On the convolution equation related to the diamond kernel
of Marcel Riesz
Amnuay Kananthai
Department of Mathematics, Chiang Mai University, Chiang Mai 50200, Thailand
Received 3 February 1998; revised 15 July 1998
Abstract
In this paper, we study the distribution e t♦k where ♦k is introduced and named as the Diamond operator iterated
k-times (k= 0;1;2; : : :) and is dended by ♦k= @
2
@t2 1
+ @ 2
@t2 2
+· · ·+ @ 2
@t2
p
2 −
@2 @t2
p+1 + @
2
@t2
p+2
+· · ·+ @ 2
@t2
p+q
2!k
;
where t= (t1; t2; : : : ; tn) is a variable and = (1; 2; : : : ; n) is a constant and both are the points in the n-dimensional Euclidean spaceRn,is the Dirac-delta distribution with ♦0=and p+q=n (the dimension ofRn)
At rst, the properties of e t♦k are studied and later we study the application of e t♦kfor solving the solutions of the convolution equation
(e t♦k)∗u(t) = e t
m
X
r=0
cr♦r:
We found that its solutions related to the Diamond Kernel of Marcel Riesz and moreover, the type of solutions such as, the classical solution (the ordinary function) or the tempered distributions depending on m; k and . c 1998 Elsevier Science B.V. All rights reserved.
AMS classication:46F10
Keywords:Diamond operator; Kernel of Marcel Riesz; Dirac delta distributions; Tempered distribution
1. Introduction
From [2, Theorem 3.1], the equation ♦ku(t) = has (−1)kS
2k(t)∗R2k(t) as an elementary solution
and is called the Diamond Kernel of Marcel Riesz where S2k(t) and R2k(t) are dened by (2.1) and
(2.2), respectively, with = 2k where is nonnegative.
Consider the convolution equation
(et♦k)∗u(t) = et m
X
r=0
cr♦r: (1.1)
In nding the type of solutions u(t) of Eq. (1.1), we use the method of convolution of the tempered distribution. Before going to that point, some denitions and basic concepts are needed.
2. Preliminaries
Denition 2.1. Let the function S(t) be dened by
S= 2−−n=2
n−
2
|t|−n
(=2); (2.1)
where is a complex parameter, n is the dimension of Rn, t= (t
1; t2; : : : ; tn)∈Rn and |t|=q(t2
1 +· · ·+t2n). Now S is an ordinary function if Re()¿n and is a distribution of if
Re()¡n.
Denition 2.2. Lett= (t1; t2; : : : ; tn) be the point of Rn and write v=t12+t22+· · ·+tp2−tp2+1−tp2+2−
· · · −t2
p+q, p+q=n. Denote by +={t∈Rn: t1¿0 and v¿0} the set of an interior of the forward
cone and is the closure of .
For any complex number , dene
R(t) =
V(−n)=2
Kn() if t∈ +;
0 if t =∈ +;
where Kn() is given by the formula
Kn() =
(n−1)=2 (2+−n
2 ) ( 1−
2 ) ()
(2+−p
2 ) (
p−
2 )
:
The function R(t) was introduced by Nozaki [3, p.72]. It is well known that R(t) is an ordinary function if Re()¿n and is a distribution of if Re()¡n. Let suppR(t) denote the support of R(t) and suppose that suppR(t)⊂ +.
Proof. Since S(t) and R(t) satisfy the Euler equation
then they are homogeneous distribution of order −n by Donoghue [1, pp. 154,155] that proved that every homogeneous distribution is a tempered distribution.
Lemma 2.2 (The convolution of tempered distribution). The convolution S(t)∗R(t) exists and is
a tempered distribution.
where L is the partial dierential operator of Diamond type and is dened by
where = Ppi=1 @2 @t2
i
−Ppj=+pq@2 @t2
j
; =Ppi=1 @2 @t2
i
; p+q=n. Actually ♦= andet♦k is a tempered
distribution of order 4k.
Proof. For k= 1, we have
het♦; ’(t)i=h;♦et’(t)i;
where ’(t)∈S the Schwartz space. By computing directly we obtain
♦et’(t) = etM’(t); (3.3)
where M is the partial dierential operator of the form (3.2) whose the coecients of the third term, the fourth term, the sixth term and the eighth term of the right-hand side of Eq. (3.2) have opposite signs.
Thus h;♦et’(t)i=h;etM’(t)i=M’(0).
By the properties of and its partial derivatives with the linear dierential operator M, we obtain
M’(0) =hL; ’i where L dened by Eq. (3.2). It follows that et♦=L. Now
(et♦)∗(et♦)∗ · · · ∗(et♦)
| {z }
k-times
= (L)∗(L)∗ · · · ∗(L)
| {z }
k-times
;
we have et(∗ ♦k) =∗(Lk). Thus et♦k=Lk. It follows that, for any k, we obtain Eq. (3.1).
Since has a compact support, hence by Schwartz [4], and Lk are tempered distributions and Lk has order 4k. It follows that et♦k is a tempered distribution of order 4k.
Lemma 3.2 (Boundedness property). |het♦k; ’(t)i|6K where K is a constant and ’∈S.
Proof.
het♦k; ’(t)i=h♦k;et’(t)i
=h♦k−1;♦et’(t)i
=h♦k−1;etM’(t)i;
where M is dened by Eq. (3.3). By keeping on operate ♦k−1 times we obtain
het♦k; ’ti=h;etMk’ti
=Mk’(0):
Since ’(0) is bounded and also Mk’(0) is bounded. It follows that |het♦k; ’(t)|=|Mk ’(0)|6K.
4. The application of et♦k
Given u(t) is an distribution and by Lemma 3.1, we have (et♦k)∗u(t) = (Lk)∗u(t) =Lku(t);
Theorem 4.1. Given the linear partial dierential equation of the form
(et♦k)∗u(t) =Lku(t) =: (4.1)
Then u(t) = et(−1)kS
2k(t)∗R2k(t) is an elementary solution of (4:1) or The Diamond Kernel of
Marcel Riesz of (4:1) where S2k(t) and R2k(t) are dened by (2:1) and (2:2) respectively with = 2k.
Proof. By Kananthai [2, Lemma 2.4], (−1)kS
2k(t) is an elementary solution of the Laplace operator k iterated k-times and also by Trione [5], R
2k(t) is an elementary solution of the ultra-hyperbolic
operator k iterated k-times (that is k(−1)kS
2k(t) = and kR2k(t) =).
Now ♦k= kk, consider et( kk)∗R
2k(t) = By Lemma 2.2 with = 2k, (−1)kS2k(t)∗R2k(t)
exists and is a tempered distribution.
Convolving both sides of the above equation by et[(−1)kS
2k(t)∗R2k(t)] we obtain
et[k(−1)kS2k(t)∗ kR2k(t)∗u(t)] = [et(−1)kS2k(t)∗R2k(t)]∗
(et)∗u(t) = et(−1)kS2k(t)∗R2k(t):
It follows that u(t) = et(−1)kS
2k(t)∗R2k(t).
Theorem 4.2. Given the convolution equation
(et♦k)∗u(t) = et m
X
r=0
cr♦r; (4.2)
then the type of solutions of (4:2) depend on the relationship between k; m and are as the following cases.
(1) If m¡k and m= 0 then (4:2) has the solution u(t) = et[c
0(−1)kS2k(t)∗R2k(t)] and u(t) is an
ordinary function for 2k¿n with any and is a tempered distribution for 2k¡nand for some
= (1; 2; : : : ; n) with i¡0 (i= 1;2; : : : ; n).
(2) If 0¡m¡k; then the solution of (4:2) is
u(t) = et
" m X
r=1
cr(−1)k−rS2k−2r(t)∗R2k−2r(t) #
which is an ordinary function for 2k −2r¿n with any and is a tempered distribution if
2k−2r¡n for some with i¡0 (i= 1;2; : : : ; n).
(3) If m¿k and for any and suppose that k6m6M; then (4:2) has u(t) = etPM
r=kcr♦r−k as
a solution which is the singular distribution.
Proof. (1) For m¡k and m= 0, then (4.2) becomes (et♦k)∗u(t) = etC
0=C0et=C0. By
Theorem 4.1 we obtain
u(t) =C0et((−1)kS2k(t)∗R2k(t)):
Now, by (2.1) and (2.2), S2k(t) and R2k(t) are ordinary functions respectively for 2k¿n. It follows
functions except at the origin and by Lemma 2.1 S2k(t) and R2k(t) are tempered distributions and
by Lemma 2.2, (−1)kS
2k(t)∗R2k(t) exists and is a tempered distribution.
Now, for some = (1; 2; : : : ; n) withi¡0 (i= 1;2; : : : ; n) we have et is a slow growth function
and also its partial derivative is a slow growth. It follows that et[(−1)kS
2k(t)∗R2k(t)] is also a
tempered distribution.
(2) For 0¡m¡k, we have
(et♦k)∗u(t) = et[c
1♦+c2♦2+· · ·+cm♦m]:
Convolving both sides by et[(−1)kS
2k(t)∗R2k(t)], we obtain
u(t) = et[c
1♦((−1)kS2k(t)∗R2k(t)) +c2♦2((−1)kS2k(t)∗R2k(t))
+· · ·+cm♦k((−1)kS2k(t)∗R2k(t))]
= et[c1(−1)k−1S2k−2(t)∗R2k−2(t) +c2(−1)
k−2S
2k−4(t)∗R2k−4(t)
+· · ·+cm(−1)k−mS
2k−2m(t)∗R2k−2m(t)]
= et
" m X
r=1
(−1)k−rS
2k−2r(t)∗R2k−2r(t) #
by Theorem 4.1 and by Kananthai [2, Theorem 3.2] for r¡k. Similarly, as in the case (1) et
[Pmr=1(−1)k−rS
2k−2r(t)∗R2k−2r(t)] is the ordinary function if 2k −2r¿n and for any , and is a
tempered distribution if 2k−2r¡n and for some with i¡0 (i= 1;2; : : : ; n).
(3) For m¿k and for any , suppose that k6m6M we have
(et♦k)∗u(t) =cket♦k+ck
+1et♦k+1+· · ·+cMet♦k:
Convoluing both sides by et((−1)kS
2k(t)∗R2k(t)) and by Kananthai [2, Theorem 3.2] fork6m6M,
we obtain
u(t) =cket+ck+1et♦+ck+2et♦2+· · ·+cMet♦M−k
oru(t) = etPM
r=kcr♦r−k. Since et♦r−k=Lr−k fork6r6M andLis dened by (3.2). ThusLr−k
is a singular distribution. It follows that u(t) is a singular distribution. That completes the proof.
Acknowledgements
References
[1] W.F. Donoghue, Distribution and Fourier Transform, Academic Press, 1969.
[2] A. Kananthai, On the solution of the n-dimensional Diamond operator, Applied Mathematics and Computation, Elsevier, 88 (1997), 27 – 37.
[3] Y. Nozaki, On the Riemann–Liouville integral of ultra-hyperbolic type Kodai Mathematical Seminar Reports 6(2) (1964) 69 – 87.
[4] L. Schwartz, Theorie des Distribution, Vols. 1 and 2 Actualite’s, Scientiques et Industrial, Hermann, Paris, 1957, 1959.
