On the Inversion of Bessel Ultrahyperbolic Kernel of Marcel Riesz

Abstract

We define the Bessel ultrahyperbolic Marcel Riesz operator on the function 𝑓 by 𝑈^𝛼(𝑓)=𝑅^𝐵𝛼∗𝑓, where 𝑅^𝐵𝛼 is Bessel ultrahyperbolic kernel of Marcel Riesz, 𝛼…ℂ, the symbol ∗ designates as the convolution, and 𝑓∈𝒮, 𝒮 is the Schwartz space of functions. Our purpose in this paper is to obtain the operator 𝐸^𝛼=(𝑈^𝛼)⁻¹ such that, if 𝑈^𝛼(𝑓)=𝜑, then 𝐸^𝛼𝜑=𝑓.

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.

Consider the linear differential equation in the form of□^𝑘𝑢(𝑥)=𝑓(𝑥),(1.2) where 𝑢(𝑥) and 𝑓(𝑥) are generalized functions and 𝑥=(𝑥₁,𝑥₂,…,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^𝐻2𝑘(𝑥), defined by (2.6) with 𝛾=2𝑘, is the unique fundamental solution of (1.2) and Téllez [3] has also proved that 𝑅^𝐻2𝑘(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◊^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), and 𝑘 is a nonnegative integer. 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=𝜕²𝜕𝑥²¹+𝜕²𝜕𝑥²²+⋯+𝜕²𝜕𝑥^2𝑛^𝑘(1.5) 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)^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(𝑥),(1.6) where 𝑅^𝐻2𝑘(𝑥) and 𝑅^𝑒2𝑘(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)^𝑘𝑅^e_2𝑘(𝑥)∗𝑅^𝐻2𝑘(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 Kananthai [5–10].

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, by𝑅^𝛼𝜑=𝐻_𝛼(𝑃±𝑖0,𝑛)∗𝜑,(1.7) where 𝜑∈𝒮, 𝒮 is the Schwartz space of functions [14] and 𝐻_𝛼(𝑃±𝑖0,𝑛) is given by𝐻_𝛼(𝑃±𝑖0,𝑛)=𝑒^{∓𝛼𝜋𝑖/2}𝑒^{±𝑞𝜋𝑖/2}Γ((𝑛−𝛼)/2)(𝑃±𝑖0)^{(𝛼−𝑛)/2}2^𝛼𝜋^𝑛/2Γ(𝛼/2).(1.8) Here, 𝑃 is defined by𝑃=𝑃(𝑥)=𝑥²¹+𝑥²²+⋯+𝑥^2𝑝−𝑥^2𝑝+1−𝑥^2𝑝+2−⋯−𝑥^2𝑝+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_𝜖→0𝑃±𝑖𝜖|𝑥|²^𝜆,(1.10) where 𝜖>0, 𝜆∈ℂ, and |𝑥|²=𝑥²¹+𝑥²²+⋯+𝑥^2𝑛; see [1]. They have also studied the inverse operator of 𝑅^𝛼, denoted by (𝑅^𝛼)⁻¹, such that, if 𝑓=𝑅^𝛼𝜑, then (𝑅^𝛼)⁻¹𝑓=𝜑.

Later, Aguirre [15] has defined the ultrahyperbolic Marcel Riesz operator 𝑀^𝛼 of the function 𝑓 by𝑀^𝛼(𝑓)=𝑅^𝐻𝛼∗𝑓,(1.11) where 𝑅^𝐻𝛼 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^𝛼𝜑=𝑓.

Let us consider the diamond kernel of Marcel Riesz 𝐾_𝛼,𝛽(𝑥) introduced by Kananthai in [6], which is given by the convolution𝐾_𝛼,𝛽(𝑥)=𝑅^𝑒𝛼∗𝑅^𝐻𝛽,(1.12) where 𝑅^𝑒𝛼 is elliptic kernel defined by (2.11) and 𝑅^𝐻𝛽 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 by𝑀^(𝛼,𝛽)(𝑓)=𝐾_𝛼,𝛽∗𝑓,(1.13) where 𝐾_𝛼,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, if 𝑀^(𝛼,𝛽)(𝑓)=𝜑, then 𝑁^(𝛼,𝛽)𝜑=𝑓.

In this paper, we define the Bessel ultrahyperbolic Marcel Riesz operator of order 𝛼 of the function 𝑓 by𝑈^𝛼(𝑓)=𝑅^𝐵𝛼∗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^𝛼=(𝑈^𝛼)⁻¹ such that, if 𝑈^𝛼(𝑓)=𝜑, then 𝐸^𝛼𝜑=𝑓.

Before we proceed to our main theorem, the following definitions and concepts require some clarifications.

2. Preliminaries

Definition 2.1. Let 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛) be a point in 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∶𝑢>0and𝑥_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 𝑅^𝐵𝛾(𝑥)=⎧⎪⎨⎪⎩𝑢𝐾^𝑛|𝜈|(𝛾)^{(𝛾−2|𝜈|−𝑛)/2},for𝑥∈Γ⁺0,for𝑥∉Γ₊,,(2.2) where 𝐾^𝑛|𝜈|(𝛾)=𝜋^{(𝑛−1+2|𝜈|)/2}Γ((2+𝛾−𝑛−2|𝜈|)/2)Γ((1−𝛾)/2)Γ(𝛾)Γ((2+𝛾−𝑝−2|𝜈|)/2)Γ((𝑝−𝛾)/2),(2.3)2𝑣_𝑖=2𝛼_𝑖+1, 𝛼_𝑖>−1/2 and |𝜈|=𝜈₁+𝜈₂+⋯+𝜈_𝑛, see [18–20].

The function 𝑅^𝐵𝛾(𝑥) is called the Bessel ultrahyperbolic kernel and was introduced by Aguirre [21]. It is well known that 𝑅^𝐵𝛾(𝑥) is an ordinary function if Re(𝛾−2|𝜈|)≥𝑛 and is a distribution of (𝛾−2|𝜈|) if Re(𝛾−2|𝜈|)<𝑛. Let supp𝑅^𝐵𝛾(𝑥) denote the support of 𝑅^𝐵𝛾(𝑥) and suppose that supp𝑅^𝐵𝛾(𝑥)⊂Γ₊ (i.e., supp𝑅^𝐵𝛾(𝑥) is compact).

Letting 𝛾=2𝑘 in (2.2) and (2.3), we obtain𝑅^𝐵2𝑘(𝑥)=𝑢^{(2𝑘−𝑛−2|𝜈|)/2}𝐾_𝑛(2𝑘),(2.4) where𝐾_𝑛(2𝑘)=𝜋^{(𝑛−1+2|𝜈|)/2}Γ((2+2𝑘−𝑛−2|𝜈|)/2)Γ((1−2𝑘)/2)Γ(2𝑘)Γ((2+2𝑘−𝑝−2|𝜈|)/2)Γ((𝑝−2𝑘)/2).(2.5) By putting |𝜈|=0 in (2.2) and (2.3), then formulae (2.2) and (2.3) reduce to𝑅^𝐻𝛾(𝑥)=⎧⎪⎨⎪⎩𝑢𝐾_𝑛(𝛾)^{(𝛾−𝑛)/2},for𝑥∈Γ⁺0,for𝑥∉Γ₊,𝐾,(2.6)_𝑛(𝛾)=𝜋^(𝑛−1)/2Γ((𝛾−𝑛)/2+1)Γ((1−𝛾)/2)Γ(𝛾)Γ((𝛾−𝑝)/2+1)Γ((𝑝−𝛾)/2).(2.7) The function 𝑅^𝐻𝛾(𝑥) is called the ultrahyperbolic kernel of Marcel Riesz and was introduced by Nozaki [22]. 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 supp𝑅^𝐻𝛾(𝑥) denote the support of 𝑅^𝐻𝛾(𝑥) and suppose that supp𝑅^𝐻𝛾(𝑥)⊂Γ₊ ( i.e., supp𝑅^𝐻𝛾(𝑥) is compact).

By putting 𝑝=1 in 𝑅^𝐻2𝑘(𝑥) and taking into account Legendre's duplication formula for Γ(𝑧), that is,Γ(2𝑧)=2^2𝑧−1𝜋^−1/2Γ(𝑧)Γ𝑧+12,(2.8) we obtain𝐼^𝐻𝛾(𝑥)=𝑣^{(𝛾−𝑛)/2}𝐻_𝑛(𝛾)(2.9) and 𝑣=𝑥²¹−𝑥²²−𝑥²³−⋯−𝑥^2𝑛, where𝐻_𝑛(𝛾)=𝜋^(𝑛−2)/22^𝛾−1Γ𝛾+2−𝑛2Γ𝛾2.(2.10) 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),(2.11) where 𝑛 is the dimension of ℝ^𝑛, 𝛾∈ℂ, and 𝑊_𝑛(𝛾)=𝜋^𝑛/22^𝛾Γ(𝛾/2)Γ((𝑛−𝛾)/2).(2.12)
Note that 𝑛=𝑝+𝑞. By putting 𝑞=0 (i.e., 𝑛=𝑝) in (2.6) and (2.7), 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).(2.13)
Using Legendre's duplication formula Γ(2𝑧)=2^2𝑧−1𝜋^−1/2Γ(𝑧)Γ𝑧+12,(2.14) and Γ12+𝑧Γ12−𝑧=𝜋sec(𝜋𝑧),(2.15) we obtain 𝐾_𝑝(𝛾)=12sec𝛾2𝜋𝑊^𝑝(𝛾).(2.16) Thus, for 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).(2.17) In addition, if 𝛾=2𝑘 for some nonnegative integer 𝑘, then 𝑅^𝐻2𝑘(𝑥)=2(−1)^𝑘𝑅^𝑒2𝑘(𝑥).(2.18)

The proofs of Lemma 2.3 are given in [2].

Lemma 2.3. The function 𝑅^𝐻𝛼(x) has the following properties: (i)𝑅^𝐻0(𝑥)=𝛿(𝑥); (ii)𝑅^𝐻−2𝑘(𝑥)=□^𝑘𝛿(𝑥); (iii)□^𝑘𝑅^𝐻𝛼(𝑥)=𝑅^{𝐻𝛼−2𝑘}(𝑥); (iv)□^𝑘𝑅^𝐻2𝑘(𝑥)=𝛿(𝑥).

Lemma 2.4. If |𝜈|≠0, then 𝑅^𝐵𝛾(𝑥)=ℎ_{𝛾,𝑝,|𝜈|}𝑅^{𝐻𝛾−2|𝜈|}(𝑥),(2.19) where 𝑅^𝐵𝛾(𝑥) and 𝑅^{𝐻𝛾−2|𝜈|}(𝑥) are defined by (2.2) and (2.6), respectively, and ℎ_{𝛾,𝑝,|𝜈|}=Γ((1−𝛾)/2+|𝜈|)Γ(𝛾−2|𝜈|)Γ((𝑝−𝛾)/2)𝜋^|𝑣|Γ((𝑝−𝛾)/2+|𝜈|)Γ((1−𝛾)/2)Γ(𝛾).(2.20)

Proof. We get (2.19) by computing directly from definition of 𝑅^𝐵𝛾(𝑥) and 𝑅^{𝐻𝛾−2|𝜈|}(𝑥).

The proof of the following lemma is given in [23].

Lemma 2.5 (the convolutions of 𝑅^𝐻𝛼(𝑥)). (i) If 𝑝 is odd, then 𝑅^𝐻𝛼(𝑥)∗𝑅^𝐻𝛽(𝑥)=𝑅^{𝐻𝛼+𝛽}(𝑥)+𝐴_𝛼,𝛽,(2.21) where 𝐴_𝛼,𝛽=−𝑖2sinsi(n𝛼(𝜋(/𝛼2)+si𝛽n)(𝜋𝛽/𝜋2)/2)𝐻^+𝛼+𝛽𝐻−𝐻^−𝛼+𝛽,(2.22)_±𝛼+𝛽=𝐻_𝛼+𝛽(𝑃±𝑖0,𝑛)(2.23) as defined by (1.8).
(ii) If 𝑝 is even, then 𝑅^𝐻𝛼(𝑥)∗𝑅^𝐻𝛽(𝑥)=𝐵_𝛼,𝛽𝑅^{𝐻𝛼+𝛽}(𝑥),(2.24) where 𝐵_𝛼,𝛽=cos(𝛼𝜋/2)cos(𝛽𝜋/2)cos((𝛼+𝛽)𝜋/2).(2.25)

Lemma 2.6 (the convolutions of 𝑅^𝐵𝛼(𝑥)). (i) If 𝑝 is odd, then 𝑅^𝐵𝛼(𝑥)∗𝑅^𝐵𝛽(𝑥)=ℎ_{𝛼,𝑝,|𝜈|}ℎ_{𝛽,𝑝,|𝜇|}𝑅^{𝐻||𝜇||𝛼+𝛽−2|𝜈|+}+𝐴_{𝛼−2|𝜈|,𝛽−2|𝜇|},(2.26) where 𝑅^𝐻𝛼(𝑥) and 𝐴_{𝛼−2|𝜈|,𝛽−2|𝜇|} are defined by (2.6) and (2.22), respectively.
(ii) If 𝑝 is even, then 𝑅^𝐵𝛼(𝑥)∗𝑅^𝐵𝛽(𝑥)=ℎ_{𝛼,𝑝,|𝜈|}ℎ_{𝛽,𝑝,|𝜇|}𝐵^{𝛼−2|𝜈|,𝛽−2|𝜇|}𝑅^{𝐻||𝜇||𝛼+𝛽−2|𝜈|+},(2.27) where 𝐵_{𝛼−2|𝜈|,𝛽−2|𝜇|} is defined by (2.25).

The proof of this lemma can be easily seen from Lemmas 2.4, 2.5 and [23].

3. The Convolution 𝑅^𝐵𝛼(𝑥)∗𝑅^𝐵𝛽(𝑥) When 𝛽=−𝛼

We will now consider the property of 𝑅^𝐵𝛼(𝑥)∗𝑅^𝐵𝛽(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 𝑅^𝐵𝛼(𝑥)∗𝑅^𝐵𝛽(𝑥)=ℎ_{𝛼,𝑝,|𝜈|}ℎ_{𝛽,𝑝,|𝜇|}𝑅^{𝐻||𝜇||𝛼+𝛽−2|𝜈|+}+𝐴_{𝛼−2|𝜈|,𝛽−2|𝜇|},(3.1) where 𝑅^𝐻𝛼(𝑥) and 𝐴_{𝛼−2|𝜈|,𝛽−2|𝜇|} are defined by (2.6) and (2.22), respectively. (2)If 𝑝 and 𝑞 are both odd, then 𝑅^𝐵𝛼(𝑥)∗𝑅^𝐵𝛽(𝑥)=ℎ_{𝛼,𝑝,|𝜈|}ℎ_{𝛽,𝑝,|𝜇|}𝑅^{𝐻||𝜇||𝛼+𝛽−2|𝜈|+}+𝐴_{𝛼−2|𝜈|,𝛽−2|𝜇|}.(3.2)(3)If 𝑝 is even and 𝑞 is odd, then 𝑅^𝐵𝛼(𝑥)∗𝑅^𝐵𝛽(𝑥)=ℎ_{𝛼,𝑝,|𝜈|}ℎ_{𝛽,𝑝,|𝜇|}cos(c(𝛼os−2𝛼|𝜈+|)𝜋𝛽/2−)2⋅co|𝜈s|+||𝛽𝜇||−2𝜋||𝜇/2||𝜋/2𝑅^{𝐻||𝜇||𝛼+𝛽−2|𝜈|+}.(3.3)(4)If 𝑝 and 𝑞 are both even, then 𝑅^𝐵𝛼(𝑥)∗𝑅^𝐵𝛽(𝑥)=ℎ_{𝛼,𝑝,|𝜈|}ℎ_{𝛽,𝑝,|𝜇|}cos(c(𝛼os−2𝛼|𝜈+|)𝜋𝛽/2−)2⋅co|𝜈s|+||𝛽𝜇||−2𝜋||𝜇/2||𝜋/2𝑅^{𝐻||𝜇||𝛼+𝛽−2|𝜈|+}.(3.4)

Moreover, it follows from (2.22) that𝐴_{𝛼−2|𝜈|,−(𝛼−2|𝜈|)}=lim_{||𝜇||𝛽−2→−(𝛼−2|𝜈|)}𝐴_{𝛼−2|𝜈|,𝛽−2|𝜇|}=−𝑖2lim^𝛾→0sin((𝛼−2|𝜈|)𝜋/2)sin((𝛾−(𝛼−2|𝜈|))𝜋/2)sin(𝛾𝜋/2)𝐻^+𝛾=−𝑖−𝐻^−𝛾2lim^𝛾→0sin((𝛼−2|𝜈|)𝜋/2)sin((𝛾−(𝛼−2|𝜈|))𝜋/2)sin(𝛾𝜋/2)⋅lim^𝛾→0𝐻^+𝛾−𝐻^−𝛾,(3.5) where 𝛾=𝛼+𝛽−2(|𝜈|+|𝜇|).

On the other hand, using (2.23) and (1.8), we havelim_𝛾→0𝐻^+𝛾−𝐻^−𝛾=Γ(𝑛/2)𝜋^𝑛/2lim_𝛾→0𝑒^{−𝛾𝜋𝑖/2}𝑒^{𝑞𝜋𝑖/2}(𝑃+𝑖0)^{(𝛾−𝑛)/2}−lim_𝛾→0Γ(𝛾/2)𝑒^{𝛾𝜋𝑖/2}𝑒^{−𝑞𝜋𝑖/2}(𝑃−𝑖0)^{(𝛾−𝑛)/2}=Γ(𝑛/2)Γ(𝛾/2)𝜋^𝑛/2lim^𝛾→0𝑒^{−𝛾𝜋𝑖/2}𝑒^{𝑞𝜋𝑖/2}⋅Res^{𝛽=−𝑛/2}(𝑃+𝑖0)^𝛽−lim_𝛾→0Res_{𝛽=−𝑛/2}Γ(𝛽+𝑛/2)𝑒^{𝛾𝜋𝑖/2}𝑒^{−𝑞𝜋𝑖/2}⋅Res^{𝛽=−𝑛/2}(𝑃−𝑖0)^𝛽Res_{𝛽=−𝑛/2}Γ(𝛽+𝑛/2).(3.6) Now, taking 𝑛 as an odd integer, we obtainRes_{𝜆=−𝑛/2−𝑘}(𝑃±𝑖0)^𝜆=𝑒^{±𝑞𝜋𝑖/2}𝜋^𝑛/22^2𝑘𝑘!Γ(𝑛/2+𝑘)□^𝑘𝛿(𝑥),(3.7) 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, thenRes_{𝜆=−𝑛/2−𝑘}(𝑃±𝑖0)^𝜆=𝑒^{±𝑞𝜋𝑖/2}𝜋^𝑛/22^2𝑘𝑘!Γ(𝑛/2+𝑘)□^𝑘𝛿(𝑥).(3.8) Nevertheless, if 𝑝 and 𝑞 are both odd, thenRes_{𝜆=−𝑛/2−𝑘}(𝑃±𝑖0)^𝜆=0.(3.9) Therefore, we havelim_𝛾→0𝐻^+𝛾−𝐻^−𝛾=Γ(𝑛/2)𝜋^𝑛/2⋅𝜋^𝑛/2Γ(𝑛/2)lim^𝛾→0𝑒^{−𝛾𝜋𝑖/2}−lim_𝛾→0𝑒^{𝛾𝜋𝑖/2}=lim_𝛾→0𝛿(𝑥)[−2𝑖sin(𝛾𝜋/2)]𝛿(𝑥).(3.10)

From (3.6) and (3.9), we havelim_𝛾→0𝐻^+𝛾−𝐻^−𝛾=0(3.11) if 𝑝 and 𝑞 are both odd (𝑛 even).

Applying (3.10) and (3.11) into (3.5), we have𝐴_{𝛼−2|𝜈|,−𝛼+2|𝜈|}=−𝑖2lim^𝛾→0sin((𝛼−2|𝜈|)𝜋/2)sin((𝛾−(𝛼−2|𝜈|))𝜋/2)sin(𝛾𝜋/2)⋅lim^𝛾→0=sin₂[−2𝑖sin(𝛾𝜋/2)]𝛿(𝑥)((𝛼−2|𝜈|)𝜋/2)𝛿(𝑥)(3.12) if 𝑝 is odd and 𝑞 is even and𝐴_{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 obtain𝑅^𝐵𝛼(𝑥)∗𝑅^𝐵−𝛼(𝑥)==ℎℎ_{𝛼,𝑝,|𝜈|𝛼,𝑝,|𝜈|}ℎ_{−𝛼,𝑝,|𝜈|}𝑅^𝐻0+𝐴_{𝛼−2|𝜈|,−𝛼+2|𝜈|}=ℎ_{𝛼,𝑝,|𝜈|}ℎ_{−𝛼,𝑝,|𝜈|}𝛿(𝑥)+sin²((𝛼−2|𝜈|)𝜋/2)𝛿(𝑥)ℎ_{−𝛼,𝑝,|𝜈|}1+sin²((𝛼−2|𝜈|)𝜋/2)𝛿(𝑥).(3.14) If 𝑝 and 𝑞 are both odd, then𝑅^𝐵𝛼(𝑥)∗𝑅^𝐵−𝛼(𝑥)==ℎℎ_{𝛼,𝑝,|𝜈|𝛼,𝑝,|𝜈|}ℎ_{−𝛼,𝑝,|𝜈|}𝑅^𝐻0+𝐴_{𝛼−2|𝜈|,−𝛼+2|𝜈|}ℎ_{−𝛼,𝑝,|𝜈|}𝛿(𝑥).(3.15) If 𝑝 is even and 𝑞 is odd, then𝑅^𝐵𝛼(𝑥)∗𝑅^𝐵−𝛼(𝑥)=ℎ_{𝛼,𝑝,|𝜈|}ℎ_{−𝛼,𝑝,|𝜈|}cos((𝛼−2|𝜈|)𝜋/2)cos((−𝛼+2|𝜈|)𝜋/2)cos((𝛼−𝛼−2|𝜈|+2|𝜈|)𝜋/2)𝑅^𝐻0=ℎ_{𝛼,𝑝,|𝜈|}ℎ_{−𝛼,𝑝,|𝜈|}cos²((𝛼−2|𝜈|)𝜋/2)𝛿(𝑥).(3.16) Finally, if 𝑝 and 𝑞 are both even, then𝑅^𝐵𝛼(𝑥)∗𝑅^𝐵−𝛼(𝑥)=ℎ_{𝛼,𝑝,|𝜈|}ℎ_{−𝛼,𝑝,|𝜈|}cos((𝛼−2|𝜈|)𝜋/2)cos((−𝛼+2|𝜈|)𝜋/2)cos((𝛼−𝛼−2|𝜈|+2|𝜈|)𝜋/2)𝑅^𝐻0=ℎ_{𝛼,𝑝,|𝜈|}ℎ_{−𝛼,𝑝,|𝜈|}cos²((𝛼−2|𝜈|)𝜋/2)𝛿(𝑥).(3.17)

4. The Main Theorem

Let 𝑀^𝛼(𝑓) be the Bessel ultrahyperbolic Marcel Riesz operator of order 𝛼 of the function 𝑓, which is defined by𝑈^𝛼(𝑓)=𝑅^𝐵𝛼∗𝑓,(4.1) where 𝑅^𝐵𝛼 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^𝛼𝜑=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^𝛼𝜑=𝑓 such that 𝐸^𝛼=(𝑈^𝛼)⁻¹=⎧⎪⎪⎨⎪⎪⎩1ℎ_{𝛼,𝑝,|𝜈|}ℎ_{−𝛼,𝑝,|𝜈|}1+sin²1((𝛼−2|𝜈|)𝜋/2)⁻¹𝑅^𝐵−𝛼if𝑝isoddand𝑞iseven,ℎ_{𝛼,𝑝,|𝜈|}ℎ_{−𝛼,𝑝,|𝜈|}1𝑅^𝐵−𝛼if𝑝and𝑞arebothodd,ℎ_{𝛼,𝑝,|𝜈|}ℎ_{−𝛼,𝑝,|𝜈|}sec²((𝛼−2|𝜈|)𝜋/2)𝑅^𝐵−𝛼if𝑝isevenwith(𝛼−2|𝜈|)/2≠2s+1(4.2) for any nonnegative integer 𝑠.

Proof. By (4.1), we have 𝑈^𝛼(𝑓)=𝑅^𝐵𝛼∗𝑓=𝜑,(4.3) where 𝑅^𝐵𝛼 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 1ℎ_{𝛼,𝑝,|𝜈|}ℎ_{−𝛼,𝑝,|𝜈|}1+sin²=1((𝛼−2|𝜈|)𝜋/2)⁻¹𝑅^𝐵−𝛼∗𝑅^𝐵𝛼∗𝑓ℎ_{𝛼,𝑝,|𝜈|}ℎ_{−𝛼,𝑝,|𝜈|}1+sin²=1((𝛼−2|𝜈|)𝜋/2)⁻¹𝑅^𝐵−𝛼∗𝑅^𝐵𝛼∗𝑓ℎ_{𝛼,𝑝,|𝜈|}ℎ_{−𝛼,𝑝,|𝜈|}1+sin²((𝛼−2|𝜈|)𝜋/2)⁻¹=×𝛿∗ℎ𝑓=𝑓_{𝛼,𝑝,|𝜈|}ℎ._{−𝛼,𝑝,|𝜈|}1+sin²((𝛼−2|𝜈|)𝜋/2)𝛿(𝑥)∗𝑓(4.4) Hence, 1ℎ_{𝛼,𝑝,|𝜈|}ℎ_{−𝛼,𝑝,|𝜈|}1+sin²((𝛼−2|𝜈|)𝜋/2)⁻¹𝑅^𝐵−𝛼=(𝑈^𝛼)⁻¹=𝑅^𝐵𝛼⁻¹(4.5) for all 𝛼∈ℂ.
Similarly, if both 𝑝 and 𝑞 are odd, then, by (3.15), we obtain 1ℎ_{𝛼,𝑝,|𝜈|}ℎ_{−𝛼,𝑝,|𝜈|}𝑅^𝐵−𝛼∗𝑅^𝐵𝛼∗𝑓=1ℎ_{𝛼,𝑝,|𝜈|}ℎ_{−𝛼,𝑝,|𝜈|}𝑅^𝐵−𝛼=1∗𝑅^𝐵𝛼∗𝑓ℎ_{𝛼,𝑝,|𝜈|}ℎ_{−𝛼,𝑝,|𝜈|}=𝑓.ℎ_{𝛼,𝑝,|𝜈|}ℎ_{−𝛼,𝑝,|𝜈|}𝛿(𝑥)∗𝑓(4.6) Hence, 1ℎ_{𝛼,𝑝,|𝜈|}ℎ_{−𝛼,𝑝,|𝜈|}𝑅^𝐵−𝛼=(𝑈^𝛼)⁻¹=𝑅^𝐵𝛼⁻¹(4.7) for all 𝛼∈ℂ.
Finally, if 𝑝 is even, then, by (3.16) and (3.17), we have 1ℎ_{𝛼,𝑝,|𝜈|}ℎ_{−𝛼,𝑝,|𝜈|}=1sec²((𝛼−2|𝜈|)𝜋/2)𝑅^𝐵−𝛼∗𝑅^𝐵𝛼∗𝑓ℎ_{𝛼,𝑝,|𝜈|}ℎ_{−𝛼,𝑝,|𝜈|}=1sec²((𝛼−2|𝜈|)𝜋/2)𝑅^𝐵−𝛼∗𝑅^𝐵𝛼∗𝑓ℎ_{𝛼,𝑝,|𝜈|}ℎ_{−𝛼,𝑝,|𝜈|}=𝛿∗𝑓=𝑓,sec²((𝛼−2|𝜈|)𝜋/2)ℎ_{𝛼,𝑝,|𝜈|}ℎ_{−𝛼,𝑝,|𝜈|}cos²((𝛼−2|𝜈|)𝜋/2)𝛿(𝑥)∗𝑓(4.8) provided that (𝛼−2|𝜈|)/2≠2𝑠+1 for any nonnegative integer 𝑠.
Hence, 1ℎ_{𝛼,𝑝,|𝜈|}ℎ_{−𝛼,𝑝,|𝜈|}sec²((𝛼−2|𝜈|)𝜋/2)𝑅^𝐵−𝛼=(𝑈^𝛼)⁻¹=𝑅^𝐵𝛼⁻¹(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.

Acknowledgments

This work is supported by the Commission on Higher Education, the Thailand Research Fund, and Khon Kaen University (Contract no. MRG5380118) and the Centre of Excellence in Mathematics, Thailand.