‘₯π‘‡πœ‘(π‘₯),(3.5) where 𝑇 is the partial differential operator defined by 𝑇≑⋄+π‘š2ξ€Έ+π‘›ξ“π‘Ÿ=1𝛼2π‘Ÿβ–‘+2π‘›ξ“π‘π‘Ÿ=1𝑖=1ξƒ©π›Όπ‘Ÿπœ•3πœ•π‘₯2π‘–πœ•π‘₯π‘Ÿ+π›Όπ‘–πœ•3πœ•π‘₯π‘–πœ•π‘₯2π‘Ÿξƒͺβˆ’2π‘›ξ“π‘Ÿ=1𝑝+π‘žξ“π‘—=𝑝+1ξƒ©π›Όπ‘Ÿπœ•3πœ•π‘₯2π‘—πœ•π‘₯π‘Ÿ+π›Όπ‘—πœ•3πœ•π‘₯π‘—πœ•π‘₯2π‘Ÿξƒͺ+4π‘›ξ“π‘Ÿ=1π›Όπ‘Ÿξƒ©π‘ξ“π‘–=1π›Όπ‘–πœ•2πœ•π‘₯π‘–πœ•π‘₯π‘Ÿβˆ’π‘+π‘žξ“π‘—=𝑝+1π›Όπ‘—πœ•2πœ•π‘₯π‘—πœ•π‘₯π‘Ÿξƒͺ+2π‘›ξ“π‘Ÿ=1𝛼2π‘Ÿξƒ©π‘ξ“π‘–=1π›Όπ‘–πœ•πœ•π‘₯π‘–βˆ’π‘+π‘žξ“π‘—=𝑝+1π›Όπ‘—πœ•πœ•π‘₯𝑗ξƒͺ+𝑝𝑖=1𝛼2π‘–βˆ’π‘+π‘žξ“π‘—=𝑝+1𝛼2𝑗ξƒͺ□+2𝑝𝑖=1𝛼2π‘–βˆ’π‘+π‘žξ“π‘—=𝑝+1𝛼2𝑗ξƒͺπ‘›ξ“π‘Ÿ=1π›Όπ‘Ÿπœ•πœ•π‘₯π‘Ÿ+𝑝𝑖=1𝛼2π‘–βˆ’π‘+π‘žξ“π‘—=𝑝+1𝛼2𝑗ξƒͺπ‘›ξ“π‘Ÿ=1𝛼2π‘Ÿ.(3.6) Thus, 𝛿,β‹„+π‘š2𝑒𝛼π‘₯ξ¬πœ‘(π‘₯)=βŸ¨π›Ώ,𝑒𝛼π‘₯π‘‡πœ‘(π‘₯)⟩=π‘‡πœ‘(0).(3.7)
Since βŸ¨π‘’π›Όπ‘₯(β‹„+π‘š2)π‘˜π›Ώ,πœ‘(π‘₯)⟩=⟨(β‹„+π‘š2)π‘˜π›Ώ,𝑒𝛼π‘₯πœ‘(π‘₯)⟩ for every πœ‘(π‘₯)βˆˆπ’Ÿ and 𝑒𝛼π‘₯(β‹„+π‘š2)π‘˜π›Ώβˆˆπ’Ÿβ€², we have ⋄+π‘š2ξ€Έπ‘˜π›Ώ,𝑒𝛼π‘₯ξ‚­=ξ‚¬ξ€·πœ‘(π‘₯)β‹„+π‘š2ξ€Έπ‘˜βˆ’1𝛿,β‹„+π‘š2𝑒𝛼π‘₯ξ‚­=ξ‚¬ξ€·πœ‘(π‘₯)β‹„+π‘š2ξ€Έπ‘˜βˆ’1𝛿,𝑒𝛼π‘₯ξ‚­=ξ‚¬ξ€·π‘‡πœ‘(π‘₯)β‹„+π‘š2ξ€Έπ‘˜βˆ’2𝛿,β‹„+π‘š2𝑒𝛼π‘₯ξ‚­=ξ‚¬ξ€·π‘‡πœ‘(π‘₯)β‹„+π‘š2ξ€Έπ‘˜βˆ’2𝛿,𝑒𝛼π‘₯ξ‚­=𝑇(π‘‡πœ‘(π‘₯))β‹„+π‘š2ξ€Έπ‘˜βˆ’2𝛿,𝑒𝛼π‘₯𝑇2ξ‚­.πœ‘(π‘₯)(3.8) Repeating this process (β‹„+π‘š2) with π‘˜βˆ’2 times, we finally obtain ⋄+π‘š2ξ€Έπ‘˜βˆ’2𝛿,𝑒𝛼π‘₯𝑇2ξ‚­=ξ«πœ‘(π‘₯)𝛿,𝑒𝛼π‘₯π‘‡π‘˜ξ¬πœ‘(π‘₯)=π‘‡π‘˜πœ‘(0),(3.9) where π‘‡π‘˜ is the operator of (3.6) iterated π‘˜ times. Now, π‘‡π‘˜ξ«πœ‘(0)=𝛿,π‘‡π‘˜ξ¬=ξ«πœ‘(π‘₯)𝐿𝛿,π‘‡π‘˜βˆ’1ξ¬πœ‘(π‘₯),(3.10) by the operator 𝐿 in (3.2) and the derivative of distribution. Continuing this process, we obtain π‘‡π‘˜πœ‘(0)=βŸ¨πΏπ‘˜π›Ώ,πœ‘(π‘₯)⟩ or βŸ¨π‘’π›Όπ‘₯(β‹„+π‘š2)π‘˜π›Ώ,πœ‘(π‘₯)⟩=βŸ¨πΏπ‘˜π›Ώ,πœ‘(π‘₯)⟩. By equality of distributions, we obtain (3.1) as required. Since 𝛿 and its partial derivatives have support {0} which is compact, hence, by Schwartz [25], πΏπ‘˜π›Ώ are tempered distributions and πΏπ‘˜π›Ώ has order 4π‘˜. It follows that 𝑒𝛼π‘₯(β‹„+π‘š2)π‘˜π›Ώ is a tempered distribution of order 4π‘˜ with point support {0} by (3.1). This completes the proof.

Lemma 3.2 (boundedness property). Let π’Ÿ be the space of testing functions and π’Ÿξ…ž the space of distributions. For every πœ‘βˆˆπ’Ÿ and 𝑒𝛼π‘₯(β‹„+π‘š2)π‘˜π›Ώβˆˆπ’Ÿξ…ž, |||𝑒𝛼π‘₯ξ€·β‹„+π‘š2ξ€Έπ‘˜ξ‚­|||𝛿,πœ‘(π‘₯)≀𝑀,(3.11) for some constant 𝑀.

Proof. Note that we have βŸ¨π‘’π›Όπ‘₯(β‹„+π‘š2)π‘˜π›Ώ,πœ‘(π‘₯)⟩=⟨(β‹„+π‘š2)π‘˜π›Ώ,𝑒𝛼π‘₯πœ‘(π‘₯)⟩ for every πœ‘(π‘₯)βˆˆπ’Ÿ and 𝑒𝛼π‘₯(β‹„+π‘š2)π‘˜π›Ώβˆˆπ’Ÿβ€². Hence, ⋄+π‘š2ξ€Έπ‘˜π›Ώ,𝑒𝛼π‘₯ξ‚­=ξ‚¬ξ€·πœ‘(π‘₯)β‹„+π‘š2ξ€Έπ‘˜βˆ’1𝛿,β‹„+π‘š2𝑒𝛼π‘₯ξ‚­=ξ‚¬ξ€·πœ‘(π‘₯)β‹„+π‘š2ξ€Έπ‘˜βˆ’1𝛿,𝑒𝛼π‘₯ξ‚­π‘‡πœ‘(π‘₯),(3.12) where 𝑇 is defined by (3.6). Continuing this process for π‘˜βˆ’1 times, we will obtain 𝑒𝛼π‘₯ξ€·β‹„+π‘š2ξ€Έπ‘˜ξ‚­=𝛿,πœ‘(π‘₯)𝛿,𝑒𝛼π‘₯π‘‡π‘˜ξ¬πœ‘(π‘₯)=π‘‡π‘˜πœ‘(0).(3.13) Since πœ‘βˆˆπ’Ÿ, so πœ‘(0) is bounded, and also π‘‡π‘˜πœ‘(0) is bounded. It then follows that |||𝑒𝛼π‘₯ξ€·β‹„+π‘š2ξ€Έπ‘˜ξ‚­|||𝛿,πœ‘(π‘₯)=π‘‡π‘˜πœ‘(0)≀𝑀,(3.14) for some constant 𝑀.

4. The Application of Distribution 𝑒𝛼π‘₯(β‹„+π‘š2)π‘˜π›Ώ

Theorem 4.1. Let 𝐿 be the partial differential operator defined by (3.2), and consider the equation 𝐿𝑒(π‘₯)=𝛿,(4.1) where 𝑒(π‘₯) is any distribution in π’Ÿξ…ž, then 𝑒(π‘₯)=𝑒𝛼π‘₯𝑇2(π‘₯,π‘š) is the fundamental solution of the operator 𝐿, where 𝑇2(π‘₯,π‘š) is defined by (2.16) with 𝛾=2.

Proof. From (3.1) and (4.1), we can write 𝑒𝛼π‘₯(β‹„+π‘š2)π›Ώβˆ—π‘’(π‘₯)=𝐿𝑒(π‘₯)=𝛿. Convolving both sides by 𝑒𝛼π‘₯𝑇2(π‘₯,π‘š), we have 𝑒𝛼π‘₯𝑇2(π‘₯,π‘š)βˆ—π‘’π›Όπ‘₯ξ€·β‹„+π‘š2ξ€Έπ›Ώβˆ—π‘’(π‘₯)=𝑒𝛼π‘₯𝑇2(π‘₯,π‘š)βˆ—π›Ώ,(4.2) then 𝑒𝛼π‘₯𝑇2ξ€·(π‘₯,π‘š)βˆ—β‹„+π‘š2ξ€Έπ›Ώξ€Έβˆ—π‘’(π‘₯)=𝑒𝛼π‘₯𝑇2(π‘₯,π‘š),(4.3) or equivalently, 𝑒𝛼π‘₯ξ€·ξ€·β‹„+π‘š2𝑇2ξ€Έ(π‘₯,π‘š)βˆ—π‘’(π‘₯)=𝑒𝛼π‘₯𝑇2(π‘₯,π‘š).(4.4) Since (β‹„+π‘š2)𝑇2(π‘₯,π‘š)=𝛿 by Lemma 2.5 with π‘˜=1, we obtain (𝑒𝛼π‘₯𝛿)βˆ—π‘’(π‘₯)=𝑒𝛼π‘₯𝑇2(π‘₯,π‘š).(4.5) Moreover, since 𝑒𝛼π‘₯𝛿=𝛿, we have π›Ώβˆ—π‘’(π‘₯)=𝑒𝛼π‘₯𝑇2(π‘₯,π‘š). It then follows that 𝑒(π‘₯)=𝑒𝛼π‘₯𝑇2(π‘₯,π‘š) is the fundamental solution of the operator 𝐿.

Theorem 4.2 (the generalization of Theorem 4.1). From Lemma 3.1, consider that 𝑒𝛼π‘₯ξ€·β‹„+π‘š2ξ€Έπ‘˜π›Ώβˆ—π‘’(π‘₯)=𝛿,(4.6) or πΏπ‘˜π‘’(π‘₯)=𝛿,(4.7) then 𝑒(π‘₯)=𝑒𝛼π‘₯𝑇2π‘˜(π‘₯,π‘š) is the fundamental solution of the operator πΏπ‘˜.

Proof. We can prove it by using either (4.6) or (4.7). If we start with (4.6), by convolving both sides by 𝑒𝛼π‘₯𝑇2π‘˜(π‘₯,π‘š), we obtain 𝑒𝛼π‘₯𝑇2π‘˜ξ‚€π‘’(π‘₯,π‘š)βˆ—π›Όπ‘₯ξ€·β‹„+π‘š2ξ€Έπ‘˜ξ‚π›Ώβˆ—π‘’(π‘₯)=𝑒𝛼π‘₯𝑇2π‘˜(π‘₯,π‘š)βˆ—π›Ώ,(4.8) or 𝑒𝛼π‘₯((β‹„+π‘š2)π‘˜π‘‡2π‘˜(π‘₯,π‘š))βˆ—π‘’(π‘₯)=𝑒𝛼π‘₯𝑇2π‘˜(π‘₯,π‘š). Since (β‹„+π‘š2)π‘˜π‘‡2π‘˜(π‘₯,π‘š)=𝛿 by Lemma 2.5, we have (𝑒𝛼π‘₯𝛿)βˆ—π‘’(π‘₯)=𝑒𝛼π‘₯𝑇2π‘˜(π‘₯,π‘š) or 𝑒(π‘₯)=𝑒𝛼π‘₯𝑇2π‘˜(π‘₯,π‘š) as required.
If we use (4.7), by convolving both sides by 𝑒𝛼π‘₯𝑇2(π‘₯,π‘š), we obtain 𝑒𝛼π‘₯𝑇2(π‘₯,π‘š)βˆ—πΏπ‘˜π‘’(π‘₯)=𝑒𝛼π‘₯𝑇2(π‘₯,π‘š)βˆ—π›Ώ,(4.9) or 𝐿(𝑒𝛼π‘₯𝑇2(π‘₯,π‘š))βˆ—πΏπ‘˜βˆ’1𝑒(π‘₯)=𝑒𝛼π‘₯𝑇2(π‘₯,π‘š). By Theorem 4.1, we obtain πΏπ‘˜βˆ’1𝑒(π‘₯)=𝑒𝛼π‘₯𝑇2(π‘₯,π‘š). Keeping on convolving 𝑒𝛼π‘₯𝑇2(π‘₯,π‘š) for π‘˜βˆ’1 times, we finally obtain 𝑒(π‘₯)=𝑒𝛼π‘₯𝑇2(π‘₯,π‘š)βˆ—π‘‡2(π‘₯,π‘š)βˆ—β‹―βˆ—π‘‡2ξ€Έ(π‘₯,π‘š)=𝑒𝛼π‘₯𝑇2π‘˜(π‘₯,π‘š),(4.10) by Lemma 2.6 and [26, page 196].

In particular, if we put 𝛼=(𝛼1,𝛼2,…,𝛼𝑛)=0 in (4.6), then (4.6) reduces to (2.21), and we obtain 𝑒(π‘₯)=𝑇2π‘˜(π‘₯,π‘š) as the fundamental solution of the diamond Klein-Gordon operator iterated π‘˜ times.

Theorem 4.3. Given the convolution equation 𝑒𝛼π‘₯ξ€·β‹„+π‘š2ξ€Έπ‘˜π›Ώβˆ—π‘’(π‘₯)=𝑒𝑀𝛼π‘₯ξ“π‘Ÿ=0πΆπ‘Ÿξ€·β‹„+π‘š2ξ€Έπ‘Ÿπ›Ώ,(4.11) where (β‹„+π‘š2)π‘˜ is the diamond Klein-Gordon operator iterated π‘˜ times defined by ξ€·β‹„+π‘š2ξ€Έπ‘˜=𝑝𝑖=1πœ•2πœ•π‘₯2π‘–βˆ’π‘+π‘žξ“π‘—=𝑝+1πœ•2πœ•π‘₯2𝑗+π‘š2ξƒͺπ‘˜,(4.12) the variable π‘₯=(π‘₯1,π‘₯2,…,π‘₯𝑛)βˆˆβ„π‘›, the constant 𝛼=(𝛼1,𝛼2,…,𝛼𝑛)βˆˆβ„π‘›, π‘š is a nonnegative real number, 𝛿 is the Dirac-delta distribution with (β‹„+π‘š2)0𝛿=𝛿,(β‹„+π‘š2)1𝛿=(β‹„+π‘š2)𝛿, and πΆπ‘Ÿ is a constant, then the type of solution 𝑒(π‘₯) of (4.11) depends on π‘˜,𝑀, and 𝛼 as follows: (1)if 𝑀<π‘˜ and 𝑀=0, then the solution of (4.11) is 𝑒(π‘₯)=𝐢0𝑒𝛼π‘₯𝑇2π‘˜(π‘₯,π‘š),(4.13) where 𝑇2π‘˜(π‘₯,π‘š) is defined by (2.16) with 𝛾=2π‘˜. If 2π‘˜β‰₯𝑛 and for any 𝛼, then 𝑒𝛼π‘₯𝑇2π‘˜(π‘₯,π‘š) is the ordinary function, (2)if 0<𝑀<π‘˜, then the solution of (4.11) is 𝑒(π‘₯)=𝑒𝑀𝛼π‘₯ξ“π‘Ÿ=1πΆπ‘Ÿπ‘‡2π‘˜βˆ’2π‘Ÿ(π‘₯,π‘š),(4.14) which is an ordinary function for 2π‘˜βˆ’2π‘Ÿβ‰₯𝑛 with any arbitrary constant 𝛼, (3)if 𝑀β‰₯π‘˜ and for any 𝛼 one supposes that π‘˜β‰€π‘€β‰€π‘, then (4.11) has 𝑒(π‘₯)=𝑒𝑁𝛼π‘₯ξ“π‘Ÿ=π‘˜πΆπ‘Ÿξ€·β‹„+π‘š2ξ€Έπ‘Ÿβˆ’π‘˜π›Ώ(4.15) as a solution which is the singular distribution.

Proof. (1)For 𝑀<π‘˜ and 𝑀=0, then (4.11) becomes 𝑒𝛼π‘₯ξ€·β‹„+π‘š2ξ€Έπ‘˜π›Ώβˆ—π‘’(π‘₯)=𝐢0𝑒𝛼π‘₯𝛿=𝐢0𝛿,(4.16) and by Theorem 4.2, we obtain 𝑒(π‘₯)=𝐢0𝑒𝛼π‘₯𝑇2π‘˜(π‘₯,π‘š).(4.17) Now, by (2.2) and (2.7), 𝑅𝑒2π‘˜(π‘₯) and 𝑅𝐻2π‘˜(π‘₯) are ordinary functions, respectively, for 2π‘˜β‰₯𝑛. It then follows that 𝐢0𝑒𝛼π‘₯𝑇2π‘˜(π‘₯,π‘š) is an ordinary function for 2π‘˜β‰₯𝑛 with any 𝛼.(2)For 0<𝑀<π‘˜, then we can write (4.11) as 𝑒𝛼π‘₯ξ€·β‹„+π‘š2ξ€Έπ‘˜π›Ώβˆ—π‘’(π‘₯)=𝑒𝛼π‘₯𝐢1ξ€·β‹„+π‘š2𝛿+𝐢2ξ€·β‹„+π‘š2ξ€Έ2𝛿+β‹―+𝐢𝑀⋄+π‘š2𝑀𝛿.(4.18) Convolving both sides by 𝑒𝛼π‘₯𝑇2π‘˜(π‘₯,π‘š) and applying Lemma 2.5, we obtain 𝑒(π‘₯)=𝑒𝛼π‘₯𝐢1ξ€·β‹„+π‘š2𝑇2π‘˜(π‘₯,π‘š)+𝐢2ξ€·β‹„+π‘š2ξ€Έ2𝑇2π‘˜(π‘₯,π‘š)+β‹―+𝐢𝑀⋄+π‘š2𝑀𝑇2π‘˜ξ‚„(π‘₯,π‘š).(4.19) It is known that (β‹„+π‘š2)π‘˜π‘‡2π‘˜(π‘₯,π‘š)=𝛿, thus (β‹„+π‘š2)π‘˜βˆ’π‘Ÿ(β‹„+π‘š2)π‘Ÿπ‘‡2π‘˜(π‘₯,π‘š)=𝛿 for π‘Ÿ<π‘˜. Convolving both sides by 𝑇2π‘˜βˆ’2π‘Ÿ(π‘₯,π‘š), we obtain 𝑇2π‘˜βˆ’2π‘Ÿξ€·(π‘₯,π‘š)βˆ—β‹„+π‘š2ξ€Έπ‘˜βˆ’π‘Ÿξ€·β‹„+π‘š2ξ€Έπ‘Ÿπ‘‡2π‘˜(π‘₯,π‘š)=𝑇2π‘˜βˆ’2π‘Ÿ(π‘₯,π‘š),(4.20) or ξ€·β‹„+π‘š2ξ€Έπ‘˜βˆ’π‘Ÿπ‘‡2π‘˜βˆ’2π‘Ÿξ€·(π‘₯,π‘š)βˆ—β‹„+π‘š2ξ€Έπ‘Ÿπ‘‡2π‘˜(π‘₯,π‘š)=𝑇2π‘˜βˆ’2π‘Ÿ(π‘₯,π‘š),(4.21) which leads to ξ€·β‹„+π‘š2ξ€Έπ‘Ÿπ‘‡2π‘˜(π‘₯,π‘š)=𝑇2π‘˜βˆ’2π‘Ÿ(π‘₯,π‘š),(4.22) for π‘Ÿ<π‘˜. It follows that 𝑒(π‘₯)=𝑒𝛼π‘₯𝐢1𝑇2π‘˜βˆ’2(π‘₯,π‘š)+𝐢2𝑇2π‘˜βˆ’4(π‘₯,π‘š)+β‹―+𝐢𝑀𝑇2π‘˜βˆ’2𝑀(π‘₯,π‘š),(4.23) or 𝑒(π‘₯)=𝑒𝑀𝛼π‘₯ξ“π‘Ÿ=1πΆπ‘Ÿπ‘‡2π‘˜βˆ’2π‘Ÿ(π‘₯,π‘š).(4.24) Similarly, by case (1), 𝑒𝛼π‘₯𝑇2π‘˜βˆ’2π‘Ÿ(π‘₯,π‘š) is the ordinary function for 2π‘˜βˆ’2π‘Ÿβ‰₯𝑛 with any 𝛼. It follows that 𝑒(π‘₯)=𝑒𝑀𝛼π‘₯ξ“π‘Ÿ=1πΆπ‘Ÿπ‘‡2π‘˜βˆ’2π‘Ÿ(π‘₯,π‘š)(4.25) is also the ordinary function with any 𝛼.(3)if 𝑀β‰₯π‘˜ and for any 𝛼, we suppose that π‘˜β‰€π‘€β‰€π‘, then (4.11) becomes 𝑒𝛼π‘₯ξ€·β‹„+π‘š2ξ€Έπ‘˜π›Ώβˆ—π‘’(π‘₯)=𝑒𝛼π‘₯ξ‚ƒπΆπ‘˜ξ€·β‹„+π‘š2ξ€Έπ‘˜π›Ώ+πΆπ‘˜+1ξ€·β‹„+π‘š2ξ€Έπ‘˜+1𝛿+β‹―+𝐢𝑁⋄+π‘š2𝑁𝛿.(4.26) Convolving both sides by 𝑒𝛼π‘₯𝑇2π‘˜(π‘₯,π‘š) and applying Lemma 2.5, we have 𝑒(π‘₯)=𝑒𝛼π‘₯ξ‚ƒπΆπ‘˜ξ€·β‹„+π‘š2ξ€Έπ‘˜π‘‡2π‘˜(π‘₯,π‘š)+πΆπ‘˜+1ξ€·β‹„+π‘š2ξ€Έπ‘˜+1𝑇2π‘˜(π‘₯,π‘š)+β‹―+𝐢𝑁⋄+π‘š2𝑁𝑇2π‘˜ξ‚„(π‘₯,π‘š).(4.27) Now, ξ€·β‹„+π‘š2𝑀𝑇2π‘˜ξ€·(π‘₯,π‘š)=β‹„+π‘š2ξ€Έπ‘€βˆ’π‘˜ξ€·β‹„+π‘š2ξ€Έπ‘˜π‘‡2π‘˜ξ€·(π‘₯,π‘š)=β‹„+π‘š2ξ€Έπ‘€βˆ’π‘˜,(4.28) for π‘˜β‰€π‘€β‰€π‘. Thus, 𝑒(π‘₯)=𝑒𝛼π‘₯ξ‚ƒπΆπ‘˜π›Ώ+πΆπ‘˜+1ξ€·β‹„+π‘š2𝛿+πΆπ‘˜+2ξ€·β‹„+π‘š2ξ€Έ2𝛿+β‹―+𝐢𝑁⋄+π‘š2ξ€Έπ‘βˆ’π‘˜π›Ώξ‚„=𝑒𝑁𝛼π‘₯ξ“π‘Ÿ=π‘˜πΆπ‘Ÿξ€·β‹„+π‘š2ξ€Έπ‘Ÿβˆ’π‘˜π›Ώ.(4.29) Now, by (3.1) and (3.2), we have 𝑒𝛼π‘₯ξ€·β‹„+π‘š2ξ€Έπ‘Ÿβˆ’π‘˜ξ€·π›Ώ=β‹„+π‘š2ξ€Έπ‘Ÿβˆ’π‘˜π›Ώ+(termsoflowerorderofpartialderivativeof𝛿)(4.30) for π‘˜β‰€π‘Ÿβ‰€π‘. Since all terms on the right-hand side of this equation are singular distribution, it follows that 𝑒(π‘₯)=𝑒𝑁𝛼π‘₯ξ“π‘Ÿ=π‘˜πΆπ‘Ÿξ€·β‹„+π‘š2ξ€Έπ‘Ÿβˆ’π‘˜π›Ώ(4.31) is the singular distribution. This completes the proof.

Acknowledgments

This work was supported by the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission, through the Cluster of Research to Enhance the Quality of Basic Education, and the Centre of Excellence in Mathematics, Thailand.

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