On the Inverse Ultrahyperbolic Klein-Gordon Kernel

Abstract

In this work, we define the ultrahyperbolic Klein-Gordon operator of order $\alpha$ on the function f by $T^{\alpha }(f)=W_{\alpha }\ast f,$ where $\alpha \in \mathbb{C},W_{\alpha }$ is the ultrahyperbolic Klein-Gordon kernel, the symbol ∗ denotes the convolution, and $f\in \mathcal{S},\mathcal{S}$ is the Schwartz space of functions. Our purpose of this work is to study the convolution of $W_{\alpha }$ and obtain the operator $L^{\alpha }={\left[T^{\alpha }\right]}^{-1}$ such that if $T^{\alpha }(f)=\phi$, then $L^{\alpha }\phi =f.$

1. Introduction

Consider the linear differential equation of the form

$${{\displaystyle \square }}^{k}u(x)=f(x),$$

(1)

where $u(x)$ and $f(x)$ are generalized functions, $x=(x_1,x_2,\dots ,x_n)\in {\mathbb{R}}^n$ and ${{\displaystyle \square }}^k$ is the n-dimensional ultra-hyperbolic operator iterated k times, which is defined by

$${{\displaystyle \square }}^k={\left(\frac{{\partial }^2}{\partial x_1^2}+\frac{{\partial }^2}{\partial x_2^2}+\cdots +\frac{{\partial }^2}{\partial x_p^2}-\frac{{\partial }^2}{\partial x_{p+1}^2}-\frac{{\partial }^2}{\partial x_{p+2}^2}-\cdots -\frac{{\partial }^2}{\partial x_{p+q}^2}\right)}^k,$$

(2)

where $p+q=n$ is the dimension of ${\mathbb{R}}^n$, 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 $R_{2k}(x)$, defined by Equation (22) with $\gamma =2k$, is the fundamental solution of Equation (1). Later, Tellez [3] also proved that $R_{2k}(x)$ 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

$${◊}^k={\left[{\left(\sum _{i=1}^p\frac{{\partial }^2}{\partial x_i^2}\right)}^2-{\left(\sum _{j=p+1}^{p+q}\frac{{\partial }^2}{\partial x_j^2}\right)}^2\right]}^k,$$

(3)

where k is a non-negative integer and $p+q=n$ is the dimension of ${\mathbb{R}}^n$. The operator ${◊}^k$ can be expressed as the product of the operators ${▵}^k$ and ${{\displaystyle \square }}^k$, that is

$${◊}^k={▵}^k{{\displaystyle \square }}^k={{\displaystyle \square }}^k{▵}^k,$$

(4)

where ${{\displaystyle \square }}^k$ is defined by Equation (2), and ${▵}^k$ is the Laplace operator iterated k times, which is defined by

$${▵}^k={\left(\frac{{\partial }^2}{\partial x_1^2}+\frac{{\partial }^2}{\partial x_2^2}+\cdots +\frac{{\partial }^2}{\partial x_n^2}\right)}^k.$$

(5)

On finding the fundamental solution of diamond operator iterated k times, Kananthai applied the convolution of functions which are fundamental solutions of the operators ${{\displaystyle \square }}^k$ and ${▵}^k$. He showed that ${(-1)}^k{S}_{2k}(x)\ast R_{2k}(x)$ is the fundamental solution of the operator ${◊}^k$. That is,

$${◊}^k\left({(-1)}^k{S}_{2k}(x)\ast R_{2k}(x)\right)=\delta ,$$

(6)

where $R_{2k}(x)$ and $S_{2k}(x)$ are defined by Equations (22) and (29), respectively, with $\gamma =2k$, and $\delta$ is the Dirac delta function. The solution ${(-1)}^k{S}_{2k}(x)\ast R_{2k}(x)$ is called the diamond kernel of Marcel Riesz. Interested readers are referred to [5,6,7,8,9,10,11,12,13] for some advance in the property of the diamond kernel of Marcel Riesz.

In 1978, Dominguez and Trione [14] introduced the distributional functions $H_{\alpha }(P\pm i0,n)$, which is defined by

$$H_{\alpha }(P\pm i0,n)=\frac{e^{\mp \alpha \pi i/2}{e}^{\pm q\pi i/2}\Gamma ((n-\alpha )/2){(P\pm i0)}^{(\alpha -n)/2}}{2^{\alpha }{\pi }^{n/2}\Gamma (\alpha /2)},$$

(7)

where

$$P=P(x)=x_1^2+x_2^2+\cdots +x_p^2-x_{p+1}^2-x_{p+2}^2-\cdots -x_{p+q}^2,$$

(8)

$p+q=n$, and q is the number of negative terms of the quadratic form P. The distributions ${(P\pm i0)}^{\lambda }$ are defined by

$${(P\pm i0)}^{\lambda }=\underset{ϵ\to 0}{lim}{(P\pm iϵ|x|}^2{)}^{\lambda },$$

(9)

where $\lambda \in \mathbb{C},ϵ>0,$ and ${\left|x\right|}^2=x_1^2+x_2^2+\cdots +x_n^2$, see [1]. They also showed the distributional functions $H_{\alpha }(P\pm i0,n)$ 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 $\alpha ,\alpha \in \mathbb{C},$ by

$$R^{\alpha }\phi =H_{\alpha }(P\pm i0,n)\ast \phi ,$$

(10)

where $\phi \in \mathcal{S},\mathcal{S}$ is the Schwartz space of functions [17], and $H_{\alpha }(P\pm i0,n)$ is defined by Equation (7). They also studied the operator ${(R^{\alpha })}^{-1}$, that is the inverse operator of $R^{\alpha }$, such that $f=R^{\alpha }\phi$ implies ${(R^{\alpha })}^{-1}f=\phi$.

In 1999, Aguirre [18] defined the ultra-hyperbolic Marcel Riesz operator $M^{\alpha }$ of the function f by

$$M^{\alpha }(f)=R_{\alpha }\ast f,$$

(11)

where $\alpha \in \mathbb{C},R_{\alpha }$ is defined by (22), and $f\in \mathcal{S}$. He also studied the operator $N^{\alpha }={\left(M^{\alpha }\right)}^{-1}$ such that $M^{\alpha }(f)=\phi$ implies $N^{\alpha }\phi =f.$

In 2000, Kananthai [8] introduced the diamond kernel of Marcel Riesz $K_{\alpha ,\beta }$, which is given by

$$K_{\alpha ,\beta }=S_{\alpha }\ast R_{\beta },$$

(12)

where $R_{\beta }$ and $S_{\alpha }$ are defined by Equations (22) and (29), respectively. Next, Tellez and Kananthai [13] proved that $K_{\alpha ,\beta }$ exists and is in the space of tempered distributions. In addition, they also showed the relationship between the convolution of the distributional families $K_{\alpha ,\beta }$ and diamond operator iterated k times.

In 2011, Maneetus and Nonlaopon [19] defined the Bessel ultra-hyperbolic Marcel Riesz operator of order $\alpha$ on the function f by

$$U^{\alpha }(f)=R_{\alpha }^B\ast f,$$

(13)

where $\alpha \in \mathbb{C},R_{\alpha }^B$ is the Bessel ultra-hyperbolic kernel of Marcel Riesz, and $f\in \mathcal{S}$. In addition, they studied the operator $E^{\alpha }={(U^{\alpha })}^{-1}$ such that $U^{\alpha }(f)=\phi$ implies $E^{\alpha }\phi =f$. Moreover, they defined the diamond Marcel Riesz operator of order $(\alpha ,\beta )$ of the function f by

$$M^{(\alpha ,\beta )}(f)=K_{\alpha ,\beta }\ast f,$$

(14)

where $\alpha ,\beta \in \mathbb{C},K_{\alpha ,\beta }$ is defined by (12), and $f\in \mathcal{S}$; see [20], for more details. In addition, they have also studied the operator $N^{(\alpha ,\beta )}={\left[M^{(\alpha ,\beta )}\right]}^{-1}$ such that $M^{(\alpha ,\beta )}(f)=\phi$ implies $N^{(\alpha ,\beta )}\phi =f.$

In 2013, Salao and Nonlaopon [21] defined the Bessel diamond kernel of Marcel Riesz by

$$K_{\alpha ,\beta }^B(x)=S_{\alpha }^B(x)\ast R_{\beta }^B(x),$$

(15)

where $S_{\alpha }^B(x)$ and $R_{\beta }^B(x)$ 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 $(\alpha ,\beta )$ on the function f by

$$U^{(\alpha ,\beta )}(f)=K_{\alpha ,\beta }^B\ast f,$$

(16)

where $\alpha ,\beta \in \mathbb{C},K_{\alpha ,\beta }^B$ is defined by (15), and $f\in \mathcal{S}$. In addition, they studied the operator $E^{(\alpha ,\beta )}={\left[U^{(\alpha ,\beta )}\right]}^{-1}$ such that $U^{(\alpha ,\beta )}(f)=\phi$ implies $E^{(\alpha ,\beta )}\phi =f$.

In 2007, Tariboon and Kananthai [22] introduced the diamond Klein-Gordon operator ${(◊+m^2)}^k$ iterated k times, which is defined by

$${(◊+m^2)}^k={\left[{\left(\sum _{i=1}^p\frac{{\partial }^2}{\partial x_i^2}\right)}^2-{\left(\sum _{j=p+1}^{p+q}\frac{{\partial }^2}{\partial x_j^2}\right)}^2+m^2\right]}^k,$$

(17)

where $m\ge 0$, k is non-negative integer, $p+q=n$ is the dimension of ${\mathbb{R}}^n$, for all $x=(x_1,x_2,\dots ,x_n)\in {\mathbb{R}}^n$. 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 $e^{\alpha x}{(◊+m^2)}^k\delta$ and showed the boundedness property of the distribution $e^{\alpha x}{(◊+m^2)}^k\delta$, where ${(◊+m^2)}^k$ is defined by Equation (17), $\alpha \in \mathbb{C},$ and $\delta$ is Dirac delta function.

In 2013, Sattaso and Nonlaopon [26] defined the diamond Klein-Gordon operator of order $\alpha$ on the function f by

$$D^{\alpha }(f)=T_{\alpha }\ast f,$$

(18)

where $\alpha \in \mathbb{C}$, and $T_{\alpha }$ is the diamond Klein-Gordon kernel. They also studied the convolution of $T_{\alpha }$ and obtain the operator $L^{\alpha }={[D^{\alpha }]}^{-1}$ such that $D^{\alpha }(f)=\phi$ implies $L^{\alpha }\phi =f.$

In 1988, Trione [27] studied the fundamental solution of the ultrahyperbolic Klein-Gordon operator ${({\displaystyle \square }+m^2)}^k$ iterated k times, which is defined by

$${({\displaystyle \square }+m^2)}^k={\left(\sum _{i=1}^p\frac{{\partial }^2}{\partial x_i^2}-\sum _{j=p+1}^{p+q}\frac{{\partial }^2}{\partial x_j^2}+m^2\right)}^k.$$

(19)

She showed that $W_{2k}(x,m),$ defined by Equation (37) with $\alpha =2k$, is the fundamental solution of the operator ${({\displaystyle \square }+m^2)}^k$, which is called the ultra-hyperbolic Klein-Gordon kernel. Next, Tellez [28] studied the convolution product of $W_{\alpha }(x,m)\ast W_{\beta }(x,m)$, where $\alpha ,\beta \in \mathbb{C}$. In addition, Trione [29] has studied the fundamental ${(P\pm i0)}^{\lambda }$-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 $W_{\alpha }(x,m)$, see [30] for more details.

In this paper, we define the Klein-Gordon operator of order $\alpha$ of the function f by

$$T^{\alpha }(f)=W_{\alpha }\ast f,$$

(20)

where $\alpha \in \mathbb{C},W_{\alpha }$ is the ultra-hyperbolic Klein-Gordon kernel defined by Equation (37), and $f\in \mathcal{S}.$ Our aim of this paper is to obtain the operator $L^{\alpha }={\left[T^{\alpha }\right]}^{-1}$ such that if $T^{\alpha }(f)=\phi$ then $L^{\alpha }\phi =f.$

Before we proceed to that point, we clarify some concepts and definitions.

2. Preliminaries

Definition 1.

Let $x=(x_1,x_2,\dots ,x_n)$ be a point of the n-dimensional Euclidean space ${\mathbb{R}}^n$ and

$$u=x_1^2+x_2^2+\cdots +x_p^2-x_{p+1}^2-x_{p+2}^2-\cdots -x_{p+q}^2$$

(21)

be the non-degenerated quadratic form, where $p+q=n$ is the dimension of ${\mathbb{R}}^n$. Let ${\Gamma }_{+}=\{x\in {\mathbb{R}}^n:x_1>0\mathit{and}u>0\}$ be the interior of a forward cone and let ${\overline{\Gamma }}_{+}$ denote its closure. For any complex number γ, we define

$$R_{\gamma }(x)=\left\{\begin{array}{cc}\frac{u^{(\gamma -n)/2}}{K_n(\gamma )},\hfill & \mathit{for}x\in {\Gamma }_{+};\hfill \\ 0,\hfill & \mathit{for}x\notin {\Gamma }_{+},\hfill \end{array}\right.$$

(22)

where

$$K_n(\gamma )=\frac{{\pi }^{(n-1)/2}\Gamma ((2+\gamma -n)/2)\Gamma ((1-\gamma )/2)\Gamma (\gamma )}{\Gamma ((2+\gamma -p)/2)\Gamma ((p-\gamma )/2)}.$$

(23)

The function $R_{\gamma }(x)$, which was introduced by Y. Nozaki [31], is called the ultra-hyperbolic kernel of Marcel Riesz. It is well known that $R_{\gamma }(x)$ is an ordinary function when $\mathrm{Re}(\gamma )\ge n$ and is a distribution of $\gamma$ otherwise. The support of $R_{\gamma }(x)$ is denoted by supp $R_{\gamma }(x)$ and suppose that supp $R_{\gamma }(x)\subset {\overline{\Gamma }}_{+}$, that is, supp $R_{\gamma }(x)$ is compact.

By putting $p=1$ in $R_{\gamma }(x)$ and taking into the Legendre’s duplication formula

$$\Gamma (2z)=2^{2z-1}{\pi }^{-1/2}\Gamma (z)\Gamma \left(z+1/2\right),$$

(24)

we obtain

$$I_{\gamma }(x)=\frac{v^{(\gamma -n)/2}}{H_n(\gamma )},$$

(25)

and $v=x_1^2-x_2^2-x_3^2-\cdots -x_n^2$, where

$$H_n(\gamma )={\pi }^{(n-2)/2}{2}^{\gamma -1}\Gamma \left((\gamma +2-n)/2\right)\Gamma \left(\gamma /2\right).$$

(26)

The function $I_{\gamma }(x)$ is called the hyperbolic kernel of Marcel Riesz.

From [2], the generalized function $R_{2k}(x)$ is the fundamental solution of the operator ${{\displaystyle \square }}^k$, that is

$${{\displaystyle \square }}^k\left(R_{2k}(x)\right)=\delta .$$

(27)

In addition, it can be shown that

$$R_{-2k}(x)={{\displaystyle \square }}^k\delta$$

(28)

for k is a nonnegative integer, see [2,13].

Definition 2.

Let $x=(x_1,x_2,\dots ,x_n)$ be a point of ${\mathbb{R}}^n$ and $\omega =x_1^2+x_2^2+\cdots +x_n^2$. The elliptic kernel of Marcel Riesz is defined by

$$S_{\gamma }(x)=\frac{{\omega }^{(\gamma -n)/2}}{U_n(\gamma )},$$

(29)

where $\gamma \in \mathbb{C},n$ is the dimension of ${\mathbb{R}}^n,$ and

$$U_n(\gamma )=\frac{{\pi }^{n/2}{2}^{\gamma }\Gamma (\gamma /2)}{\Gamma ((n-\gamma )/2)}.$$

(30)

Note that $n=p+q$. By putting $q=0$ (i.e., $n=p$) in (22) and (23), we can reduce $u^{(\gamma -n)/2}$ to ${\omega }_p^{(\gamma -p)/2}$, where ${\omega }_p=x_1^2+x_2^2+\cdots +x_p^2$, and reduce $K_n(\gamma )$ to

$$K_p(\gamma )=\frac{{\pi }^{(p-1)/2}\Gamma ((1-\gamma )/2)\Gamma (\gamma )}{\Gamma ((p-\gamma )/2)}.$$

Using the Legendre’s duplication formula (Equation (24)) and

$$\Gamma \left(1/2+z\right)\Gamma \left(1/2-z\right)=\pi sec(\pi z),$$

(31)

we obtain

$$K_p(\gamma )=\frac{1}{2}sec\left(\gamma \pi /2\right)U_p(\gamma ).$$

(32)

Thus, for $q=0$, we have

$$R_{\gamma }(x)=\frac{u^{(\gamma -p)/2}}{K_p(\gamma )}=2cos\left(\gamma \pi /2\right)\frac{u^{(\gamma -p)/2}}{U_p(\gamma )}=2cos\left(\gamma \pi /2\right)S_{\gamma }(x).$$

(33)

In addition, if $\gamma =2k$ for some non-negative integer k, then

$$R_{2k}(x)=2{(-1)}^k{S}_{2k}(x).$$

(34)

Next, we consider the function

$$W_{\alpha }(x,m)=\left\{\begin{array}{cc}\frac{{(m^{-2}u)}^{(\alpha -n)/4}}{2^{(\alpha +n-2)/2}{\pi }^{(n-2)/2}\Gamma \left(\alpha /2\right)}{J}_{\frac{\alpha -n}{2}}(m^2{u}^{1/2}),\hfill & \mathrm{for}x\in {\Gamma }_{+};\hfill \\ 0,\hfill & \mathrm{for}x\notin {\Gamma }_{+},\hfill \end{array}\right.$$

(35)

where $\alpha \in \mathbb{C},u$ is defined by Equation (21), m a real non-negative number, n is the dimension of ${\mathbb{R}}^n$, and $J_{\nu }(z)$ is Bessel function of the first kind, which is defined by

$$J_{\nu }(z)=\sum _{k=0}^{\infty }\frac{{(-1)}^k{(z/2)}^{2k+\nu }}{k!\Gamma (k+\nu +1)}.$$

(36)

It is well known that $W_{\alpha }(x,m)$ is an ordinary function when $\mathrm{Re}(\alpha )\ge n$ and is a distribution otherwise. In addition, $W_{\alpha }(x,m)$ can be expressed as an infinitely linear combination of $R_{\alpha }(x)$ of different orders, that is

$$W_{\alpha }(x,m)=\sum _{\nu =0}^{\infty }\left(\genfrac{}{}{0pt}{}{-\alpha /2}{\nu }\right)m^{2\nu }{R}_{\alpha +2\nu }(x),$$

(37)

where $\alpha \in \mathbb{C},R_{\alpha }(x)$ is defined by Equation (22), see [27,29,30], for more details.

From Equation (37) and by putting $\alpha =-2k$, for k is non-negative integer, we have

$$W_{-2k}(x,m)=\sum _{\nu =0}^{\infty }\left(\genfrac{}{}{0pt}{}{k}{\nu }\right)m^{2\nu }{R}_{-2(k-\nu )}(x).$$

(38)

Since the operator ${({\displaystyle \square }+m^2)}^k$ defined by Equation (19) is linearly continuous and injective mapping of this possess its own inverse. From Equation (28), we obtain

$$W_{-2k}(x,m)=\sum _{\nu =0}^{\infty }\left(\genfrac{}{}{0pt}{}{k}{\nu }\right)m^{2\nu }{{\displaystyle \square }}^{k-\nu }\delta ={({\displaystyle \square }+m^2)}^k\delta .$$

(39)

Substituting $k=0$ in Equation (39), yields $W_0(x,m)=\delta$. On the other hand, by putting $\alpha =2k$ in Equation (37), yields

$$W_{2k}(x,m)=\left(\genfrac{}{}{0pt}{}{-k}{0}\right)m^{2(0)}{R}_{2k+0}(x)+\sum _{\nu =1}^{\infty }\left(\genfrac{}{}{0pt}{}{-k}{\nu }\right)m^{2\nu }{R}_{2k+2\nu }(x).$$

(40)

The second summand of the right-hand side of Equation (40) vanishes when $m^2=0$. Therefore, we obtain

$$W_{2k}(x,m=0)=R_{2k}(x),$$

is the fundamental solution of the ultra-hyperbolic operator ${{\displaystyle \square }}^k$. For the convenience, we will denote $W_{\alpha }(x,m)$ by $W_{\alpha }$.

The proof of Lemmas 1 and 2 are given in [28].

Lemma 1.

The function $W_{\alpha }$ has the following properties:

(i) 

$W_0=\delta$;

(ii) 

$W_{-2k}={({\displaystyle \square }+m^2)}^k\delta ;$

(iii) 

${({\displaystyle \square }+m^2)}^k{W}_{\alpha }=W_{\alpha -2k};$

(iv) 

${({\displaystyle \square }+m^2)}^k{W}_{2k}=\delta ;$

(v) 

$W_{\alpha }\ast W_{-2k}={({\displaystyle \square }+m^2)}^k{W}_{\alpha }$.

Lemma 2.

(The convolutions of $W_{\alpha }$)

(i) 

If p is odd, then

$$W_{\alpha }\ast W_{\beta }=W_{\alpha +\beta }+A_{\alpha ,\beta },$$

(41)

where

$$A_{\alpha ,\beta }=\frac{i}{2}\frac{sin(\alpha \pi /2)sin(\beta \pi /2)}{sin((\alpha +\beta )\pi /2)}\sum _{\nu =0}^{\infty }{(m^2)}^{\nu }\left(\genfrac{}{}{0pt}{}{-(\alpha +\beta )/2}{\nu }\right)\left[H_{\alpha +\beta +2\nu }^{+}-H_{\alpha +\beta +2\nu }^{-}\right]$$

(42)

and

$$H_{\alpha +\beta +2\nu }^{\pm }=H_{\alpha +\beta +2\nu }(P\pm i0,n)$$

(43)

as defined by Equation (7) with $\gamma =\alpha +\beta +2\nu$.

(ii) 

If p is even, then

$$W_{\alpha }\ast W_{\beta }=B_{\alpha ,\beta }{W}_{\alpha +\beta },$$

(44)

where

$$B_{\alpha ,\beta }=\frac{cos(\alpha \pi /2)cos(\beta \pi /2)}{cos\left((\alpha +\beta )\pi /2\right)}.$$

3. The Convolution of ${\mathit{W}}_{\mathit{\alpha }}\ast {\mathit{W}}_{\mathit{\beta }}$ when $\mathit{\beta }=-\mathit{\alpha }$

In this section, we will consider the property of $W_{\alpha }\ast W_{\beta }$ when $\beta =-\alpha$.

From Equations (41) and (44), we immediately obtain the following properties:

• If p is odd and q is even, then

  $$W_{\alpha }\ast W_{\beta }=W_{\alpha +\beta }+A_{\alpha ,\beta },$$

  (45)

  where $A_{\alpha ,\beta }$ is defined by Equation (42).
• If p and q are both odd, then

  $$W_{\alpha }\ast W_{\beta }=W_{\alpha +\beta }+A_{\alpha ,\beta }.$$

  (46)
• If p is even and q is odd, then

  $$W_{\alpha }\ast W_{\beta }=\frac{cos(\alpha \pi /2)cos(\beta \pi /2)}{cos\left((\alpha +\beta )\pi /2\right)}{W}_{\alpha +\beta }.$$

  (47)
• If p and q are both even, then

  $$W_{\alpha }\ast W_{\beta }=\frac{cos(\alpha \pi /2)cos(\beta \pi /2)}{cos\left((\alpha +\beta )\pi /2\right)}{W}_{\alpha +\beta }.$$

  (48)

Moreover, it follows from Equation (42) that

$$\begin{array}{cc}\hfill A_{\alpha ,-\alpha }& =\underset{\beta \to -\alpha }{lim}{A}_{\alpha ,\beta }\hfill \\ & =\frac{i}{2}\underset{\gamma \to 0}{lim}\frac{sin(\alpha \pi /2)sin((\gamma -\alpha )\pi /2)}{sin(\gamma \pi /2)}\sum _{\nu =0}^{\infty }{(m^2)}^{\nu }\left(\genfrac{}{}{0pt}{}{-\gamma /2}{\nu }\right)\left[H_{\gamma +2\nu }^{+}-H_{\gamma +2\nu }^{-}\right]\hfill \\ & =\frac{i}{2}\underset{\gamma \to 0}{lim}\frac{sin(\alpha \pi /2)sin((\gamma -\alpha )\pi /2)}{sin(\gamma \pi /2)}\hfill \\ & \times \underset{\gamma \to 0}{lim}\left\{\left[H_{\gamma }^{+}-H_{\gamma }^{-}\right]+\sum _{\nu =1}^{\infty }{(m^2)}^{\nu }\left(\genfrac{}{}{0pt}{}{-\gamma /2}{k}\right)\left[H_{\gamma +2\nu }^{+}-H_{\gamma +2\nu }^{-}\right]\right\}\hfill \\ & =\frac{i}{2}\underset{\gamma \to 0}{lim}\frac{sin(\alpha \pi /2)sin((\gamma -\alpha )\pi /2)}{sin(\gamma \pi /2)}·\underset{\gamma \to 0}{lim}\left[H_{\gamma }^{+}-H_{\gamma }^{-}\right],\hfill \end{array}$$

(49)

where $\gamma =\alpha +\beta$.

On the other hand, using Equations (43) and (7), we have

$$\begin{array}{cc}\hfill \underset{\gamma \to 0}{lim}[H_{\gamma }^{+}-H_{\gamma }^{-}]& =\frac{\Gamma (n/2)}{{\pi }^{n/2}}\left[\underset{\gamma \to 0}{lim}{e}^{-\gamma \pi i/2}{e}^{q\pi i/2}\frac{{(P+i0)}^{(\gamma -n)/2}}{\Gamma (\gamma /2)}\right.\hfill \\ & \left.-\underset{\gamma \to 0}{lim}{e}^{\gamma \pi i/2}{e}^{-q\pi i/2}\frac{{(P-i0)}^{(\gamma -n)/2}}{\Gamma (\gamma /2)}\right]\hfill \\ & =\frac{\Gamma (n/2)}{{\pi }^{n/2}}\left[\underset{\gamma \to 0}{lim}{e}^{-\gamma \pi i/2}{e}^{q\pi i/2}·\frac{\underset{\beta =-n/2}{\mathrm{Res}}{(P+i0)}^{\beta }}{\underset{\beta =-n/2}{\mathrm{Res}}\Gamma (\beta +n/2)}\right.\hfill \\ & \left.-\underset{\gamma \to 0}{lim}{e}^{\gamma \pi i/2}{e}^{-q\pi i/2}·\frac{\underset{\beta =-n/2}{\mathrm{Res}}{(P-i0)}^{\beta }}{\underset{\beta =-n/2}{\mathrm{Res}}\Gamma (\beta +n/2)}\right].\hfill \end{array}$$

(50)

Now, taking n as an odd integer, yields

$$\underset{\lambda =-n/2-k}{\mathrm{Res}}{(P\pm i0)}^{\lambda }=\frac{e^{\pm q\pi i/2}{\pi }^{n/2}}{2^{2k}k!\Gamma \left(n/2+k\right)}{{\displaystyle \square }}^k\delta ,$$

(51)

where ${{\displaystyle \square }}^k$ is defined by (2), $p+q=n$, and k is non-negative integer; see [32,33]. If p and q are both even, then

$$\underset{\lambda =-n/2-k}{\mathrm{Res}}{(P\pm i0)}^{\lambda }=\frac{e^{\pm q\pi i/2}{\pi }^{n/2}}{2^{2k}k!\Gamma \left(n/2+k\right)}{{\displaystyle \square }}^k\delta .$$

(52)

Nevertheless, if p and q are both odd, then

$$\underset{\lambda =-n/2-k}{\mathrm{Res}}{(P\pm i0)}^{\lambda }=0,$$

(53)

Therefore, we have

$$\begin{array}{cc}\hfill \underset{\gamma \to 0}{lim}[H_{\gamma }^{+}-H_{\gamma }^{-}]& =\frac{\Gamma \left(n/2\right)}{{\pi }^{n/2}}·\frac{{\pi }^{n/2}}{\Gamma \left(n/2\right)}\left[\underset{\gamma \to 0}{lim}{e}^{-\gamma \pi i/2}-\underset{\gamma \to 0}{lim}{e}^{\gamma \pi i/2}\right]\delta \hfill \\ & =\underset{\gamma \to 0}{lim}\left[-2isin\left(\gamma \pi /2\right)\right]\delta .\hfill \end{array}$$

(54)

From Equations (50) and (53), we have

$$\underset{\gamma \to 0}{lim}[H_{\gamma }^{+}-H_{\gamma }^{-}]=0$$

(55)

if p and q are both odd (n even).

Applying Equations (54) and (55) into Equation (49), we have

$$\begin{array}{cc}\hfill A_{\alpha ,-\alpha }& =\frac{i}{2}\underset{\gamma \to 0}{lim}\frac{sin(\alpha \pi /2)sin((\gamma -\alpha )\pi /2)}{sin(\gamma \pi /2)}·\underset{\gamma \to 0}{lim}\left[-2isin\left(\gamma \pi /2\right)\right]\delta (x)\hfill \\ & ={sin}^2\left(\alpha \pi /2\right)\delta \hfill \end{array}$$

(56)

if p is odd and q is even, and

$$\begin{array}{c}\hfill A_{\alpha ,-\alpha }=0\end{array}$$

(57)

if p and q are both odd.

From Equations (45)–(48) and using Lemmas 1, 2 and Equations (56) and (57), if p is odd and q is even, then we obtain

$$\begin{array}{cc}\hfill W_{\alpha }\ast W_{-\alpha }& =W_0+A_{\alpha ,-\alpha }\hfill \\ & =\delta +{sin}^2\left(\alpha \pi /2\right)\delta \hfill \\ & =\left[1+{sin}^2\left(\alpha \pi /2\right)\right]\delta .\hfill \end{array}$$

(58)

If p and q are both odd, then

$$\begin{array}{cc}\hfill W_{\alpha }\ast W_{-\alpha }& =W_0=\delta .\hfill \end{array}$$

(59)

If p is even and q is odd, then

$$\begin{array}{cc}\hfill W_{\alpha }\ast W_{-\alpha }& =\frac{cos\left(\alpha \pi /2\right)cos\left(-\alpha \pi /2\right)}{cos\left((\alpha -\alpha )\pi /2\right)}{W}_0\hfill \\ & ={cos}^2\left(\alpha \pi /2\right)\delta .\hfill \end{array}$$

(60)

Finally, if p and q are both even, then

$$\begin{array}{cc}\hfill W_{\alpha }\ast W_{-\alpha }& =\frac{cos\left(\alpha \pi /2\right)cos\left(-\alpha \pi /2\right)}{cos\left((\alpha -\alpha )\pi /2\right)}{W}_0\hfill \\ & ={cos}^2\left(\alpha \pi /2\right)\delta .\hfill \end{array}$$

(61)

4. The Main Theorem

Let $T^{\alpha }(f)$ be the ultrahyperbolic Klein-Gordon operator of order $\alpha$ on the function f, which is defined by

$$T^{\alpha }(f)=W_{\alpha }\ast f,$$

(62)

where $\alpha \in \mathbb{C},W_{\alpha }$ is defined by Equation (37), and $f\in \mathcal{S}$.

Recall that our objective is to obtain the operator $L^{\alpha }={\left[T^{\alpha }\right]}^{-1}$ such that if $T^{\alpha }(f)=\phi$, then $L^{\alpha }\phi =f$ for all $\alpha \in \mathbb{C}$.

Theorem 1.

If $T^{\alpha }(f)=\phi$, then $L^{\alpha }\phi =f$ such that

$$L^{\alpha }={\left[T^{\alpha }\right]}^{-1}=\left\{\begin{array}{cc}{\left[1+{sin}^2(\alpha \pi /2)\right]}^{-1}{W}_{-\alpha },\hfill & \mathit{if}p\mathit{is}\mathit{odd}\mathit{and}q\mathit{is}\mathit{even};\hfill \\ W_{-\alpha },\hfill & \mathit{if}p\mathit{and}q\mathit{are}\mathit{both}\mathit{odd};\hfill \\ {sec}^2\left(\alpha \pi /2\right)W_{-\alpha },\hfill & \mathit{if}p\mathit{is}\mathit{even}\mathit{with}\alpha /2\ne 2s+1\hfill \end{array}\right.$$

for any non-negative integer s.

Proof. 

By Equation (62), we have

$$T^{\alpha }(f)=W_{\alpha }\ast f=\phi ,$$

where $\alpha \in \mathbb{C},W_{\alpha }$ is defined by Equation (37), and $f\in \mathcal{S}$. If p is odd and q is even, then, in view of Equation (58), we obtain

$$\begin{array}{cc}\hfill {\left[1+{sin}^2(\alpha \pi /2)\right]}^{-1}{W}_{-\alpha }\ast \left(W_{\alpha }\ast f\right)& ={\left[1+{sin}^2(\alpha \pi /2)\right]}^{-1}\left(W_{-\alpha }\ast W_{\alpha }\right)\ast f\hfill \\ & ={\left[1+{sin}^2(\alpha \pi /2)\right]}^{-1}\left\{\left[1+{sin}^2(\alpha \pi /2)\right]\delta \right\}\ast f\hfill \\ & =\delta \ast f=f.\hfill \end{array}$$

Therefore,

$${\left[1+{sin}^2(\alpha \pi /2)\right]}^{-1}{W}_{-\alpha }={\left[T^{\alpha }\right]}^{-1}={\left(W_{\alpha }\right)}^{-1}$$

(63)

for all $\alpha \in \mathbb{C}$.

Similarly, if both p and q are odd, then by Equation (59), we obtain

$$W_{-\alpha }\ast \left(W_{\alpha })\ast f\right)=\left(W_{-\alpha }\ast W_{\alpha }\right)\ast f=\delta \ast f=f.$$

Therefore,

$$W_{-\alpha }={\left[T^{\alpha }\right]}^{-1}={\left(W_{\alpha }\right)}^{-1}$$

(64)

for all $\alpha \in \mathbb{C}$.

Finally, if p is even, then by Equations (60) and (61), we have

$$\begin{array}{cc}\hfill {sec}^2\left(\alpha \pi /2\right)W_{-\alpha }\ast \left(W_{\alpha }\ast f\right)& ={sec}^2\left(\alpha \pi /2\right)\left(W_{-\alpha }\ast W_{\alpha }\right)\ast f\hfill \\ & ={sec}^2\left(\alpha \pi /2\right)\left\{{cos}^2\left(\alpha \pi /2\right)\delta \right\}\ast f\hfill \\ & =\delta \ast f=f,\hfill \end{array}$$

provided that $\beta /2\ne 2s+1$ for any non-negative integer s. Therefore,

$${sec}^2\left(\alpha \pi /2\right)W_{-\alpha }={\left[T^{\alpha }\right]}^{-1}={\left(W_{\alpha }\right)}^{-1}$$

(65)

for all $\alpha \in \mathbb{C}$ with $\alpha /2\ne 2s+1$ for any non-negative integer s.

Therefore, we have the desired results in Equations (63)–(65). □

5. Conlusions

In this work, we have considered the property of convolution of the ultrahyperbolic Klein-Gordon kernel in the form $W_{\alpha }\ast W_{\beta }$ when $\beta =-\alpha$. We have obtained the inverse ultrahyperbolic Klein-Gordon kernel, that is, the operator $L^{\alpha }={\left[T^{\alpha }\right]}^{-1}$ such that if $T^{\alpha }(f)=\phi$, then $L^{\alpha }\phi =f$ for all $\alpha \in \mathbb{C}$. It is expected that this work may stimulate further research in this field.