Abstract and Applied AnalysisVolume 2011 (2011), Article ID 908491, 16 pagesdoi:10.1155/2011/908491

Research Article

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

Amphon Liangprom¹ and Kamsing Nonlaopon1,2

¹Department of Mathematics, Khon Kaen University, Khon Kaen 40002, Thailand²Centre 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: Elena Braverman

Copyright © 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 for , where is the diamond Klein-Gordon operator iterated times, is the Dirac delta distribution, is a variable in , and is a constant. In particular, we study the application of 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 and .

1. Introduction

The -dimensional ultrahyperbolic operator iterated times is defined by where is the dimension of , and is a nonnegative integer. We consider the linear differential equation of the form where and are generalized functions, and .

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 defined by (2.2) with is the unique fundamental solution of (1.2). Tellez [3] has also proved that exists only when with odd .

Kananthai [4] has first introduced the operator called the diamond operator iterated times, which is defined by where is the dimension of , for all and nonnegative integers . The operator can be expressed in the form where is defined by (1.1), and is the Laplace operator iterated times defined by Note that in case , 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 and . He found that the convolution is the fundamental solution of the operator , that is, where and are defined by (2.2) and (2.7), respectively (with ), and is the Dirac-delta distribution. The fundamental solution 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 and the application of the distribution 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 and its application for solving the convolution equation 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 for solving the convolution equation which is related to the ultrahyperbolic equation.

In 2009, Sasopa and Nonlaopon [16] have studied the properties of the distribution and its application to solve the convolution equation Here, is the operator related to the ultrahyperbolic type operator iterated times, which is defined by where is the dimension of .

In 1988, Trione [17] has studied the fundamental solution of the ultrahyperbolic Klein-Gordon operator iterated times, which is defined by The fundamental solution of the operator is given by where is defined by (2.2) with . Next, Tellez [18] has studied the convolution product of , where and are any complex parameter. In addition, Trione [19] has studied the fundamental -ultrahyperbolic solution of the Klein-Gordon operator iterated times and the convolution of such fundamental solution.

Liangprom and Nonlaopon [20] have studied the properties of the distribution and its application for solving the convolution equation where is defined by (1.11).

In 2007, Tariboon and Kananthai [21] have introduced the operator called diamond Klein-Gordon operator iterated times, which is defined by where is the dimension of , for all and nonnegative integers . Later, Lunnaree and Nonlaopon [22, 23] have studied the fundamental solution of operator , 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 and the application of for solving the convolution equation where is defined by (1.14), is the generalized function, and is a constant. On finding the type of solution 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 be a point of the -dimensional Euclidean space . Let be the nondegenerated quadratic form, where is the dimension of . Let be the interior of a forward cone, and let denote its closure. For any complex number , we define the function where the constant is given by

The function is called the ultrahyperbolic kernel of Marcel Riesz, which was introduced by Nozaki [24]. It is well known that is an ordinary function if and is a distribution of if . Let supp denote the support of and suppose that supp , that is, supp is compact.

By putting in and taking into account Legendre's duplication formula we obtain, where The function is called the hyperbolic kernel of Marcel Riesz.

Definition 2.2. Let be a point of and . The elliptic kernel of Marcel Riesz is defined by where is the dimension of , , and

Note that . By putting (i.e., ) in (2.2) and (2.3), we can reduce to , where , and reduce to

Using the Legendre’s duplication formula we obtain Thus, in case , we have In addition, if for some nonnegative integer , then

Lemma 2.3. The convolution is the fundamental solution of the diamond operator iterated times, that is,

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

It can be shown that , for all nonnegative integers .

Definition 2.4. Let be a point of . The function is defined by where is a complex parameter, and is a nonnegative real number. Here, and are defined by (2.2) and (2.7), respectively.

From the definition of , by putting , we have Since the operator defined by (1.14) is linearly continuous and has 1-1 mapping, this possesses its own inverses. From Lemma 2.3, we obtain

Substituting in (2.18) yields that we have . On the other hand, putting in (2.16) yields The second summand of the right-hand side of (2.19) vanishes when . 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 is the diamond Klein-Gordon operator iterated times, defined by with a nonnegative integer and the Dirac-delta distribution , then is the fundamental solution of the diamond Klein-Gordon operator iterated times , where is defined by (2.16) with .

Lemma 2.6. Let be the diamond Klein-Gordon kernel defined by (2.16), then is a tempered distribution and can be expressed by where is a nonnegative integer and . Moreover, if one puts and , then one obtains for .

3. Properties of the Distribution

Lemma 3.1. The following equality holds: and is the tempered distribution of order with support , where is the partial differential operator and is defined by As before, is the ultrahyperbolic operator defined by (1.1) (with ), and is the Laplace operator defined by

Proof. Let be the space of testing functions which are infinitely differentiable with compact supports, and let be the space of distributions. Now, for . A direct computation shows that where is the partial differential operator defined by Thus, Since for every and , we have Repeating this process with times, we finally obtain where is the operator of (3.6) iterated times. Now, by the operator in (3.2) and the derivative of distribution. Continuing this process, we obtain or . By equality of distributions, we obtain (3.1) as required. Since and its partial derivatives have support which is compact, hence, by Schwartz [25], are tempered distributions and has order . It follows that is a tempered distribution of order with point support by (3.1). This completes the proof.

Lemma 3.2 (boundedness property). Let be the space of testing functions and the space of distributions. For every and , for some constant .

Proof. Note that we have for every and . Hence, where is defined by (3.6). Continuing this process for times, we will obtain Since , so is bounded, and also is bounded. It then follows that for some constant .

4. The Application of Distribution

Theorem 4.1. Let be the partial differential operator defined by (3.2), and consider the equation where is any distribution in , then is the fundamental solution of the operator , where is defined by (2.16) with .

Proof. From (3.1) and (4.1), we can write . Convolving both sides by , we have then or equivalently, Since by Lemma 2.5 with , we obtain Moreover, since , we have . It then follows that is the fundamental solution of the operator .

Theorem 4.2 (the generalization of Theorem 4.1). From Lemma 3.1, consider that or then is the fundamental solution of the operator .

Proof. We can prove it by using either (4.6) or (4.7). If we start with (4.6), by convolving both sides by , we obtain or . Since by Lemma 2.5, we have or as required.If we use (4.7), by convolving both sides by , we obtain or . By Theorem 4.1, we obtain . Keeping on convolving for times, we finally obtain by Lemma 2.6 and [26, page 196].

In particular, if we put in (4.6), then (4.6) reduces to (2.21), and we obtain as the fundamental solution of the diamond Klein-Gordon operator iterated times.

Theorem 4.3. Given the convolution equation where is the diamond Klein-Gordon operator iterated times defined by the variable , the constant , is a nonnegative real number, is the Dirac-delta distribution with , and is a constant, then the type of solution of (4.11) depends on , and as follows: (1)if and , then the solution of (4.11) is where is defined by (2.16) with . If and for any , then is the ordinary function, (2)if , then the solution of (4.11) is which is an ordinary function for with any arbitrary constant , (3)if and for any one supposes that , then (4.11) has as a solution which is the singular distribution.

Proof. (1)For and , then (4.11) becomes and by Theorem 4.2, we obtain Now, by (2.2) and (2.7), and are ordinary functions, respectively, for . It then follows that is an ordinary function for with any .(2)For , then we can write (4.11) as Convolving both sides by and applying Lemma 2.5, we obtain It is known that , thus for . Convolving both sides by , we obtain or which leads to for . It follows that or Similarly, by case (1), is the ordinary function for with any . It follows that is also the ordinary function with any .(3)if and for any , we suppose that , then (4.11) becomes Convolving both sides by and applying Lemma 2.5, we have Now, for . Thus, Now, by (3.1) and (3.2), we have for . Since all terms on the right-hand side of this equation are singular distribution, it follows that 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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