On the Inverse Ultrahyperbolic Klein-Gordon Kernel

: In this work, we deﬁne the ultrahyperbolic Klein-Gordon operator of order α on the function f by T α p f q “ W α ˚ f , where α P C , W α is the ultrahyperbolic Klein-Gordon kernel, the symbol ˚ denotes the convolution, and f P S , S is the Schwartz space of functions. Our purpose of this work is to study the convolution of W α and obtain the operator L α “ r T α s ´ 1 such that if T α p f q “ ϕ , then L α ϕ “ f .


Introduction
Consider the linear differential equation of the form where upxq and f pxq are generalized functions, x " px 1 , x 2 , . . . , x n q P R n and˝k is the n-dimensional ultra-hyperbolic operator iterated k times, which is defined by k "˜B where p`q " n is the dimension of R n , and k is non-negative integer. The fundamental solution of Equation (1) was first introduced by Gelfand and Shilov [1] but the form is complicated and Trione [2] showed that the generalized function R 2k pxq, defined by Equation (22) with γ " 2k, is the fundamental solution of Equation (1). Later, Tellez [3] also proved that R 2k pxq exists only when p is odd with p`q " n.
In 1997, Kananthai [4] introduced the diamond operator ♦ k iterated k times, which is defined by where k is a non-negative integer and p`q " n is the dimension of R n . The operator ♦ k can be expressed as the product of the operators k and˝k, that is where˝k is defined by Equation (2), and k is the Laplace operator iterated k times, which is defined by On finding the fundamental solution of diamond operator iterated k times, Kananthai applied the convolution of functions which are fundamental solutions of the operators˝k and k . He showed that p´1q k S 2k pxq˚R 2k pxq is the fundamental solution of the operator ♦ k . That is, where R 2k pxq and S 2k pxq are defined by Equations (22) and (29), respectively, with γ " 2k, and δ is the Dirac delta function. The solution p´1q k S 2k pxq˚R 2k pxq is called the diamond kernel of Marcel Riesz.
In 1978, Dominguez and Trione [14] introduced the distributional functions H α pP˘i0, nq, which is defined by where p`q " n, and q is the number of negative terms of the quadratic form P. The distributions pP˘i0q λ are defined by where λ P C, ą 0, and |x| 2 " x 2 1`x 2 2`¨¨¨`x 2 n , see [1]. They also showed the distributional functions H α pP˘i0, nq are causal (anticausal) analogues of the elliptic kernel of Marcel Riesz [15]. Next, Cerutti and Trione [16] defined the causal (anticausal) generalized Marcel Riesz potentials of order α, α P C, by where ϕ P S, S is the Schwartz space of functions [17], and H α pP˘i0, nq is defined by Equation (7). They also studied the operator pR α q´1, that is the inverse operator of R α , such that f " R α ϕ implies pR α q´1 f " ϕ.
In 1999, Aguirre [18] defined the ultra-hyperbolic Marcel Riesz operator M α of the function f by where α P C, R α is defined by (22), and f P S. He also studied the operator N α " pM α q´1 such that M α p f q " ϕ implies N α ϕ " f . In 2000, Kananthai [8] introduced the diamond kernel of Marcel Riesz K α,β , which is given by where R β and S α are defined by Equations (22) and (29), respectively. Next, Tellez and Kananthai [13] proved that K α,β exists and is in the space of tempered distributions. In addition, they also showed the relationship between the convolution of the distributional families K α,β and diamond operator iterated k times. In 2011, Maneetus and Nonlaopon [19] defined the Bessel ultra-hyperbolic Marcel Riesz operator of order α on the function f by where α P C, R B α is the Bessel ultra-hyperbolic kernel of Marcel Riesz, and f P S. In addition, they studied the operator E α " pU α q´1 such that U α p f q " ϕ implies E α ϕ " f . Moreover, they defined the diamond Marcel Riesz operator of order pα, βq of the function f by where α, β P C, K α,β is defined by (12), and f P S; see [20], for more details. In addition, they have also studied the operator N pα,βq " such that M pα,βq p f q " ϕ implies N pα,βq ϕ " f . In 2013, Salao and Nonlaopon [21] defined the Bessel diamond kernel of Marcel Riesz by where S B α pxq and R B β pxq are the Bessel elliptic kernel of Marcel Riesz and the Bessel ultra-hyperbolic kernel of Marcel Riesz, respectively. They also defined the Bessel diamond Marcel Riesz operator of order pα, βq on the function f by where α, β P C, K B α,β is defined by (15), and f P S. In addition, they studied the operator E pα,βq " In 2007, Tariboon and Kananthai [22] introduced the diamond Klein-Gordon operator p♦`m 2 q k iterated k times, which is defined by where m ě 0, k is non-negative integer, p`q " n is the dimension of R n , for all x " px 1 , x 2 , . . . , x n q P R n . Next, Nonlaopon et al. [23] studied the fundamental solution of diamond Klein-Gordon operator iterated k times, which is called the diamond Klein-Gordon kernel, and studied the Fourier transform of the diamond Klein-Gordon kernel and its convolution [24]. In 2011, Liangprom and Nonlaopon [25] studied some properties of the distribution e αx p♦`m 2 q k δ and showed the boundedness property of the distribution e αx p♦`m 2 q k δ, where p♦`m 2 q k is defined by Equation (17), α P C, and δ is Dirac delta function.
In 2013, Sattaso and Nonlaopon [26] defined the diamond Klein-Gordon operator of order α on the function f by where α P C, and T α is the diamond Klein-Gordon kernel. They also studied the convolution of T α and obtain the operator L α " rD α s´1 such that D α p f q " ϕ implies L α ϕ " f . In 1988, Trione [27] studied the fundamental solution of the ultrahyperbolic Klein-Gordon operator p˝`m 2 q k iterated k times, which is defined by She showed that W 2k px, mq, defined by Equation (37) with α " 2k, is the fundamental solution of the operator p˝`m 2 q k , which is called the ultra-hyperbolic Klein-Gordon kernel. Next, Tellez [28] studied the convolution product of W α px, mq˚W β px, mq, where α, β P C. In addition, Trione [29] has studied the fundamental pP˘i0q λ -ultrahyperbolic solution of the Klein-Gordon operator iterated k times and the convolution of such fundamental solution. She also studied the integral representation of the kernel W α px, mq, see [30] for more details.
In this paper, we define the Klein-Gordon operator of order α of the function f by where α P C, W α is the ultra-hyperbolic Klein-Gordon kernel defined by Equation (37), and f P S. Our aim of this paper is to obtain the operator L α " rT α s´1 such that if T α p f q " ϕ then L α ϕ " f . Before we proceed to that point, we clarify some concepts and definitions.

Preliminaries
Definition 1. Let x " px 1 , x 2 , . . . , x n q be a point of the n-dimensional Euclidean space R n and be the non-degenerated quadratic form, where p`q " n is the dimension of R n . Let Γ`" tx P R n : x 1 ą 0 and u ą 0u be the interior of a forward cone and let Γ`denote its closure. For any complex number γ, we define where The function R γ pxq, which was introduced by Y. Nozaki [31], is called the ultra-hyperbolic kernel of Marcel Riesz. It is well known that R γ pxq is an ordinary function when Repγq ě n and is a distribution of γ otherwise. The support of R γ pxq is denoted by supp R γ pxq and suppose that supp R γ pxq Ă Γ`, that is, supp R γ pxq is compact.
By putting p " 1 in R γ pxq and taking into the Legendre's duplication formula we obtain and v " x 2 The function I γ pxq is called the hyperbolic kernel of Marcel Riesz. From [2], the generalized function R 2k pxq is the fundamental solution of the operator˝k, that is In addition, it can be shown that R´2 k pxq "˝kδ (28) for k is a nonnegative integer, see [2,13]. Definition 2. Let x " px 1 , x 2 , . . . , x n q be a point of R n and ω " x 2 1`x 2 2`¨¨¨`x 2 n . The elliptic kernel of Marcel Riesz is defined by where γ P C, n is the dimension of R n , and U n pγq " π n{2 2 γ Γpγ{2q Γppn´γq{2q .
Thus, for q " 0, we have In addition, if γ " 2k for some non-negative integer k, then R 2k pxq " 2p´1q k S 2k pxq.
Next, we consider the function where α P C, u is defined by Equation (21), m a real non-negative number, n is the dimension of R n , and J ν pzq is Bessel function of the first kind, which is defined by It is well known that W α px, mq is an ordinary function when Repαq ě n and is a distribution otherwise. In addition, W α px, mq can be expressed as an infinitely linear combination of R α pxq of different orders, that is where α P C, R α pxq is defined by Equation (22), see [27,29,30], for more details. From Equation (37) and by putting α "´2k, for k is non-negative integer, we have Since the operator p˝`m 2 q k defined by Equation (19) is linearly continuous and injective mapping of this possess its own inverse. From Equation (28), we obtain W´2 k px, mq " Substituting k " 0 in Equation (39), yields W 0 px, mq " δ. On the other hand, by putting α " 2k in Equation (37), yields The second summand of the right-hand side of Equation (40) vanishes when m 2 " 0. Therefore, we obtain W 2k px, m " 0q " R 2k pxq, is the fundamental solution of the ultra-hyperbolic operator˝k. For the convenience, we will denote W α px, mq by W α . The proof of Lemmas 1 and 2 are given in [28].

The Convolution of W α˚Wβ when β "´α
In this section, we will consider the property of W α˚Wβ when β "´α. From Equations (41) and (44), we immediately obtain the following properties: 1.
If p is odd and q is even, then where A α,β is defined by Equation (42).
If p is even and q is odd, then 4.
If p and q are both even, then Moreover, it follows from Equation (42) that where γ " α`β.

The Main Theorem
Let T α p f q be the ultrahyperbolic Klein-Gordon operator of order α on the function f , which is defined by where α P C, W α is defined by Equation (37), and f P S. Recall that our objective is to obtain the operator L α " rT α s´1 such that if T α p f q " ϕ, then L α ϕ " f for all α P C.
W´α, if p is odd and q is even; W´α, if p and q are both odd; sec 2 pαπ{2q W´α, if p is even with α{2 " 2s`1 for any non-negative integer s. (62), we have T α p f q " W α˚f " ϕ,

Conlusions
In this work, we have considered the property of convolution of the ultrahyperbolic Klein-Gordon kernel in the form W α˚Wβ when β "´α. We have obtained the inverse ultrahyperbolic Klein-Gordon kernel, that is, the operator L α " rT α s´1 such that if T α p f q " ϕ, then L α ϕ " f for all α P C. It is expected that this work may stimulate further research in this field.
Funding: This research received no external funding.