On the convolution product of the distributional kernel Kα,β,γ,ν

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PII. S0161171203106151 http://ijmms.hindawi.com © Hindawi Publishing Corp.

ON THE CONVOLUTION PRODUCT

OF THE DISTRIBUTIONAL

KERNEL

K

α,β,γ,ν

A. KANANTHAI and S. SUANTAI

Received 18 June 2001 and in revised form 19 February 2002

We introduce a distributional kernelKα,β,γ,ν which is related to the operator⊕k iterated k times and deﬁned by ⊕k₌_[(p

r=1∂2/∂xr2)4−(

p+q

j=p+1∂2/∂xj2)4]k, where p+q= nis the dimension of the space Rn _{of the}_n_-dimensional Eu-clidean space,x=(x1,x2,...,xn)∈Rn,kis a nonnegative integer, andα,β,γ, andνare complex parameters. It is found that the existence of the convolution Kα,β,γ,ν∗Kα,β,γ,νis depending on the conditions ofpandq.

2000 Mathematics Subject Classiﬁcation: 46F10, 46F12.

1. Introduction. The operator⊕k_{can be factorized in the form}

⊕k₌     p

r=1 ∂2 ∂x2 r   2 −   p+q

j=p+1 ∂2 ∂x2 j   2   k ×  p

r=1 ∂2 ∂x2

r +i p+q

j=p+1 ∂2 ∂x2 j   k p

r=1 ∂2 ∂x2

r−i p+q

j=p+1 ∂2 ∂x2 j   k , (1.1)

wherep+q=nis the dimension of the spaceRn_, _i₌√₋_{1, and}_k_{is a}

non-negative integer. The operator(pr=1∂2/∂x2r)2−( p+q

j=p+1∂2/∂x2j)2is ﬁrst

in-troduced by Kananthai [2] and named the Diamond operator denoted by

♦=  p

r=1 ∂2 ∂x2r

  2

−   p+q

j=p+1 ∂2 ∂x2j

  2

. (1.2)

We denote the operatorsL1andL2by

L1= p

r=1 ∂2 ∂xr2+i

p+q

j=p+1 ∂2 ∂xj2

,

L2= p

r=1 ∂2 ∂x2

r−i p+q

j=p+1 ∂2 ∂x2

j .

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Thus (1.1) can be written by

⊕k₌_♦k_Lk_1Lk_2. _(1.4)

Now consider the convolutionRH

α(u)∗Rβe(v)∗Sγ(w)∗Tν(z)whereRαH(u), Rβe(v),Sγ(w), andTν(z)are deﬁned by (2.2), (2.4), (2.6), and (2.7), respectively.

We deﬁned the distributional kernelKα,β,γ,ν by

Kα,β,γ,ν=RH

α(u)∗Rβe(v)∗Sγ(w)∗Tν(z). (1.5)

Since the functionsRH

α(u),Reβ(v),Sγ(w), andTν(z)are all tempered

distribu-tions and the supports ofRH

α(u)andRβe(v)are compact (see [2, pages 30–31]

and [1, pages 152–153]), then the convolution on the right-hand side of (1.5) exists and also is a tempered distributions. ThusKα,β,γ,ν is well deﬁned and also is a tempered distribution.

Forα=β=γ=ν=2k, we obtain(−1)k_K2k,2k,2k,2k_{as an elementary solution}

of the operator⊕k_{, see [}₃_{]. That is}_⊕k₍₋₁₎k_{K2k,2k,2k,2k(x)}₌_δ _where_δ _{is the}

Dirac-delta distribution and⊕k_{is deﬁned by (}_1.4_).

2. Preliminaries

Definition2.1. Letx=(x1,x2,...,xn)∈Rn_{and write}

x=x2

1+x22+···+xp2−xp2+1−x2p+2−···−x2p+q, p+q=n. (2.1)

Denote byΓ+= {x∈Rn:x1>0 andu >0}the interior of forward cone and Γ+denote its closure. For any complex numberα, we deﬁne the function

RH α(x)=

      

u(α−n)/2

Kn(α) , ifx∈Γ+,

0, ifx∉Γ+,

(2.2)

where the constantKn(α)is given by the formula

Kn(α)=π(n−1)/2Γ

(2+α−n)/2Γ(1−α)/2Γ(α)

Γ(2+α−p)/2Γ(p−α)/2 , (2.3)

the functionRH

α is ﬁrst introduced by Nozaki [4, page 72] and is called the

ultra-hyperbolic kernel of Marcel Riesz. HenceRH

α(x)is an ordinary function

if Re(α)≥nand is a distribution ofαif Re(α) < n. Let suppRH

α(u)⊂Γ+where

suppRH

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...

Definition2.2. Letx=(x1,x2,...,xn)∈Rn_{and write}_v₌_x2

1+x22+···+ x2

n.

For any complex numberβ, deﬁne the function

Reβ(v)=v (β−n)/2

Wn(β) , (2.4)

whereWn(β)=πn/2₂β_Γ_(β)/_Γ_((n₋_β)/₂₎_{, the function}_Re

β(v)is called the

el-liptic kernel of Marcel Riesz and is ordinary function if Re(β)≥nand is a distribution ofβif Re(β) < n.

Definition2.3. Letx=(x1,x2,...,xn)be a point of the spaceRn. Write

w=x2

1+x22+···+x2p−i

x2

p+1+xp2+2+···+xp+q

,

z=x2

1+x22+···+x2p+i

x2

p+1+xp2+2+···+xp+q, p+q=n, i=

−1.

(2.5)

For any complex numbersγandν, deﬁne

Sγ(ω)=ω(γ−n)/2

Wn(γ) , (2.6)

Tν(z)=z(ν−n)/2

Wn(ν), (2.7)

where

Wn(γ)=π_Γn/2 2γΓ(γ/2)

(n−γ)/2 , Wn(ν)=

πn/2₂ν_Γ_(ν/₂₎

Γ(n−ν)/2 . (2.8)

Lemma2.4(the convolution product ofRβ(v)e ). The convolutionRβe∗Reβ= Re

β+β whereReβandRβare given by (2.2). Proof. See [5, page 20].

Lemma2.5(the convolution product ofRH

α(x)). The convolution product is given by

(i)

RHα∗RHα=cos

α(π/2)cosα(π/2)

cos(α+β)/2π ·R H

α+α, (2.9)

whereRH

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(ii) RH

α∗RHα=RαH+α+Aα,αforpodd, where

Aα,α=2πi

4

C(−α−α)/2

C(−α/2)C(−α/2)

Hα++α−Hα−+α

,

C(r )=Γ(r )Γ(1−r ),

Hr±=Hr(x±i0,n)=e∓r (π/2)ie±q(π/2)ia

r

2

(u±i0)(r−n)/2_,

a

r

2

=Γ

n−r

2

2rπn/2Γ

r

2

−1 ,

(u±i0,n)λ₌_lim

→0(u±i,n) λ_,

(2.10)

u=u(x)is defined by (2.1) and|x| =(x12+x22+···+x2n)1/2in particu-larRH

α∗R−H2k=RHα−2kandRαH∗R2kH =RHα+2k.

The proof of this lemma is given by Téllez [6, pages 121–123].

Lemma2.6(the convolutions product ofSγ(w)and Tν(z)). The convolu-tions product is given by

(i) Sγ∗Sγ=_(i)q/2_Sγ₊_γ,

(ii) Tν∗Tν=₍−_i)q/2_Tν₊_ν whereSγ andTν are defined by (2.6) and (2.7), respectively.

Proof. (i) Now

Sγ(w),ϕ(x)= 1 Wn(γ)

Rnω

(γ−n)/2_ϕ(x)dx, _(2.11)

whereϕ∈Ᏸthe space of inﬁnitely diﬀerentiable function with compact sup-ports. We haveω=x12+x22+···+x2p−i(xp2+1+xp2+2+···+x2p+q), p+q=n.

By changing the variablesx1=y1, x2=y2,...,xp=yp, xp+1=yp+1/√−i,

xp+2=yp+2/ √

−i,..., andxp+q=yp+q/√−i. Thus we obtainω=y12+y22+ ··· +y2

p+yp2+1+yp2+2+ ··· +yp2+q. Letr2=y12+y22+ ··· +yp2+q, p+q=n.

Thus (2.11) can be written in the form

Sγ(w),ϕ(x)= 1 Wn(γ)

Rnr γ−n_ϕ∂

x1,x2,...,xn

∂y1,y2,...,yndy1dy2···dyn

= 1

(−i)q/2

1

Wn(γ)

Rnr

γ−n_{ϕ dy}

= (i)q/2 Wn(γ)

rγ−n_,ϕ_.

(2.12)

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...

Consider the convolutionSγ∗Sγ. We have

Sγ∗Sγ=_(i)q/2_R_γ(w)e ∗_(i)q/2_Re_γ(w) =(i)q_Re

γ+γ(w) byLemma 2.4and [1, pages 157–159] =(i)q/2_(i)q/2_Re

γ+γ(w) =(i)q/2_Sγ

+γ.

(2.13)

Similarly, for (ii) we also have

Tν∗Tν=₍−_i)q/2_Tν₊_ν_. (2.14)

3. Main results

Theorem3.1. LetKα,β,γ,ν be the distributional kernel defined by (1.5). Then we obtain the following:

(i) Kα,β,γ,ν∗Kα_,β_,γ_,ν=_Kα₊_α_,β₊_β_,γ₊_γ_,ν₊_ν forα,β,γ,ν,α,β,γ, and ν positive even numbers withα=β=γ=ν,α=β=γ=ν;

(ii) Kα,β,γ,ν∗Kα_,β_,γ_,ν=_Bα,α·_Kα₊_α_,β₊_β_,γ₊_γ_,ν₊_ν forpeven,α,β,γ,ν,α, β,γ, andνany complex numbers, andBα,α=cos(απ/2)cos(απ/2)/

cos((α+α)/2)π;

(iii) Kα,β,γ,ν∗Kα,β,γ,ν=_Kα₊α,β+β,γ+γ,ν+ν+_Aα,α∗_Re_β₊_β∗Sγ+γ∗_Tν₊ν forpodd,α,β,γ,ν,α,β,γ, andνany complex numbersRe

β,Sγ, and Tνdefined by (2.4), (2.6), and (2.7), respectively. And

Aα,α=C

(−α−α)/2

C(−α/2) C

−α

2

·2πi

4

Hα++α−Hα−+α

,

C(r )=Γ(r )Γ(1−r ),

Hr±=Hr(u±i0,n)=e∓r (π/2)ie±q(π/2)ia

r

2

(u±i0)r−n/2,

a

r

2

=Γ

n−r

2

2r_πn/2_Γ

r

2

−1 ,

(u±i0)λ=lim

→0

u+i∈ |x|2λ ,

(3.1)

whereu=u(x)is defined by (2.1) and

|x| =x12+x22+···+xn2 1/2

. (3.2)

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References

[1] W. F. Donoghue,Distributions and Fourier Transforms, Academic Press, New York, 1969.

[2] A. Kananthai,On the solutions of then-dimensional diamond operator, Appl. Math. Comput.88(1997), no. 1, 27–37.

[3] A. Kananthai, S. Suantai, and V. Longani,On the weak solutions of the equation related to the diamond operator, Vychisl. Tekhnol.5(2000), no. 5, 61–67. [4] Y. Nozaki,On Riemann-Liouville integral of ultra-hyperbolic type, K¯odai Math. Sem.

Rep.16(1964), 69–87.

[5] M. Riesz,L’intégrale de Riemann-Liouville et le problème de Cauchy, Acta Math.

81(1949), 1–223 (French).

[6] M. A. Téllez and S. E. Trione,The distributional convolution products of Marcel Riesz’ ultra-hyperbolic kernel, Rev. Un. Mat. Argentina39(1995), no. 3-4, 115–124.

A. Kananthai: Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand

E-mail address:[email protected]

S. Suantai: Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand