On the Inversion of Bessel Ultrahyperbolic Kernel of Marcel Riesz

and Applied Analysis 3 Here, P is defined by P P x x2 1 x 2 2 · · · x2 p − x2 p 1 − x2 p 2 − · · · − x2 p q, 1.9 where q is the number of negative terms of the quadratic form P . The distributions P ± i0 λ are defined by P ± i0 λ lim → 0 ( P ± i |x| )λ , 1.10 where > 0, λ ∈ C, and |x|2 x2 1 x2 2 · · · x2 n; see 1 . They have also studied the inverse operator of R, denoted by R −1, such that, if f Rφ, then R −1f φ. Later, Aguirre 15 has defined the ultrahyperbolic Marcel Riesz operator M of the function f by M ( f ) Rα ∗ f, 1.11 where Rα is defined by 2.6 and f ∈ S. He has also studied the operator N M −1 such that, ifM f φ, thenNφ f . Let us consider the diamond kernel of Marcel Riesz Kα,β x introduced by Kananthai in 6 , which is given by the convolution Kα,β x Rα ∗ RHβ , 1.12 where Rα is elliptic kernel defined by 2.11 and R H β is the ultrahyperbolic kernel defined by 2.6 . Tellez and Kananthai 16 have proved that Kα,β x exists and is in the space of rapidly decreasing distributions. Moreover, they have also shown that the convolution of the distributional families Kα,β x relates to the diamond operator. Later, Maneetus and Nonlaopon 17 have defined the diamondMarcel Riesz operator of order α, β of the function f by M α,β ( f ) Kα,β ∗ f, 1.13 where Kα,β is defined by 1.12 , α, β ∈ C, and f ∈ S. They have also studied the operator N α,β M α,β −1 such that, ifM α,β f φ, thenN α,β φ f . In this paper, we define the Bessel ultrahyperbolic Marcel Riesz operator of order α of the function f by U ( f ) Rα ∗ f, 1.14 where α ∈ C and f ∈ S, S is the Schwartz space of functions. Our aim in this paper is to obtain the operator E U −1 such that, if U f φ, then Eφ f . Before we proceed to ourmain theorem, the following definitions and concepts require some clarifications. 4 Abstract and Applied Analysis 2. Preliminaries Definition 2.1. Let x x1, x2, . . . , xn be a point in the n-dimensional Euclidean space R. Let u x2 1 x 2 2 · · · x2 p − x2 p 1 − x2 p 2 − · · · − x2 p q 2.1 be the nondegenerated quadratic form, where p q n is the dimension of R. Let Γ {x ∈ R n : u > 0 and xi > 0 i 1, 2, . . . , p } be the interior of a forward cone, and let Γ denote its closure. For any complex number γ , we define Rγ x ⎪⎨ ⎪⎩ u γ−2|ν|−n /2 K |ν| n ( γ ) , for x ∈ Γ , 0, for x / ∈ Γ , 2.2


Introduction
The n-dimensional ultrahyperbolic operator k iterated k times is defined by where p q n is the dimension of R n and k is a nonnegative integer.Consider the linear differential equation in the form of where u x and f x are generalized functions and x x 1 , x 2 , . . ., x n ∈ R n .Gel fand and Shilov 1 have first introduced the fundamental solution of 1.2 , which is a complicated form.Later, Trione 2 has shown that the generalized function R H 2k x , Abstract and Applied Analysis defined by 2.6 with γ 2k, is the unique fundamental solution of 1.2 and Téllez 3 has also proved that R H 2k x exists only when n p q with odd p. Next, Kananthai 4 has first introduced the operator ♦ k called the diamond operator iterated k times, which is defined by where n p q is the dimension of R n , for all x x 1 , x 2 , . . ., x n , and k is a nonnegative integer.The operator ♦ k can be expressed in the form where k is defined by 1.1 , and is the Laplace operator iterated k times.On finding the fundamental solution of this product, Kananthai uses the convolution of functions which are fundamental solutions of the operators k and k .He found that the convolution −1 k R e 2k x * R H 2k x is the fundamental solution of the operator ♦ k , that is, where R H 2k x and R e 2k x are defined by 2.6 and 2.11 , respectively with γ 2k and δ x is the Dirac delta distribution.The fundamental solution −1 k R e 2k x * R H 2k x is called the diamond kernel of Marcel Riesz.A wealth of some effective works on the diamond kernel of Marcel Riesz have been presented by  In 1978, Domínguez and Trione 11 have introduced the distributional functions H α P ± i0, n which are causal anticausal analogues of the elliptic kernel of Riesz 12 .Next, Cerutti and Trione 13 have defined the causal anticausal generalized Marcel Riesz potentials of order α, α ∈ C, by where ϕ ∈ S, S is the Schwartz space of functions 14 and H α P ± i0, n is given by Abstract and Applied Analysis 3 Here, P is defined by where q is the number of negative terms of the quadratic form P .The distributions P ± i0 λ are defined by where > 0, λ ∈ C, and Later, Aguirre 15 has defined the ultrahyperbolic Marcel Riesz operator M α of the function f by where R H α is defined by 2.6 and f ∈ S.He has also studied the operator N α M α −1 such that, if M α f ϕ, then N α ϕ f.Let us consider the diamond kernel of Marcel Riesz K α,β x introduced by Kananthai in 6 , which is given by the convolution where R e α is elliptic kernel defined by 2.11 and R H β is the ultrahyperbolic kernel defined by 2.6 .Tellez and Kananthai 16 have proved that K α,β x exists and is in the space of rapidly decreasing distributions.Moreover, they have also shown that the convolution of the distributional families K α,β x relates to the diamond operator.
Later, Maneetus and Nonlaopon 17 have defined the diamond Marcel Riesz operator of order α, β of the function f by where K α,β is defined by 1.12 , α, β ∈ C, and f ∈ S. They have also studied the operator In this paper, we define the Bessel ultrahyperbolic Marcel Riesz operator of order α of the function f by where α ∈ C and f ∈ S, S is the Schwartz space of functions.Our aim in this paper is to obtain the operator E α U α −1 such that, if U α f ϕ, then E α ϕ f.Before we proceed to our main theorem, the following definitions and concepts require some clarifications.

Preliminaries
2.1 be the nondegenerated quadratic form, where p q n is the dimension of R n .Let Γ {x ∈ R n : u > 0 and x i > 0 i 1, 2, . . ., p } be the interior of a forward cone, and let Γ denote its closure.For any complex number γ, we define where where By putting |ν| 0 in 2.2 and 2.3 , then formulae 2.2 and 2.3 reduce to The function R H γ x is called the ultrahyperbolic kernel of Marcel Riesz and was introduced by Nozaki 22 we obtain , where

2.10
The function I H γ x is called the hyperbolic kernel of Marcel Riesz.
Definition 2.2.Let x x 1 , x 2 , . . ., x n be a point of R n and ω x 2 The elliptic kernel of Marcel Riesz is defined by where n is the dimension of R n , γ ∈ C, and Note that n p q.By putting q 0 i.e., n p in 2.6 and 2.7 , we can reduce p , and reduce K n γ to

2.13
Using Legendre's duplication formula and we obtain Thus, for q 0, we have 2.17 In addition, if γ 2k for some nonnegative integer k, then

2.18
The proofs of Lemma 2.3 are given in 2 .

Lemma 2.3. The function R H α x has the following properties:
i where R B γ x and R H γ−2|ν| x are defined by 2.2 and 2.6 , respectively, and Proof.We get 2.19 by computing directly from definition of R B γ x and R H γ−2|ν| x .
The proof of the following lemma is given in 23 .The proof of this lemma can be easily seen from Lemmas 2.4, 2.5 and 23 .

The Convolution R B α x * R B β x When β −α
We will now consider the property of R B α x * R B β x when β −α.From 2.26 and 2.27 , we immediately obtain the following properties.
1 If p is odd and q is even, then where R H α x and A α−2|ν|,β−2|μ| are defined by 2.6 and 2.22 , respectively. 8 Abstract and Applied Analysis 2 If p and q are both odd, then 3 If p is even and q is odd, then 4 If p and q are both even, then

3.4
Moreover, it follows from 2.22 that

3.6
Abstract and Applied Analysis 9 Now, taking n as an odd integer, we obtain where k is defined by 1.1 , p q n, and k is nonnegative integer; see 24, 25 .If p and q are both even, then Nevertheless, if p and q are both odd, then Therefore, we have

3.10
From 3.6 and 3.9 , we have if p and q are both odd n even .Applying 3.10 and 3.11 into 3.5 , we have 12 if p is odd and q is even and A α−2|ν|,−α 2|ν| 0 3.13 if p and q are both odd.
From 3.1 -3.4 and using Lemmas 2.3, and 2.6 and formulae 3.12 and 3.13 , if p is odd and q is even, then we obtain

3.14
If p and q are both odd, then

3.15
If p is even and q is odd, then

3.16
Finally, if p and q are both even, then 3.17

The Main Theorem
Let M α f be the Bessel ultrahyperbolic Marcel Riesz operator of order α of the function f, which is defined by where R B α is defined by 2.2 , α ∈ C, and f ∈ S. Recall that our objective is to obtain the operator E α U α −1 such that, if U α f ϕ, then E α ϕ f for all α ∈ C.
We are now ready to state our main theorem.
Theorem 4.1.If U α f ϕ (where U α f is defined by 4.1 and f ∈ S), then E α ϕ f such that if p is odd and q is even, if p and q are both odd, for any nonnegative integer s.
Proof.By 4.1 , we have where R B α is defined by 2.2 , α ∈ C, and f ∈ S. If p is odd and q is even, then, in view of 3.14 , we obtain 4.9 for all α ∈ C with α − 2|ν| /2 / 2s 1 for any nonnegative integer s.In this conclusion, formulae 4.5 , 4.7 , and 4.9 are the desired results, and this completes the proof.