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

and Applied Analysis 3 In 2000, Kananthai 15 has studied the application of the distribution e δ for solving the convolution equation e δ ∗ u x e m ∑ r 0 Cr δ, 1.8 which is related to the ultrahyperbolic equation. In 2009, Sasopa and Nonlaopon 16 have studied the properties of the distribution e cδ and its application to solve the convolution equation e cδ ∗ u x e m ∑ r 0 Cr cδ. 1.9 Here, c is the operator related to the ultrahyperbolic type operator iterated k times, which is defined by c ⎛ ⎝ 1 c2 p ∑ i 1 ∂2 ∂x2 i − p q ∑ j p 1 ∂2 ∂x2 j ⎞ ⎠ k , 1.10 where p q n is the dimension of R. In 1988, Trione 17 has studied the fundamental solution of the ultrahyperbolic KleinGordon operator iterated k times, which is defined by ( m2 )k ⎛ ⎝ p ∑ i 1 ∂2 ∂x2 i − p q ∑ j p 1 ∂2 ∂x2 j m2 ⎞ ⎠ k . 1.11 The fundamental solution of the operator m2 k is given by W2k x,m ∞ ∑ r 0 −1 Γ k r r!Γ k ( m2 )r −1 rR2k 2r x , 1.12 where RH2k 2r x is defined by 2.2 with γ 2k 2r. Next, Tellez 18 has studied the convolution product of Wα x,m ∗ Wβ x,m , where α and β are any complex parameter. In addition, Trione 19 has studied the fundamental P ± i0 -ultrahyperbolic solution of the Klein-Gordon operator iterated k times and the convolution of such fundamental solution. Liangprom andNonlaopon 20 have studied the properties of the distribution e m2 δ and its application for solving the convolution equation e ( m2 )k δ ∗ u x e M ∑ r 0 Cr ( m2 )r δ, 1.13 where m2 k is defined by 1.11 . 4 Abstract and Applied Analysis In 2007, Tariboon and Kananthai 21 have introduced the operator ♦ m2 k called diamond Klein-Gordon operator iterated k times, which is defined by ( ♦ m2 )k ⎡ ⎢⎣ ( p ∑ i 1 ∂2 ∂x2 i )2 − ⎛ ⎝ p q ∑ j p 1 ∂2 ∂x2 j ⎞ ⎠ 2 m2 ⎤ ⎥⎦ k


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.We consider the linear differential equation of the form where u x and f x are generalized functions, and x x 1 , x 2 , . . ., x n ∈ R n .
Abstract and Applied Analysis 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 R H 2k x defined by 2.2 with γ 2k is the unique fundamental solution of 1.2 .Tellez 3 has also proved that R H 2k x exists only when n p q with odd p. 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 ∈ R n and nonnegative integers k.The operator ♦ k can be expressed in the form where k is defined by 1.1 , and k is the Laplace operator iterated k times defined by Note that in case k 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 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 2.2 and 2.7 , respectively with γ 2k , and δ 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 number of effective results on the diamond kernel of Marcel Riesz have been presented by  In 1997, Kananthai 13 has studied the properties of the distribution e αx k δ and the application of the distribution e αx k δ 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 e αx ♦ k δ and its application for solving the convolution equation Recently, Nonlaopon gave some generalizations of his paper 6 ; see 14 for more details.Here, k c is the operator related to the ultrahyperbolic type operator iterated k times, which is defined by where p q n is the dimension of R n .In 1988, Trione 17 has studied the fundamental solution of the ultrahyperbolic Klein-Gordon operator iterated k times, which is defined by 1.11 The fundamental solution of the operator m 2 k is given by where R H 2k 2r x is defined by 2.2 with γ 2k 2r.Next, Tellez 18 has studied the convolution product of W α x, m * W β x, m , where α and β are any complex parameter.In addition, Trione 19 has studied the fundamental P ± i0 λ -ultrahyperbolic solution of the Klein-Gordon operator iterated k times and the convolution of such fundamental solution.
Liangprom and Nonlaopon 20 have studied the properties of the distribution e αx m 2 k δ and its application for solving the convolution equation where m 2 k is defined by 1.11 .
In 2007, Tariboon and Kananthai 21 have introduced the operator ♦ m 2 k called diamond Klein-Gordon operator iterated k times, which is defined by where p q n is the dimension of R n , for all x x 1 , x 2 , . . ., x n ∈ R n , m ≥ 0 and nonnegative integers k.Later, Lunnaree and Nonlaopon 22, 23 have studied the fundamental solution of operator ♦ m 2 k , 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 e αx ♦ m 2 k δ and the application of e αx ♦ m 2 k δ for solving the convolution equation where ♦ m 2 k is defined by 1.14 , u x is the generalized function, and C r is a constant.On finding the type of solution u x 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.

Preliminaries
2.1 be the nondegenerated quadratic form, where p q n is the dimension of R n .Let Γ {x ∈ R n : x 1 > 0 and u > 0} be the interior of a forward cone, and let Γ denote its closure.For any complex number γ, we define the function where the constant K n γ is given by The function R H γ x is called the ultrahyperbolic kernel of Marcel Riesz, which was introduced by Nozaki 24 we obtain , where The function I H γ x is called the hyperbolic kernel of Marcel Riesz.
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 2.2 and 2.3 , we can reduce Using the Legendre's duplication formula we obtain Thus, in case q 0, we have

2.13
In addition, if γ 2k for some nonnegative integer k, then x is the fundamental solution of the diamond operator iterated k times, that is, where γ is a complex parameter, and m is a nonnegative real number.Here, R H γ 2r x and R e γ 2r x are defined by 2.2 and 2.7 , respectively.
From the definition of T γ x, m , by putting γ −2k, we have

2.17
Since the operator ♦ m 2 k defined by 1.14 is linearly continuous and has 1-1 mapping, this possesses its own inverses.From Lemma 2.3, we obtain Substituting k 0 in 2.18 yields that we have T 0 x, m δ.On the other hand, putting γ 2k in 2.16 yields

2.19
The second summand of the right-hand side of 2.19 vanishes when m 0. Hence, we obtain 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
where ♦ m 2 k is the diamond Klein-Gordon operator iterated k times, defined by with a nonnegative integer k and the Dirac-delta distribution δ, then u x T 2k x, m is the fundamental solution of the diamond Klein-Gordon operator iterated k times ♦ m 2 k , where T 2k x, m is defined by 2.16 with γ 2k.Lemma 2.6.Let T 2k x, m be the diamond Klein-Gordon kernel defined by 2.16 , then T 2k x, m is a tempered distribution and can be expressed by where v is a nonnegative integer and v < k.Moreover, if one puts l k − v and h v, then one obtains for l h k.

Properties of the Distribution e αx ♦ m 2 k δ
Lemma 3.1.The following equality holds: 1 and e αx ♦ m 2 k δ is the tempered distribution of order 4k with support {0}, where L is the partial differential operator and is defined by

3.2
As before, is the ultrahyperbolic operator defined by 1.1 (with k 1), and is the Laplace operator defined by Proof.Let ϕ ∈ D be the space of testing functions which are infinitely differentiable with compact supports, and let D be the space of distributions.Now, where T is the partial differential operator defined by

3.8
Repeating this process ♦ m 2 with k − 2 times, we finally obtain where T k is the operator of 3.6 iterated k times.Now, by the operator L in 3.2 and the derivative of distribution.Continuing this process, we obtain T k ϕ 0 L k δ, ϕ x or e αx ♦ m 2 k δ, ϕ x L k δ, ϕ x .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 , L k δ are tempered distributions and L k δ has order 4k.It follows that e αx ♦ m 2 k δ is a tempered distribution of order 4k with point support {0} by 3.1 .This completes the proof.Lemma 3.2 boundedness property .Let D be the space of testing functions and D the space of distributions.For every ϕ ∈ D and e αx ♦ m 2 k δ ∈ D , for some constant M.
Proof.Note that we have e αx ♦ m 2 k δ, ϕ x ♦ m 2 k δ, e αx ϕ x for every ϕ x ∈ D and e αx ♦ m 2 k δ ∈ D .Hence, where T is defined by 3.6 .Continuing this process for k − 1 times, we will obtain e αx ♦ m 2 k δ, ϕ x δ, e αx T k ϕ x T k ϕ 0 .

3.13
Since ϕ ∈ D, so ϕ 0 is bounded, and also T k ϕ 0 is bounded.It then follows that for some constant M.

The Application of
Convolving both sides by e αx T 2k x, m and applying Lemma 2.5, we obtain

4.26
Convolving both sides by e αx T 2k x, m and applying Lemma 2.5, we have is the singular distribution.This completes the proof.

Abstract and Applied Analysis 3 In 2000 ,
Kananthai 15  has studied the application of the distribution e αx k δ for solving the convolution equation e αx k δ * u to the ultrahyperbolic equation.In 2009, Sasopa and Nonlaopon 16 have studied the properties of the distribution e αx k c δ and its application to solve the convolution equation

4 . 19 It 1 C 1 C
is known that ♦ m 2 k T 2k x, m δ, thus ♦ m 2 k−r ♦ m 2 r T 2k x, m δ for r < k.Convolving both sides by T 2k−2r x, m , we obtainT 2k−2r x, m * ♦ m 2 k−r ♦ m 2 r T 2k x, m T 2k−2r x, m , 4.20 or ♦ m 2 k−r T 2k−2r x, m * ♦ m 2 r T 2k x, m T 2k−2r x, m , 4.21 which leads to ♦ m 2 r T 2k x, m T 2k−2r x, m , 4.22 for r < k.It follows that u x e αx C 1 T 2k−2 x, m C 2 T 2k−4 x, m • • • C M T 2k−2M x, m , r T 2k−2r x, m .4.24Similarly, by case 1 , e αx T 2k−2r x, m is the ordinary function for 2k − 2r ≥ n with any α.r T 2k−2r x, m 4.25 is also the ordinary function with any α.3 if M ≥ k and for any α, we suppose that k ≤ M ≤ N, then 4.11 becomes

♦ m 2
M T 2k x, m ♦ m 2 M−k ♦ m 2 k T 2k x, m ♦ m 2 M−k , 4.28 for k ≤ M ≤ N. Thus, u x e αx C k δ C k 1 ♦ m 2 δ C k 2 ♦ m 2 2 δ • • • C N ♦ m 2 N3.1 and 3.2 , we have e αx ♦ m 2 r−k δ ♦ m 2 r−k δ terms of lower order of partial derivative of δ 4.30 for k ≤ r ≤ N. Since all terms on the right-hand side of this equation are singular distribution, . It is well known that R H γ x is an ordinary function if Re γ ≥ n and is a distribution of γ if Re γ < n.Let supp R H γ x denote the support of R H Distribution e αx ♦ m 2 k δ Proof.From 3.1 and 4.1 , we can write e αx ♦ m 2 δ * u x Lu x δ.Convolving both sides by e αx T 2 x, m , we have e αx T 2 x, m * e αx ♦ m 2 δ * u x e αx T 2 x, m * δ, we have δ * u x e αx T 2 x, m .It then follows that u x e αx T 2 x, m is the fundamental solution of the operator L. T 2 x, m * L k−1 u x e αx T 2 x, m .By Theorem 4.1, we obtain L k−1 u x e αx T 2 x, m .Keeping on convolving e αx T 2 x, m for k − 1 times, we finally obtain where u x is any distribution in D , then u x e αx T 2 x, m is the fundamental solution of the operator L, where T 2 x, m is defined by 2.16 with γ 2. αx and C r is a constant, then the type of solution u x of 4.11 depends on k, M, and α as follows: It then follows that C 0 e αx T 2k x, m is an ordinary function for 2k ≥ n with any α.