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

Abstract

We study the distribution 𝑒^𝛼𝑥(⋄+𝑚²)^𝑘𝛿 for 𝑚≥0, where (⋄+𝑚²)^𝑘 is the diamond Klein-Gordon operator iterated 𝑘 times, 𝛿 is the Dirac delta distribution, 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛) is a variable in ℝ^𝑛, and 𝛼=(𝛼₁,𝛼₂,…,𝛼_𝑛) is a constant. In particular, we study the application of 𝑒^𝛼𝑥(⋄+𝑚²)^𝑘𝛿 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 𝑘 and 𝑀.

1. Introduction

The 𝑛-dimensional ultrahyperbolic operator □^𝑘 iterated 𝑘 times is defined by□^𝑘=𝜕²𝜕𝑥²¹+𝜕²𝜕𝑥²²+⋯+𝜕²𝜕𝑥^2𝑝−𝜕²𝜕𝑥^2𝑝+1−𝜕²𝜕𝑥^2𝑝+2−⋯−𝜕²𝜕𝑥^2𝑝+𝑞^𝑘,(1.1) 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□^𝑘𝑢(𝑥)=𝑓(𝑥),(1.2) where 𝑢(𝑥) and 𝑓(𝑥) are generalized functions, and 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛)∈ℝ^𝑛.

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^𝐻2𝑘(𝑥) defined by (2.2) with 𝛾=2𝑘 is the unique fundamental solution of (1.2). Tellez [3] has also proved that 𝑅^𝐻2𝑘(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⋄^𝑘=⎡⎢⎢⎣^𝑝_𝑖=1𝜕²𝜕𝑥^2𝑖²−^𝑝+𝑞_𝑗=_𝑝+1𝜕²𝜕𝑥^2𝑗²⎤⎥⎥⎦^𝑘,(1.3) where 𝑛=𝑝+𝑞 is the dimension of ℝ^𝑛, for all 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛)∈ℝ^𝑛 and nonnegative integers 𝑘. The operator ⋄^𝑘 can be expressed in the form⋄^𝑘=▵^𝑘□^𝑘=□^𝑘▵^𝑘,(1.4) where □^𝑘 is defined by (1.1), and ▵^𝑘 is the Laplace operator iterated 𝑘 times defined by▵^𝑘=𝜕²𝜕𝑥²¹+𝜕²𝜕𝑥²²+⋯+𝜕²𝜕𝑥^2𝑛^𝑘.(1.5) 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)^𝑘𝑅^𝑒2𝑘(𝑥)∗𝑅^𝐻2𝑘(𝑥) is the fundamental solution of the operator ⋄^𝑘, that is,⋄^𝑘(−1)^𝑘𝑅^𝑒2𝑘(𝑥)∗𝑅^𝐻2𝑘(𝑥)=𝛿,(1.6) where 𝑅^𝐻2𝑘(𝑥) and 𝑅^𝑒2𝑘(𝑥) are defined by (2.2) and (2.7), respectively (with 𝛾=2𝑘), and 𝛿 is the Dirac-delta distribution. The fundamental solution (−1)^𝑘𝑅^𝑒2𝑘(𝑥)∗𝑅^𝐻2𝑘(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 Kananthai [6–12].

In 1997, Kananthai [13] has studied the properties of the distribution 𝑒^𝛼𝑥□^𝑘𝛿 and the application of the distribution 𝑒^𝛼𝑥□^𝑘𝛿 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⋄^𝑘𝛿 and its application for solving the convolution equation𝑒^𝛼𝑥⋄^𝑘𝛿∗𝑢(𝑥)=𝑒^𝑚𝛼𝑥_𝑟=0𝐶_𝑟⋄^𝑟𝛿.(1.7) 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 𝑒^𝛼𝑥□^𝑘𝛿 for solving the convolution equation𝑒^𝛼𝑥□^𝑘𝛿∗𝑢(𝑥)=𝑒^𝑚𝛼𝑥_𝑟=0𝐶_𝑟□^𝑟𝛿,(1.8) which is related to the ultrahyperbolic equation.

In 2009, Sasopa and Nonlaopon [16] have studied the properties of the distribution 𝑒^𝛼𝑥□^𝑘𝑐𝛿 and its application to solve the convolution equation𝑒^𝛼𝑥□^𝑘𝑐𝛿∗𝑢(𝑥)=𝑒^𝑚𝛼𝑥_𝑟=0𝐶_𝑟□^𝑟𝑐𝛿.(1.9) Here, □^𝑘𝑐 is the operator related to the ultrahyperbolic type operator iterated 𝑘 times, which is defined by□^𝑘𝑐=1𝑐^2𝑝_𝑖=1𝜕²𝜕𝑥^2𝑖−^𝑝+𝑞_𝑗=_𝑝+1𝜕²𝜕𝑥^2𝑗^𝑘,(1.10) 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□+𝑚²^𝑘=^𝑝_𝑖=1𝜕²𝜕𝑥^2𝑖−^𝑝+𝑞_𝑗=_𝑝+1𝜕²𝜕𝑥^2𝑗+𝑚²^𝑘.(1.11) The fundamental solution of the operator (□+𝑚²)^𝑘 is given by𝑊_2𝑘(𝑥,𝑚)=^∞_𝑟=0(−1)^𝑟Γ(𝑘+𝑟)𝑟!Γ(𝑘)𝑚²^𝑟(−1)^𝑟𝑅^{𝐻2𝑘+2𝑟}(𝑥),(1.12) where 𝑅^{𝐻2𝑘+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(□+𝑚²)^𝑘𝛿 and its application for solving the convolution equation𝑒^𝛼𝑥□+𝑚²^𝑘𝛿∗𝑢(𝑥)=𝑒^𝑀𝛼𝑥_𝑟=0𝐶_𝑟□+𝑚²^𝑟𝛿,(1.13) where (□+𝑚²)^𝑘 is defined by (1.11).

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𝜕𝑥^2𝑖²−^𝑝+𝑞_𝑗=_𝑝+1𝜕²𝜕𝑥^2𝑗²+𝑚²⎤⎥⎥⎦^𝑘,(1.14) where 𝑝+𝑞=𝑛 is the dimension of ℝ^𝑛, for all 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛)∈ℝ^𝑛,𝑚≥0 and nonnegative integers 𝑘. Later, Lunnaree and Nonlaopon [22, 23] have studied the fundamental solution of operator (⋄+𝑚²)^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(⋄+𝑚²)^𝑘𝛿 and the application of 𝑒^𝛼𝑥(⋄+𝑚²)^𝑘𝛿 for solving the convolution equation𝑒^𝛼𝑥⋄+𝑚²^𝑘𝛿∗𝑢(𝑥)=𝑒^𝑀𝛼𝑥_𝑟=0𝐶_𝑟⋄+𝑚²^𝑟𝛿,(1.15) where (⋄+𝑚²)^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.

2. Preliminaries

Definition 2.1. Let 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛) be a point of the 𝑛-dimensional Euclidean space ℝ^𝑛. Let 𝑢=𝑥²¹+𝑥²²+⋯+𝑥^2𝑝−𝑥^2𝑝+1−𝑥^2𝑝+2−⋯−𝑥^2𝑝+𝑞(2.1) be the nondegenerated quadratic form, where 𝑝+𝑞=𝑛 is the dimension of ℝ^𝑛. Let Γ₊={𝑥∈ℝ^𝑛∶𝑥₁>0and𝑢>0} be the interior of a forward cone, and let Γ₊ denote its closure. For any complex number 𝛾, we define the function 𝑅^𝐻𝛾(𝑥)=⎧⎪⎨⎪⎩𝑢𝐾_𝑛(𝛾)^{(𝛾−𝑛)/2},for𝑥∈Γ⁺0,for𝑥∉Γ₊,,(2.2) where the constant 𝐾_𝑛(𝛾) is given by 𝐾_𝑛(𝛾)=𝜋^(𝑛−1)/2Γ((2+𝛾−𝑛)/2)Γ((1−𝛾)/2)Γ(𝛾)Γ((2+𝛾−𝑝)/2)Γ((𝑝−𝛾)/2).(2.3)

The function 𝑅^𝐻𝛾(𝑥) is called the ultrahyperbolic kernel of Marcel Riesz, which was introduced by Nozaki [24]. It is well known that 𝑅^𝐻𝛾(𝑥) is an ordinary function if Re(𝛾)≥𝑛 and is a distribution of 𝛾 if Re(𝛾)<𝑛. Let supp 𝑅^𝐻𝛾(𝑥) denote the support of 𝑅^𝐻𝛾(𝑥) and suppose that supp 𝑅^𝐻𝛾(𝑥)⊂Γ₊, that is, supp 𝑅^𝐻𝛾(𝑥) is compact.

By putting 𝑝=1 in 𝑅^𝐻2𝑘(𝑥) and taking into account Legendre's duplication formulaΓ(2𝑧)=2^2𝑧−1𝜋^−1/2Γ(𝑧)Γ𝑧+12,(2.4) we obtain𝐼^𝐻𝛾(𝑥)=𝑣^{(𝛾−𝑛)/2}𝐻_𝑛(𝛾),(2.5)𝑣=𝑥²¹−𝑥²²−𝑥²³−⋯−𝑥^2𝑛, where𝐻_𝑛(𝛾)=𝜋^(𝑛−2)/22^𝛾−1Γ𝛾+2−𝑛2Γ𝛾2.(2.6) The function 𝐼^𝐻𝛾(𝑥) is called the hyperbolic kernel of Marcel Riesz.

Definition 2.2. Let 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛) be a point of ℝ^𝑛 and 𝜔=𝑥²¹+𝑥²²+⋯+𝑥^2𝑛. The elliptic kernel of Marcel Riesz is defined by 𝑅^𝑒𝛾(𝑥)=𝜔^{(𝛾−𝑛)/2}𝑊_𝑛(𝛾),(2.7) where 𝑛 is the dimension of ℝ^𝑛, 𝛾∈ℂ, and 𝑊_𝑛(𝛾)=𝜋^𝑛/22^𝛾Γ(𝛾/2)Γ((𝑛−𝛾)/2).(2.8)

Note that 𝑛=𝑝+𝑞. By putting 𝑞=0 (i.e., 𝑛=𝑝) in (2.2) and (2.3), we can reduce 𝑢^{(𝛾−𝑛)/2} to 𝜔^{𝑝(𝛾−𝑝)/2}, where 𝜔_𝑝=𝑥²¹+𝑥²²+⋯+𝑥^2𝑝, and reduce 𝐾_𝑛(𝛾) to 𝐾_𝑝(𝛾)=𝜋^(𝑝−1)/2Γ((1−𝛾)/2)Γ(𝛾)Γ((𝑝−𝛾)/2).(2.9)

Using the Legendre’s duplication formula Γ(2𝑧)=2^2𝑧−1𝜋^−1/2Γ(𝑧)Γ𝑧+1Γ12,(2.10)2+𝑧Γ12−𝑧=𝜋sec(𝜋𝑧),(2.11) we obtain 𝐾_𝑝(𝛾)=12sec𝛾2𝜋𝑊^𝑝(𝛾).(2.12) Thus, in case 𝑞=0, we have 𝑅^𝐻𝛾(𝑥)=𝑢^{(𝛾−𝑝)/2}𝐾_𝑝(𝛾)=2cos𝛾2𝜋𝑢𝑊_𝑝(𝛾^{(𝛾−𝑝)/2})=2cos𝛾2𝜋𝑅^𝑒𝛾(𝑥).(2.13) In addition, if 𝛾=2𝑘 for some nonnegative integer 𝑘, then 𝑅^𝐻2𝑘(𝑥)=2(−1)^𝑘𝑅^𝑒2𝑘(𝑥).(2.14)

Lemma 2.3. The convolution (−1)^𝑘𝑅^𝑒2𝑘(𝑥)∗𝑅^𝐻2𝑘(𝑥) is the fundamental solution of the diamond operator iterated 𝑘 times, that is, ⋄^𝑘(−1)^𝑘𝑅^𝑒2𝑘(𝑥)∗𝑅^𝐻2𝑘(𝑥)=𝛿.(2.15)

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

It can be shown that 𝑅^𝑒−2𝑘(𝑥)∗𝑅^𝐻−2𝑘(𝑥)=(−1)^𝑘⋄^𝑘𝛿, for all nonnegative integers 𝑘.

Definition 2.4. Let 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛) be a point of ℝ^𝑛. The function 𝑇_𝛾(𝑥,𝑚) is defined by 𝑇_𝛾(𝑥,𝑚)=^∞_𝑟=0⎛⎜⎜⎝−𝛾𝑟2⎞⎟⎟⎠𝑚²^𝑟(−1)^𝛾/2+𝑟𝑅^{𝑒𝛾+2𝑟}(𝑥)∗𝑅^{𝐻𝛾+2𝑟}(𝑥),(2.16) where 𝛾 is a complex parameter, and 𝑚 is a nonnegative real number. Here, 𝑅^{𝐻𝛾+2𝑟}(𝑥) and 𝑅^{𝑒𝛾+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 𝑇_−2𝑘(𝑥,𝑚)=^∞_𝑟=0⎛⎜⎜⎝𝑘𝑟⎞⎟⎟⎠𝑚₂^𝑟(−1)^−𝑘+𝑟𝑅^{𝑒2(−𝑘+𝑟)}(𝑥)∗𝑅^{𝐻2(−𝑘+𝑟)}(𝑥).(2.17) Since the operator (⋄+𝑚²)^𝑘 defined by (1.14) is linearly continuous and has 1-1 mapping, this possesses its own inverses. From Lemma 2.3, we obtain𝑇_−2𝑘(𝑥,𝑚)=^∞_𝑟=0⎛⎜⎜⎝𝑘𝑟⎞⎟⎟⎠𝑚₂^𝑟⋄^𝑘−𝑟𝛿=⋄+𝑚²^𝑘𝛿.(2.18)

Substituting 𝑘=0 in (2.18) yields that we have 𝑇₀(𝑥,𝑚)=𝛿. On the other hand, putting 𝛾=2𝑘 in (2.16) yields𝑇_2𝑘(𝑥,𝑚)=+⎛⎜⎜⎝−0𝑘⎞⎟⎟⎠𝑚²⁰(−1)^𝑘+0𝑅^𝑒2𝑘+0(𝑥)∗𝑅^𝐻2𝑘+0(𝑥)_∞_𝑟=1⎛⎜⎜⎝−𝑟𝑘⎞⎟⎟⎠𝑚₂^𝑟(−1)^𝑘+𝑟𝑅^{𝑒2𝑘+2𝑟}(𝑥)∗𝑅^{𝐻2𝑘+2𝑟}(𝑥).(2.19) The second summand of the right-hand side of (2.19) vanishes when 𝑚=0. Hence, we obtain 𝑇_2𝑘(𝑥,𝑚=0)=(−1)^𝑘𝑅^𝑒2𝑘(𝑥)∗𝑅^𝐻2𝑘(𝑥),(2.20) 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 ⋄+𝑚²^𝑘𝑢(𝑥)=𝛿,(2.21) where (⋄+𝑚²)^𝑘 is the diamond Klein-Gordon operator iterated 𝑘 times, defined by ⋄+𝑚²^𝑘=⎡⎢⎢⎣^𝑝_𝑖=1𝜕²𝜕𝑥^2𝑖²−^𝑝+𝑞_𝑗=_𝑝+1𝜕²𝜕𝑥^2𝑗²+𝑚²⎤⎥⎥⎦^𝑘(2.22) 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 (⋄+𝑚²)^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 𝑇_2𝑘(𝑥,𝑚)=𝑇_{2𝑘−2𝑣}(𝑥,𝑚)∗𝑇_2𝑣(𝑥,𝑚),(2.23) where 𝑣 is a nonnegative integer and 𝑣<𝑘. Moreover, if one puts 𝑙=𝑘−𝑣 and ℎ=𝑣, then one obtains 𝑇_2𝑙(𝑥,𝑚)∗𝑇_2ℎ(𝑥,𝑚)=𝑇_2𝑙+2ℎ(𝑥,𝑚)(2.24) for 𝑙+ℎ=𝑘.

3. Properties of the Distribution 𝑒^𝛼𝑥(⋄+𝑚²)^𝑘𝛿

Lemma 3.1. The following equality holds: 𝑒^𝛼𝑥⋄+𝑚²^𝑘𝛿=𝐿^𝑘𝛿,(3.1) and 𝑒^𝛼𝑥(⋄+𝑚²)^𝑘𝛿 is the tempered distribution of order 4𝑘 with support {0}, where 𝐿 is the partial differential operator and is defined by 𝐿≡⋄+𝑚²+^𝑛_𝑟=1𝛼^2𝑟□−2^𝑛_𝑝𝑟=1_𝑖=1𝛼^𝑟𝜕³𝜕𝑥^2𝑖𝜕𝑥_𝑟+𝛼_𝑖𝜕³+2_𝑛𝜕𝑥_𝑖𝜕𝑥^2𝑟_{𝑟=1𝑝+𝑞𝑗=𝑝+1}𝛼^𝑟𝜕³𝜕𝑥^2𝑗𝜕𝑥_𝑟+𝛼_𝑗𝜕³𝜕𝑥_𝑗𝜕𝑥^2𝑟+4^𝑛_𝑟=1𝛼_𝑟^𝑝_𝑖=1𝛼_𝑖𝜕²𝜕𝑥_𝑖𝜕𝑥_𝑟−^𝑝+𝑞_𝑗=_𝑝+1𝛼_𝑗𝜕²−2_𝑛𝜕𝑥_𝑗𝜕𝑥_𝑟_𝑟=1𝛼^2𝑟^𝑝_𝑖=1𝛼_𝑖𝜕𝜕𝑥_𝑖−^𝑝+𝑞_𝑗=_𝑝+1𝛼_𝑗𝜕𝜕𝑥_𝑗+^𝑝_𝑖=1𝛼^2𝑖−^𝑝+𝑞_𝑗=_𝑝+1𝛼^2𝑗−2_𝑝▵_𝑖=1𝛼^2𝑖−^𝑝+𝑞_𝑗=_𝑝+1𝛼²_j^𝑛_𝑟=1𝛼_𝑟𝜕𝜕𝑥_𝑟+^𝑝_𝑖=1𝛼^2𝑖−^𝑝+𝑞_𝑗=_𝑝+1𝛼^2𝑗^𝑛_𝑟=1𝛼^2𝑟.(3.2) As before, □ is the ultrahyperbolic operator defined by (1.1) (with 𝑘=1), and ▵ is the Laplace operator defined by ▵=𝜕²𝜕𝑥₁+𝜕²𝜕𝑥₂+⋯+𝜕²𝜕𝑥_𝑛.(3.3)

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, 𝑒^𝛼𝑥⋄+𝑚²𝛿,𝜑(𝑥)=𝛿,⋄+𝑚²𝑒^𝛼𝑥𝜑(𝑥),(3.4) for 𝑒^𝛼𝑥(⋄+𝑚²)𝛿∈𝒟′. A direct computation shows that ⋄+𝑚²𝑒^𝛼𝑥𝜑(𝑥)=𝑒^𝛼𝑥𝑇𝜑(𝑥),(3.5) where 𝑇 is the partial differential operator defined by 𝑇≡⋄+𝑚²+^𝑛_𝑟=1𝛼^2𝑟□+2^𝑛_𝑝𝑟=1_𝑖=1𝛼^𝑟𝜕³𝜕𝑥^2𝑖𝜕𝑥_𝑟+𝛼_𝑖𝜕³−2_𝑛𝜕𝑥_𝑖𝜕𝑥^2𝑟_{𝑟=1𝑝+𝑞𝑗=𝑝+1}𝛼^𝑟𝜕³𝜕𝑥^2𝑗𝜕𝑥_𝑟+𝛼_𝑗𝜕³𝜕𝑥_𝑗𝜕𝑥^2𝑟+4^𝑛_𝑟=1𝛼_𝑟^𝑝_𝑖=1𝛼_𝑖𝜕²𝜕𝑥_𝑖𝜕𝑥_𝑟−^𝑝+𝑞_𝑗=_𝑝+1𝛼_𝑗𝜕²+2_𝑛𝜕𝑥_𝑗𝜕𝑥_𝑟_𝑟=1𝛼^2𝑟^𝑝_𝑖=1𝛼_𝑖𝜕𝜕𝑥_𝑖−^𝑝+𝑞_𝑗=_𝑝+1𝛼_𝑗𝜕𝜕𝑥_𝑗+^𝑝_𝑖=1𝛼^2𝑖−^𝑝+𝑞_𝑗=_𝑝+1𝛼^2𝑗+2_𝑝▵_𝑖=1𝛼^2𝑖−^𝑝+𝑞_𝑗=_𝑝+1𝛼^2𝑗^𝑛_𝑟=1𝛼_𝑟𝜕𝜕𝑥_𝑟+^𝑝_𝑖=1𝛼^2𝑖−^𝑝+𝑞_𝑗=_𝑝+1𝛼^2𝑗^𝑛_𝑟=1𝛼^2𝑟.(3.6) Thus, 𝛿,⋄+𝑚²𝑒^𝛼𝑥𝜑(𝑥)=⟨𝛿,𝑒^𝛼𝑥𝑇𝜑(𝑥)⟩=𝑇𝜑(0).(3.7)
Since ⟨𝑒^𝛼𝑥(⋄+𝑚²)^𝑘𝛿,𝜑(𝑥)⟩=⟨(⋄+𝑚²)^𝑘𝛿,𝑒^𝛼𝑥𝜑(𝑥)⟩ for every 𝜑(𝑥)∈𝒟 and 𝑒^𝛼𝑥(⋄+𝑚²)^𝑘𝛿∈𝒟′, we have ⋄+𝑚²^𝑘𝛿,𝑒^𝛼𝑥𝜑(𝑥)==⋄⋄++𝑚𝑚²₂^𝑘−1𝛿,⋄+𝑚²𝑒^𝛼𝑥𝜑(𝑥)=⋄+𝑚₂^𝑘−1𝛿,𝑒^𝛼𝑥𝑇𝜑(𝑥)=⋄+𝑚₂^𝑘−2𝛿,⋄+𝑚²𝑒^𝛼𝑥𝑇𝜑(𝑥)=⋄+𝑚₂^𝑘−2𝛿,𝑒^𝛼𝑥𝑇(𝑇𝜑(𝑥))^𝑘−2𝛿,𝑒^𝛼𝑥𝑇²𝜑(𝑥).(3.8) Repeating this process (⋄+𝑚²) with 𝑘−2 times, we finally obtain ⋄+𝑚²^𝑘−2𝛿,𝑒^𝛼𝑥𝑇²𝜑(𝑥)=𝛿,𝑒^𝛼𝑥𝑇^𝑘𝜑(𝑥)=𝑇^𝑘𝜑(0),(3.9) where 𝑇^𝑘 is the operator of (3.6) iterated 𝑘 times. Now, 𝑇^𝑘𝜑(0)=𝛿,𝑇^𝑘𝜑(𝑥)=𝐿𝛿,𝑇^𝑘−1𝜑(𝑥),(3.10) by the operator 𝐿 in (3.2) and the derivative of distribution. Continuing this process, we obtain 𝑇^𝑘𝜑(0)=⟨𝐿^𝑘𝛿,𝜑(𝑥)⟩ or ⟨𝑒^𝛼𝑥(⋄+𝑚²)^𝑘𝛿,𝜑(𝑥)⟩=⟨𝐿^𝑘𝛿,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^𝑘𝛿 are tempered distributions and 𝐿^𝑘𝛿 has order 4𝑘. It follows that 𝑒^𝛼𝑥(⋄+𝑚²)^𝑘𝛿 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(⋄+𝑚²)^𝑘𝛿∈𝒟^, |||𝑒_𝛼𝑥⋄+𝑚²^𝑘𝛿,𝜑(𝑥)|||≤𝑀,(3.11) for some constant 𝑀.

Proof. Note that we have ⟨𝑒^𝛼𝑥(⋄+𝑚²)^𝑘𝛿,𝜑(𝑥)⟩=⟨(⋄+𝑚²)^𝑘𝛿,𝑒^𝛼𝑥𝜑(𝑥)⟩ for every 𝜑(𝑥)∈𝒟 and 𝑒^𝛼𝑥(⋄+𝑚²)^𝑘𝛿∈𝒟′. Hence, ⋄+𝑚²^𝑘𝛿,𝑒^𝛼𝑥𝜑(𝑥)=⋄+𝑚²^𝑘−1𝛿,⋄+𝑚²𝑒^𝛼𝑥𝜑(𝑥)=⋄+𝑚²^𝑘−1𝛿,𝑒^𝛼𝑥𝑇𝜑(𝑥),(3.12) where 𝑇 is defined by (3.6). Continuing this process for 𝑘−1 times, we will obtain 𝑒^𝛼𝑥⋄+𝑚²^𝑘𝛿,𝜑(𝑥)=𝛿,𝑒^𝛼𝑥𝑇^𝑘𝜑(𝑥)=𝑇^𝑘𝜑(0).(3.13) Since 𝜑∈𝒟, so 𝜑(0) is bounded, and also 𝑇^𝑘𝜑(0) is bounded. It then follows that |||𝑒_𝛼𝑥⋄+𝑚²^𝑘𝛿,𝜑(𝑥)|||=𝑇^𝑘𝜑(0)≤𝑀,(3.14) for some constant 𝑀.

4. The Application of Distribution 𝑒^𝛼𝑥(⋄+𝑚²)^𝑘𝛿

Theorem 4.1. Let 𝐿 be the partial differential operator defined by (3.2), and consider the equation 𝐿𝑢(𝑥)=𝛿,(4.1) where 𝑢(𝑥) is any distribution in 𝒟^, then 𝑢(𝑥)=𝑒^𝛼𝑥𝑇₂(𝑥,𝑚) is the fundamental solution of the operator 𝐿, where 𝑇₂(𝑥,𝑚) is defined by (2.16) with 𝛾=2.

Proof. From (3.1) and (4.1), we can write 𝑒^𝛼𝑥(⋄+𝑚²)𝛿∗𝑢(𝑥)=𝐿𝑢(𝑥)=𝛿. Convolving both sides by 𝑒^𝛼𝑥𝑇₂(𝑥,𝑚), we have 𝑒^𝛼𝑥𝑇₂(𝑥,𝑚)∗𝑒^𝛼𝑥⋄+𝑚²𝛿∗𝑢(𝑥)=𝑒^𝛼𝑥𝑇₂(𝑥,𝑚)∗𝛿,(4.2) then 𝑒^𝛼𝑥𝑇₂(𝑥,𝑚)∗⋄+𝑚²𝛿∗𝑢(𝑥)=𝑒^𝛼𝑥𝑇₂(𝑥,𝑚),(4.3) or equivalently, 𝑒^𝛼𝑥⋄+𝑚²𝑇₂(𝑥,𝑚)∗𝑢(𝑥)=𝑒^𝛼𝑥𝑇₂(𝑥,𝑚).(4.4) Since (⋄+𝑚²)𝑇₂(𝑥,𝑚)=𝛿 by Lemma 2.5 with 𝑘=1, we obtain (𝑒^𝛼𝑥𝛿)∗𝑢(𝑥)=𝑒^𝛼𝑥𝑇₂(𝑥,𝑚).(4.5) Moreover, since 𝑒^𝛼𝑥𝛿=𝛿, we have 𝛿∗𝑢(𝑥)=𝑒^𝛼𝑥𝑇₂(𝑥,𝑚). It then follows that 𝑢(𝑥)=𝑒^𝛼𝑥𝑇₂(𝑥,𝑚) is the fundamental solution of the operator 𝐿.

Theorem 4.2 (the generalization of Theorem 4.1). From Lemma 3.1, consider that 𝑒^𝛼𝑥⋄+𝑚²^𝑘𝛿∗𝑢(𝑥)=𝛿,(4.6) or 𝐿^𝑘𝑢(𝑥)=𝛿,(4.7) then 𝑢(𝑥)=𝑒^𝛼𝑥𝑇_2𝑘(𝑥,𝑚) is the fundamental solution of the operator 𝐿^𝑘.

Proof. We can prove it by using either (4.6) or (4.7). If we start with (4.6), by convolving both sides by 𝑒^𝛼𝑥𝑇_2𝑘(𝑥,𝑚), we obtain 𝑒^𝛼𝑥𝑇_2𝑘(𝑥,𝑚)∗𝑒^𝛼𝑥⋄+𝑚²^𝑘𝛿∗𝑢(𝑥)=𝑒^𝛼𝑥𝑇_2𝑘(𝑥,𝑚)∗𝛿,(4.8) or 𝑒^𝛼𝑥((⋄+𝑚²)^𝑘𝑇_2𝑘(𝑥,𝑚))∗𝑢(𝑥)=𝑒^𝛼𝑥𝑇_2𝑘(𝑥,𝑚). Since (⋄+𝑚²)^𝑘𝑇_2𝑘(𝑥,𝑚)=𝛿 by Lemma 2.5, we have (𝑒^𝛼𝑥𝛿)∗𝑢(𝑥)=𝑒^𝛼𝑥𝑇_2𝑘(𝑥,𝑚) or 𝑢(𝑥)=𝑒^𝛼𝑥𝑇_2𝑘(𝑥,𝑚) as required.
If we use (4.7), by convolving both sides by 𝑒^𝛼𝑥𝑇₂(𝑥,𝑚), we obtain 𝑒^𝛼𝑥𝑇₂(𝑥,𝑚)∗𝐿^𝑘𝑢(𝑥)=𝑒^𝛼𝑥𝑇₂(𝑥,𝑚)∗𝛿,(4.9) or 𝐿(𝑒^𝛼𝑥𝑇₂(𝑥,𝑚))∗𝐿^𝑘−1𝑢(𝑥)=𝑒^𝛼𝑥𝑇₂(𝑥,𝑚). By Theorem 4.1, we obtain 𝐿^𝑘−1𝑢(𝑥)=𝑒^𝛼𝑥𝑇₂(𝑥,𝑚). Keeping on convolving 𝑒^𝛼𝑥𝑇₂(𝑥,𝑚) for 𝑘−1 times, we finally obtain 𝑢(𝑥)=𝑒^𝛼𝑥𝑇₂(𝑥,𝑚)∗𝑇₂(𝑥,𝑚)∗⋯∗𝑇₂(𝑥,𝑚)=𝑒^𝛼𝑥𝑇_2𝑘(𝑥,𝑚),(4.10) by Lemma 2.6 and [26, page 196].

In particular, if we put 𝛼=(𝛼₁,𝛼₂,…,𝛼_𝑛)=0 in (4.6), then (4.6) reduces to (2.21), and we obtain 𝑢(𝑥)=𝑇_2𝑘(𝑥,𝑚) as the fundamental solution of the diamond Klein-Gordon operator iterated 𝑘 times.

Theorem 4.3. Given the convolution equation 𝑒^𝛼𝑥⋄+𝑚²^𝑘𝛿∗𝑢(𝑥)=𝑒^𝑀𝛼𝑥_𝑟=0𝐶_𝑟⋄+𝑚²^𝑟𝛿,(4.11) where (⋄+𝑚²)^𝑘 is the diamond Klein-Gordon operator iterated 𝑘 times defined by ⋄+𝑚²^𝑘=^𝑝_𝑖=1𝜕²𝜕𝑥^2𝑖−^𝑝+𝑞_𝑗=_𝑝+1𝜕²𝜕𝑥^2𝑗+𝑚²^𝑘,(4.12) the variable 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛)∈ℝ^𝑛, the constant 𝛼=(𝛼₁,𝛼₂,…,𝛼_𝑛)∈ℝ^𝑛, 𝑚 is a nonnegative real number, 𝛿 is the Dirac-delta distribution with (⋄+𝑚²)⁰𝛿=𝛿,(⋄+𝑚²)¹𝛿=(⋄+𝑚²)𝛿, and 𝐶_𝑟 is a constant, then the type of solution 𝑢(𝑥) of (4.11) depends on 𝑘,𝑀, and 𝛼 as follows: (1)if 𝑀<𝑘 and 𝑀=0, then the solution of (4.11) is 𝑢(𝑥)=𝐶₀𝑒^𝛼𝑥𝑇_2𝑘(𝑥,𝑚),(4.13) where 𝑇_2𝑘(𝑥,𝑚) is defined by (2.16) with 𝛾=2𝑘. If 2𝑘≥𝑛 and for any 𝛼, then 𝑒^𝛼𝑥𝑇_2𝑘(𝑥,𝑚) is the ordinary function, (2)if 0<𝑀<𝑘, then the solution of (4.11) is 𝑢(𝑥)=𝑒^𝑀𝛼𝑥_𝑟=1𝐶_𝑟𝑇_{2𝑘−2𝑟}(𝑥,𝑚),(4.14) which is an ordinary function for 2𝑘−2𝑟≥𝑛 with any arbitrary constant 𝛼, (3)if 𝑀≥𝑘 and for any 𝛼 one supposes that 𝑘≤𝑀≤𝑁, then (4.11) has 𝑢(𝑥)=𝑒^𝑁𝛼𝑥_𝑟=𝑘𝐶_𝑟⋄+𝑚²^𝑟−𝑘𝛿(4.15) as a solution which is the singular distribution.

Proof. (1)For 𝑀<𝑘 and 𝑀=0, then (4.11) becomes 𝑒^𝛼𝑥⋄+𝑚²^𝑘𝛿∗𝑢(𝑥)=𝐶₀𝑒^𝛼𝑥𝛿=𝐶₀𝛿,(4.16) and by Theorem 4.2, we obtain 𝑢(𝑥)=𝐶₀𝑒^𝛼𝑥𝑇_2𝑘(𝑥,𝑚).(4.17) Now, by (2.2) and (2.7), 𝑅^𝑒2𝑘(𝑥) and 𝑅^𝐻2𝑘(𝑥) are ordinary functions, respectively, for 2𝑘≥𝑛. It then follows that 𝐶₀𝑒^𝛼𝑥𝑇_2𝑘(𝑥,𝑚) is an ordinary function for 2𝑘≥𝑛 with any 𝛼.(2)For 0<𝑀<𝑘, then we can write (4.11) as 𝑒^𝛼𝑥⋄+𝑚²^𝑘𝛿∗𝑢(𝑥)=𝑒^𝛼𝑥𝐶¹⋄+𝑚²𝛿+𝐶₂⋄+𝑚²²𝛿+⋯+𝐶_𝑀⋄+𝑚²^𝑀𝛿.(4.18) Convolving both sides by 𝑒^𝛼𝑥𝑇_2𝑘(𝑥,𝑚) and applying Lemma 2.5, we obtain 𝑢(𝑥)=𝑒^𝛼𝑥𝐶¹⋄+𝑚²𝑇_2𝑘(𝑥,𝑚)+𝐶₂⋄+𝑚²²𝑇_2𝑘(𝑥,𝑚)+⋯+𝐶_𝑀⋄+𝑚²^𝑀𝑇_2𝑘(𝑥,𝑚).(4.19) It is known that (⋄+𝑚²)^𝑘𝑇_2𝑘(𝑥,𝑚)=𝛿, thus (⋄+𝑚²)^𝑘−𝑟(⋄+𝑚²)^𝑟𝑇_2𝑘(𝑥,𝑚)=𝛿 for 𝑟<𝑘. Convolving both sides by 𝑇_{2𝑘−2𝑟}(𝑥,𝑚), we obtain 𝑇_{2𝑘−2𝑟}(𝑥,𝑚)∗⋄+𝑚²^𝑘−𝑟⋄+𝑚²^𝑟𝑇_2𝑘(𝑥,𝑚)=𝑇_{2𝑘−2𝑟}(𝑥,𝑚),(4.20) or ⋄+𝑚²^𝑘−𝑟𝑇_{2𝑘−2𝑟}(𝑥,𝑚)∗⋄+𝑚²^𝑟𝑇_2𝑘(𝑥,𝑚)=𝑇_{2𝑘−2𝑟}(𝑥,𝑚),(4.21) which leads to ⋄+𝑚²^𝑟𝑇_2𝑘(𝑥,𝑚)=𝑇_{2𝑘−2𝑟}(𝑥,𝑚),(4.22) for 𝑟<𝑘. It follows that 𝑢(𝑥)=𝑒^𝛼𝑥𝐶₁𝑇_2𝑘−2(𝑥,𝑚)+𝐶₂𝑇_2𝑘−4(𝑥,𝑚)+⋯+𝐶_𝑀𝑇_{2𝑘−2𝑀}(𝑥,𝑚),(4.23) or 𝑢(𝑥)=𝑒^𝑀𝛼𝑥_𝑟=1𝐶_𝑟𝑇_{2𝑘−2𝑟}(𝑥,𝑚).(4.24) Similarly, by case (1), 𝑒^𝛼𝑥𝑇_{2𝑘−2𝑟}(𝑥,𝑚) is the ordinary function for 2𝑘−2𝑟≥𝑛 with any 𝛼. It follows that 𝑢(𝑥)=𝑒^𝑀𝛼𝑥_𝑟=1𝐶_𝑟𝑇_{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 𝑒^𝛼𝑥⋄+𝑚²^𝑘𝛿∗𝑢(𝑥)=𝑒^𝛼𝑥𝐶^𝑘⋄+𝑚²^𝑘𝛿+𝐶_𝑘+1⋄+𝑚²^𝑘+1𝛿+⋯+𝐶_𝑁⋄+𝑚²^𝑁𝛿.(4.26) Convolving both sides by 𝑒^𝛼𝑥𝑇_2𝑘(𝑥,𝑚) and applying Lemma 2.5, we have 𝑢(𝑥)=𝑒^𝛼𝑥𝐶^𝑘⋄+𝑚²^𝑘𝑇_2𝑘(𝑥,𝑚)+𝐶_𝑘+1⋄+𝑚²^𝑘+1𝑇_2𝑘(𝑥,𝑚)+⋯+𝐶_𝑁⋄+𝑚²^𝑁𝑇_2𝑘(𝑥,𝑚).(4.27) Now, ⋄+𝑚²^𝑀𝑇_2𝑘(𝑥,𝑚)=⋄+𝑚²^𝑀−𝑘⋄+𝑚²^𝑘𝑇_2𝑘(𝑥,𝑚)=⋄+𝑚²^𝑀−𝑘,(4.28) for 𝑘≤𝑀≤𝑁. Thus, 𝑢(𝑥)==𝑒𝑒^𝛼𝑥_𝑁𝛼𝑥𝐶^𝑘𝛿+𝐶_𝑘+1⋄+𝑚²𝛿+𝐶_𝑘+2⋄+𝑚²²𝛿+⋯+𝐶_𝑁⋄+𝑚²^𝑁−𝑘𝛿_𝑟=𝑘𝐶_𝑟⋄+𝑚²^𝑟−𝑘𝛿.(4.29) Now, by (3.1) and (3.2), we have 𝑒^𝛼𝑥⋄+𝑚²^𝑟−𝑘𝛿=⋄+𝑚²^𝑟−𝑘𝛿+(termsoflowerorderofpartialderivativeof𝛿)(4.30) for 𝑘≤𝑟≤𝑁. Since all terms on the right-hand side of this equation are singular distribution, it follows that 𝑢(𝑥)=𝑒^𝑁𝛼𝑥_𝑟=𝑘𝐶_𝑟⋄+𝑚²^𝑟−𝑘𝛿(4.31) 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.