# On the Inversion of Bessel Ultrahyperbolic Kernel of Marcel Riesz

Authors: Darunee Maneetus; Kamsing Nonlaopon
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

We define the Bessel ultrahyperbolic Marcel Riesz operator on the functionf by Uαf=RαB*f, where RαB 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 Eα=Uα-1 such that, if Uαf=φ, then Eαφ=f.


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## Body

## 1. Introduction

Then-dimensional ultrahyperbolic operator □k iterated k times is defined by(1.1)□k=(∂2∂x12+∂2∂x22+⋯+∂2∂xp2-∂2∂xp+12-∂2∂xp+22-⋯-∂2∂xp+q2)k,
where p+q=n is the dimension of ℝn and k is a nonnegative integer.Consider the linear differential equation in the form of(1.2)□ku(x)=f(x),
where u(x) and f(x) are generalized functions and x=(x1,x2,…,xn)∈ℝn.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 R2kH(x), defined by (2.6) with γ=2k, is the unique fundamental solution of (1.2) and Téllez [3] has also proved that R2kH(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(1.3)◊k=[(∑i=1p∂2∂xi2)2-(∑j=p+1p+q∂2∂xj2)2]k,
where n=p+q is the dimension of ℝn, for all x=(x1,x2,…,xn), and k is a nonnegative integer. The operator ◊k can be expressed in the form(1.4)◊k=▵k□k=□k▵k,
where □k is defined by (1.1), and(1.5)▵k=(∂2∂x12+∂2∂x22+⋯+∂2∂xn2)k
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)kR2ke(x)*R2kH(x) is the fundamental solution of the operator ◊k, that is,(1.6)◊k((-1)kR2ke(x)*R2kH(x))=δ(x),
where R2kH(x) and R2ke(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)kR2ke(x)*R2kH(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 Hα(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(1.7)Rαφ=Hα(P±i0,n)*φ,
where φ∈𝒮, 𝒮 is the Schwartz space of functions [14] and Hα(P±i0,n) is given by(1.8)Hα(P±i0,n)=e∓απi/2e±qπi/2Γ((n-α)/2)(P±i0)(α-n)/22απn/2Γ(α/2).
Here, P is defined by(1.9)P=P(x)=x12+x22+⋯+xp2-xp+12-xp+22-⋯-xp+q2,
where q is the number of negative terms of the quadratic form P. The distributions (P±i0)λ are defined by(1.10)(P±i0)λ=lim⁡ϵ→0(P±iϵ|x|2)λ,
where ϵ>0, λ∈ℂ, and |x|2=x12+x22+⋯+xn2; see [1]. They have also studied the inverse operator of Rα, denoted by (Rα)-1, such that, if f=Rαφ, then (Rα)-1f=φ.Later, Aguirre [15] has defined the ultrahyperbolic Marcel Riesz operator Mα of the function f by(1.11)Mα(f)=RαH*f,
where RαH is defined by (2.6) and f∈𝒮. He has also studied the operator Nα=(Mα)-1 such that, if Mα(f)=φ, then Nαφ=f.Let us consider the diamond kernel of Marcel RieszKα,β(x) introduced by Kananthai in [6], which is given by the convolution(1.12)Kα,β(x)=Rαe*RβH,
where Rαe is elliptic kernel defined by (2.11) and RβH 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 diamond Marcel Riesz operator of order (α,β) of the function f by(1.13)M(α,β)(f)=Kα,β*f,
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, if M(α,β)(f)=φ, then N(α,β)φ=f.In this paper, we define the Bessel ultrahyperbolic Marcel Riesz operator of orderα of the function f by(1.14)Uα(f)=RαB*f,
where α∈ℂ and f∈𝒮, 𝒮 is the Schwartz space of functions. Our aim in this paper is to obtain the operator Eα=(Uα)-1 such that, if Uα(f)=φ, then Eαφ=f.Before we proceed to our main theorem, the following definitions and concepts require some clarifications.


## 2. Preliminaries

Definition 2.1.
Letx=(x1,x2,…,xn) be a point in the n-dimensional Euclidean space ℝn. Let
(2.1)u=x12+x22+⋯+xp2-xp+12-xp+22-⋯-xp+q2
be the nondegenerated quadratic form, where p+q=n is the dimension of ℝn. Let Γ+={x∈ℝn:u>0andxi>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
(2.2)RγB(x)={u(γ-2|ν|-n)/2Kn|ν|(γ),forx∈Γ+,0,forx∉Γ+,
where
(2.3)Kn|ν|(γ)=π(n-1+2|ν|)/2Γ((2+γ-n-2|ν|)/2)Γ((1-γ)/2)Γ(γ)Γ((2+γ-p-2|ν|)/2)Γ((p-γ)/2),2vi=2αi+1, αi>-1/2 and |ν|=ν1+ν2+⋯+νn, see [18–20].The functionRγB(x) is called the Bessel ultrahyperbolic kernel and was introduced by Aguirre [21]. It is well known that RγB(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 suppRγB(x) denote the support of RγB(x) and suppose that suppRγB(x)⊂Γ¯+ (i.e., suppRγB(x) is compact).Lettingγ=2k in (2.2) and (2.3), we obtain(2.4)R2kB(x)=u(2k-n-2|ν|)/2Kn(2k),
where(2.5)Kn(2k)=π(n-1+2|ν|)/2Γ((2+2k-n-2|ν|)/2)Γ((1-2k)/2)Γ(2k)Γ((2+2k-p-2|ν|)/2)Γ((p-2k)/2).
By putting |ν|=0 in (2.2) and (2.3), then formulae (2.2) and (2.3) reduce to(2.6)RγH(x)={u(γ-n)/2Kn(γ),forx∈Γ+,0,forx∉Γ+,(2.7)Kn(γ)=π(n-1)/2Γ((γ-n)/2+1)Γ((1-γ)/2)Γ(γ)Γ((γ-p)/2+1)Γ((p-γ)/2).
The function RγH(x) is called the ultrahyperbolic kernel of Marcel Riesz and was introduced by Nozaki [22]. It is well known that RγH(x) is an ordinary function if Re(γ)≥n and is a distribution of γ if Re(γ)<n. Let suppRγH(x) denote the support of RγH(x) and suppose that suppRγH(x)⊂Γ¯+ ( i.e., suppRγH(x) is compact).By puttingp=1 in R2kH(x) and taking into account Legendre's duplication formula for Γ(z), that is,(2.8)Γ(2z)=22z-1π-1/2Γ(z)Γ(z+12),
we obtain(2.9)IγH(x)=v(γ-n)/2Hn(γ)
and v=x12-x22-x32-⋯-xn2, where(2.10)Hn(γ)=π(n-2)/22γ-1Γ(γ+2-n2)Γ(γ2).
The function IγH(x) is called the hyperbolic kernel of Marcel Riesz.Definition 2.2.
Letx=(x1,x2,…,xn) be a point of ℝn and ω=x12+x22+⋯+xn2. The elliptic kernel of Marcel Riesz is defined by
(2.11)Rγe(x)=ω(γ-n)/2Wn(γ),
where n is the dimension of ℝn, γ∈ℂ, and
(2.12)Wn(γ)=πn/22γΓ(γ/2)Γ((n-γ)/2).
Note thatn=p+q. By putting q=0 (i.e., n=p) in (2.6) and (2.7), we can reduce u(γ-n)/2 to ωp(γ-p)/2, where ωp=x12+x22+⋯+xp2, and reduce Kn(γ) to
(2.13)Kp(γ)=π(p-1)/2Γ((1-γ)/2)Γ(γ)Γ((p-γ)/2).
Using Legendre's duplication formula(2.14)Γ(2z)=22z-1π-1/2Γ(z)Γ(z+12),
and
(2.15)Γ(12+z)Γ(12-z)=πsec(πz),
we obtain
(2.16)Kp(γ)=12sec(γπ2)Wp(γ).
Thus, for q=0, we have
(2.17)RγH(x)=u(γ-p)/2Kp(γ)=2cos⁡⁡(γπ2)u(γ-p)/2Wp(γ)=2cos⁡⁡(γπ2)Rγe(x).
In addition, if γ=2k for some nonnegative integer k, then
(2.18)R2kH(x)=2(-1)kR2ke(x).The proofs of Lemma2.3 are given in [2].Lemma 2.3.
The functionRαH(x) has the following properties:  (i)
R0H(x)=δ(x);(ii)
R-2kH(x)=□kδ(x);(iii)
□kRαH(x)=Rα-2kH(x);(iv)
□kR2kH(x)=δ(x).Lemma 2.4.
If|ν|≠0, then
(2.19)RγB(x)=hγ,p,|ν|Rγ-2|ν|H(x),
where RγB(x) and Rγ-2|ν|H(x) are defined by (2.2) and (2.6), respectively, and
(2.20)hγ,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).Proof.
We get (2.19) by computing directly from definition of RγB(x) and Rγ-2|ν|H(x).The proof of the following lemma is given in [23].Lemma 2.5 (the convolutions ofRαH(x)).
(i) Ifp is odd, then
(2.21)RαH(x)*RβH(x)=Rα+βH(x)+Aα,β,
where
(2.22)Aα,β=-i2sin⁡(απ/2)sin⁡(βπ/2)sin⁡((α+β)π/2)[Hα+β+-Hα+β-],(2.23)Hα+β±=Hα+β(P±i0,n)
as defined by (1.8).
(ii) Ifp is even, then
(2.24)RαH(x)*RβH(x)=Bα,βRα+βH(x),
where
(2.25)Bα,β=cos⁡(απ/2)cos⁡(βπ/2)cos⁡((α+β)π/2).Lemma 2.6 (the convolutions ofRαB(x)).
(i) Ifp is odd, then
(2.26)RαB(x)*RβB(x)=hα,p,|ν|hβ,p,|μ|(Rα+β-2(|ν|+|μ|)H+Aα-2|ν|,β-2|μ|),
where RαH(x) and Aα-2|ν|,β-2|μ| are defined by (2.6) and (2.22), respectively.
(ii) Ifp is even, then
(2.27)RαB(x)*RβB(x)=hα,p,|ν|hβ,p,|μ|(Bα-2|ν|,β-2|μ|Rα+β-2(|ν|+|μ|)H),
where Bα-2|ν|,β-2|μ| is defined by (2.25).The proof of this lemma can be easily seen from Lemmas2.4, 2.5 and [23].


## 3. The ConvolutionRαB(x)*RβB(x) When β=-α

We will now consider the property ofRαB(x)*RβB(x) when β=-α.From (2.26) and (2.27), we immediately obtain the following properties.  (1)
Ifp is odd and q is even, then
(3.1)RαB(x)*RβB(x)=hα,p,|ν|hβ,p,|μ|(Rα+β-2(|ν|+|μ|)H+Aα-2|ν|,β-2|μ|),
where RαH(x) and Aα-2|ν|,β-2|μ| are defined by (2.6) and (2.22), respectively.(2)
Ifp and q are both odd, then
(3.2)RαB(x)*RβB(x)=hα,p,|ν|hβ,p,|μ|(Rα+β-2(|ν|+|μ|)H+Aα-2|ν|,β-2|μ|).(3)
Ifp is even and q is odd, then
(3.3)RαB(x)*RβB(x)=hα,p,|ν|hβ,p,|μ|(cos⁡((α-2|ν|)π/2)⋅cos⁡((β-2|μ|)π/2)cos⁡((α+β-2(|ν|+|μ|))π/2)Rα+β-2(|ν|+|μ|)H).(4)
Ifp and q are both even, then
(3.4)RαB(x)*RβB(x)=hα,p,|ν|hβ,p,|μ|(cos⁡((α-2|ν|)π/2)⋅cos⁡((β-2|μ|)π/2)cos⁡((α+β-2(|ν|+|μ|))π/2)Rα+β-2(|ν|+|μ|)H).Moreover, it follows from (2.22) that(3.5)Aα-2|ν|,-(α-2|ν|)=lim⁡β-2|μ|→-(α-2|ν|)Aα-2|ν|,β-2|μ|=-i2lim⁡γ→0sin⁡((α-2|ν|)π/2)sin⁡((γ-(α-2|ν|))π/2)sin⁡(γπ/2)[Hγ+-Hγ-]=-i2lim⁡γ→0sin⁡((α-2|ν|)π/2)sin⁡((γ-(α-2|ν|))π/2)sin⁡(γπ/2)⋅lim⁡γ→0[Hγ+-Hγ-],
where γ=α+β-2(|ν|+|μ|).On the other hand, using (2.23) and (1.8), we have(3.6)lim⁡γ→0[Hγ+-Hγ-]=Γ(n/2)πn/2[lim⁡γ→0e-γπi/2eqπi/2(P+i0)(γ-n)/2Γ(γ/2)-lim⁡γ→0eγπi/2e-qπi/2(P-i0)(γ-n)/2Γ(γ/2)]=Γ(n/2)πn/2[lim⁡γ→0e-γπi/2eqπi/2⋅Resβ=-n/2(P+i0)βResβ=-n/2Γ(β+n/2)-lim⁡γ→0eγπi/2e-qπi/2⋅Resβ=-n/2(P-i0)βResβ=-n/2Γ(β+n/2)].
Now, taking n as an odd integer, we obtain(3.7)Resλ=-n/2-k(P±i0)λ=e±qπi/2πn/222kk!Γ(n/2+k)□kδ(x),
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(3.8)Resλ=-n/2-k(P±i0)λ=e±qπi/2πn/222kk!Γ(n/2+k)□kδ(x).
Nevertheless, if p and q are both odd, then(3.9)Resλ=-n/2-k(P±i0)λ=0.
Therefore, we have(3.10)lim⁡γ→0[Hγ+-Hγ-]=Γ(n/2)πn/2⋅πn/2Γ(n/2)[lim⁡γ→0e-γπi/2-lim⁡γ→0eγπi/2]δ(x)=lim⁡γ→0[-2isin⁡⁡(γπ/2)]δ(x).From (3.6) and (3.9), we have(3.11)lim⁡γ→0[Hγ+-Hγ-]=0
if p and q are both odd (n even).Applying (3.10) and (3.11) into (3.5), we have(3.12)Aα-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[-2isin⁡(γπ/2)]δ(x)=sin⁡2((α-2|ν|)π/2)δ(x)
if p is odd and q is even and(3.13)Aα-2|ν|,-α+2|ν|=0
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)RαB(x)*R-αB(x)=hα,p,|ν|h-α,p,|ν|(R0H+Aα-2|ν|,-α+2|ν|)=hα,p,|ν|h-α,p,|ν|[δ(x)+sin⁡2((α-2|ν|)π/2)δ(x)]=hα,p,|ν|h-α,p,|ν|[1+sin⁡2((α-2|ν|)π/2)]δ(x).
If p and q are both odd, then(3.15)RαB(x)*R-αB(x)=hα,p,|ν|h-α,p,|ν|(R0H+Aα-2|ν|,-α+2|ν|)=hα,p,|ν|h-α,p,|ν|δ(x).
If p is even and q is odd, then(3.16)RαB(x)*R-αB(x)=hα,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)R0H=hα,p,|ν|h-α,p,|ν|cos⁡2((α-2|ν|)π/2)δ(x).
Finally, if p and q are both even, then(3.17)RαB(x)*R-αB(x)=hα,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)R0H=hα,p,|ν|h-α,p,|ν|cos⁡2((α-2|ν|)π/2)δ(x).


## 4. The Main Theorem

LetMα(f) be the Bessel ultrahyperbolic Marcel Riesz operator of order α of the function f, which is defined by(4.1)Uα(f)=RαB*f,
where RαB is defined by (2.2), α∈ℂ, and f∈𝒮.Recall that our objective is to obtain the operatorEα=(Uα)-1 such that, if Uα(f)=φ, then Eαφ=f for all α∈ℂ.We are now ready to state our main theorem.Theorem 4.1.
IfUα(f)=φ (where Uα(f) is defined by (4.1) and f∈𝒮), then Eαφ=f such that
(4.2)Eα=(Uα)-1={1hα,p,|ν|h-α,p,|ν|[1+sin⁡2((α-2|ν|)π/2)]-1R-αBifpisoddandqiseven,1hα,p,|ν|h-α,p,|ν|R-αBifpandqarebothodd,1hα,p,|ν|h-α,p,|ν|sec2((α-2|ν|)π/2)R-αBifpisevenwith(α-2|ν|)/2≠2s+1
for any nonnegative integer s.Proof.
By (4.1), we have
(4.3)Uα(f)=RαB*f=φ,
where RαB 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
(4.4)1hα,p,|ν|h-α,p,|ν|[1+sin⁡2((α-2|ν|)π/2)]-1R-αB*(RαB*f)=1hα,p,|ν|h-α,p,|ν|[1+sin⁡2((α-2|ν|)π/2)]-1(R-αB*RαB)*f=1hα,p,|ν|h-α,p,|ν|[1+sin⁡2((α-2|ν|)π/2)]-1×{hα,p,|ν|h-α,p,|ν|[1+sin⁡2((α-2|ν|)π/2)]δ(x)}*f=δ*f=f.
Hence,
(4.5)1hα,p,|ν|h-α,p,|ν|[1+sin⁡2((α-2|ν|)π/2)]-1R-αB=(Uα)-1=(RαB)-1
for all α∈ℂ.
Similarly, if bothp and q are odd, then, by (3.15), we obtain
(4.6)1hα,p,|ν|h-α,p,|ν|R-αB*(RαB*f)=1hα,p,|ν|h-α,p,|ν|(R-αB*RαB)*f=1hα,p,|ν|h-α,p,|ν|hα,p,|ν|h-α,p,|ν|δ(x)*f=f.
Hence,
(4.7)1hα,p,|ν|h-α,p,|ν|R-αB=(Uα)-1=(RαB)-1
for all α∈ℂ.
Finally, ifp is even, then, by (3.16) and (3.17), we have
(4.8)1hα,p,|ν|h-α,p,|ν|sec2((α-2|ν|)π/2)R-αB*(RαB*f)=1hα,p,|ν|h-α,p,|ν|sec2((α-2|ν|)π/2)(R-αB*RαB)*f=1hα,p,|ν|h-α,p,|ν|sec2((α-2|ν|)π/2){hα,p,|ν|h-α,p,|ν|cos⁡2((α-2|ν|)π/2)δ(x)}*f=δ*f=f,
provided that (α-2|ν|)/2≠2s+1 for any nonnegative integer s.
Hence,(4.9)1hα,p,|ν|h-α,p,|ν|sec2((α-2|ν|)π/2)R-αB=(Uα)-1=(RαB)-1
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.



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*Source: 419157-2011-11-17.xml*