Revista de la Uni6n M atem ática Argentina V o lum en 3 4 , 1 9 8 8 .

INVERSION Q F ULTRAHYPERBOLIC BESSEL OPERATORS

S U SA N A E L E N A T R IO N E

69

AB S T RAC T . Let G� = Ga ( P ± io , m , n) be the c ausal ( anticausal ) d i s tribution de fined by

l( a -n) G ( P ± . ) ' H ( ) ( P ± l' o ) 2 2 a o l , m , n = a m , n

1

Kn _ a [m ( p ± io ) '2] ,

-2-where m i s a po s i t ive real numb er , a E e , K� des i gnates the modified B e s s e l func tion of the third kind and Ha (m , n) i s the cons tant defined by

Ha (m , n) ± . 'Ir l'2 Q

e . e • 'Ir 1'2 a

n ( 2 '1r) "2

1 - � 2 2

1 (n - a) (m2) 2" ""2

r (�) 2

The d i s tr ibutions G 2k ( P ± io , m , n) , wher e n = inte ger � 2 and k = 1 , 2 , . . . , are e l ementary caus al ( ant icaus al ) so lut ions o f t h e ul trahyperbol ic Kl ein - Gordon operator , i tera ted k - t imes :

+ • • • +

Let Baf b e the ul trahyperbol ic Be s s e l operator de fined by the formula

f e S .

70

Our prob1 em eóns ls ts in the obta inment · of an operator

To. = {B o,) - 1 sueh that if

then

In this. No te we prove ( Theorem I I I . 1 , formula ( I 1 I , 7) ) that

for a 1 1 a E C . We obs erve tha t the dis tr ibution Go. ( P ± io , m , n) i s a causal ( ant ieausa 1 ) ana10gue of the kernel due to N . Arons za j n - K . T . Smi th and A . P . Calderón (e f . [ 1 ] and [ 2 ] , resp ee t ive1y) . The par t icular radial case o f our prob 1em was s o l ved by No gin , for a t= 1 , 2 ; 3 , . . . ( e f . [3 ] ) .

l . I N T R O D U C T I O N

L e t x = ( x 1 , x Z " " ' xn) be a po int o f the n -d imens ional Eue 1 i ­

dean spaee Rn • Cons ider a non -degenerate quadratie form i n n var iab le s o f the form

Z P = P (x) = x l + Z - x p +q .

where n • A p+q . The d i s tr ibut ion ( P ± 1 0 ) i s defined by

( P ± io ) ;'

wher e E > O , I x l Z

1 im { P ± i E j x l Z } A E�O

Z Z . X l + . . • + xn ' A e C .

( I , 1 )

( I , 2 )

The d i s tr ibut ions ( P ± iO ) A ar e ana1ytie in A everywhere exe ept

at A = - I - k , k = 0 , 1 , . . . ; where they have s imp l e po 1es

( e f . [4 ] , p . 2 7 S ) .

7 1

The d i s tr ibu t ions ( m2 + Q ± i O ) A a r e de f ined i n an ana l o gue

manner a s the d i s tr ib u t i ons ( P ± i o ) A . "L e t u s put ( c f . [4 ] ,

. p . 2 8 9 )

( m2 + Q ± i O ) A = l im ( m2 + Q e:-+O

' 2 A ± i e: l y l ) ( I , 3 )

whe re m i s a po s i t ive r e a l numb e r , A E e , e: i s an arb i tr ary p o s i t ive numb er . I n the formu l a ( 1 , 3 ) we have wr i t ten

Q Q ( y)

p + q n ,

and

2 2 Y I + • • • + yp

2 - y p +q ( I , 4 )

2 • • • + y n

I t i s us e fu l to s ta t e an e qu iva l en t de f in i t i o n o f the d i s tr i ­butions ( m2 + Q ' ± i o ) A. .

I n th i s defin i t i o n app ear the d i s tr ibut ions

( m 2 + Q ) A i f 2

+ Q ;;;. O 2 + Q) :

m ( m 2 O i f m + Q < O

( I , 5 )

O i f 2 + Q > O 2

+ Q) � m

( m 2 Q) A 2 ( - m i f m + Q .;;;; O ( I , 6 )

We can prove , w ':' �:. _ _ . " ,.l � rficul ty , tha t the fo l l ow i n g formula i s va l id ( c f . [7 ] , p . 5 6 6 )

From thi s formula we c o nc lude ílmmed iat e l y that

2 . A 2 ( m + Q + i o ) = ( m + Q

when A = k po s i t ive inte ger .

• ) A ( 2 - 10 = m +

( 1 , 7 )

( 1 , 8 )

We o b s e rve tha t ( m2 + Q ± i o ) A are ent i r e d i s t r ibut ional func ­t ions o f A .

L e t Ga ( P ± i o , m , n) b e the caus al ( an t i c aus a l ) d i s tr ibu t i on de ­f ined by

72

io ) ] , ( I , 9 )

where m is a pos it ive real number , a E e , K� d e s ignat e s the wel l - known modif ied B � s s e l funct ion of the third kind ( cf . [ 5 ] ,

p . 7 8 , formulae ( 6 ) and ( 7 ) ) :

K ( z ) v

00 (7) 2m+v

l v ( z ) = L m= O m ! r (m+v + l )

and Ha (m , n ) i s the constant def ined by

+ 1T . . 1T 1 - � len - a) - z Q1 e

1z a H (m , n) e 2 2 (m2 ) 2 2

a n ( 21T ) Z r (�) 2

( 1 , 1 0 )

(i , 1 1 )

( 1 , 1 2 )

The fol lowing formula i s val id (cf . [ 6 ] , p . 3 5 , formula ( I I , 1 . 8 ) ) :

n ( 21T ) Z

. a a 11T - - -e 2 (m2 + Q ± io ) 2

Here A deno t e s the Four ier transform o f a distr ibut ion .

( I , 1 3 )

We obs erve that the r ight - hand member of ( 1 , 1 3 ) is an ent ire di str ibut ion ' o f a ; t her efo r e Ga i s a l so an ent ir e d i s tr ibut io ­nal funct ion of a .

1 1 . T H E P RO P E RT I E S O F G ( P ± io , m , n ) a

The Be s s e l pot ent ial o f order a (a b e ing any compl ex number ) of a t emperat e d i s tr ibut ion f , denoted by Jaf is def ined by

a (JQf ) A = ( 1 + 41T 2 J X J 2 )

-Z fA ( I l , l )

73

wa s int r o due ed by N . Arons z a j n - K . T . Sm i t h and A . P . C a l der6n ( e í . [ 1 ] and [ 2 ] , r e s p e e t iv e l y ) .

A . P . C a l d er6n pro v e s in [ 2 ] , Theo r em 1 , t ha t

wher e

Ga (x ) n - a - l

y ( a ) e - I x l J: e - I x l t ( t + �)---2---

dt

Re a < n + 1 . , and

[y ( a ) ] - 1 n - l

( 2 ) -2- r (a ) r (n - a2

+ 1 ) . 1T . 2"

( l l , 2 )

( l l , 3 )

( l l , 4 )

We s t a r t by o b s erv ing t hat t h e d i s t r ibut ional fun e t ion Ga ( P ± io , m , n ) ( e f . fo rmu l a ( 1 , 9 ) ) is a11 ( c au s a l , ant i e au s a l ) an a l o gu e o í t h e kernel d e f ined by t h e fo rmu l a ( 1 1 , 3 ) .

The d i s t r i but ions Ga = Ga ( P ± io , m , n ) s ha r e many proper t i e s w i t h t h e B e s s e l kern e l o f wh i e h t h ey ar e ( c au s a l , ant ieaus a l ) ana l o gu e s .

The ío l l ow ing t h eo r ems ho l d :

THEO REM 1 1 . 1 . L e t us put a E C , k = 0 , 1 , . . . , then

n A 2" A A { Ga * G - 2 k } . = ( 2 1T ) { Ga } . { G - 2 k } .

Here * d e s ignat e s , a s usual , t h e eonvo l ut ion .

THEOREM 1 1 . 2 . The fo Z Z owing fopmu Za i s v a Z i d

when a E C , k = 0 , 1 , 2 , . . , .

Mo pe gen epa Hy,- t he fo H o wing fo pmu Z a e a p e v a Z i d fo p a H a , a E C ,

G ( P ± i o , m , n ) o

( l l , S )

( l l , 6 )

( l l , 7 )

74

( I r" 8 )

and

( I r , 9 )

Let us define the n - d imens ional ul trahyperbo l ic Kl e in - Gordon operator , it erated 2 - t imes :

K2 = � a 2

+ + a2 a2 a 2 m2 f 2

-2 . . . ax� - -2- - - -2- -

aX 1 aXp+ 1 aXp + q

= { O - m2

} 2 ( I r , 1 0 )

where p+q n , m E R+ , 2 = 1 , 2 , . . .

From the pre ced ing r e sul t s we deduce the expl ic it expr e s s ion of a family of el ementary ( causal , ant icausal ) so lut ion of t he ultrahyperbo l ic Kl e in - Gordon operator , i t erated k - t imes .

In fact , the fo l lowing propo s it ion i s val id o

THEOREM 1 1 . 3 . The di s tribut iona l fun c t i o n s G 2 k ( P ± io , m , n) where n = i n t eger > 2 an d k = 1 , 2 , . . . , are e l im e n t ary causa l

(an t i c a u s a l ) so l u t i o n s of t h e u l tra hyp erbo l i c Kl e in-Go rdon

operator, i t e r a t e d k- times :

Kk { G 2 k ( P ± io , m , n ) } = 15 . ( I r , 1 1 )

The proofs o f the formulae ( I r , S ) , ( I r , 6 ) , ( I r , 7 ) , ( I r , 8 ) , ( I r , 9 ) and ( l l , 1 1 ) appear in [ 6 ] .

I t may be o b s erved that the el ementary so lut ions G Z k ( P ± io , m , n ) have t he same form for all n > 2 . This do e s not happen for o t her e l ementary so lut ion , who s e form depends e s s ent ial ly on the par ity o f n ( cf . [ 7 ] , p . S 8 0 and [ 8 ] , p . 4 0 3 ) .

We obs erve that the part icular case o f Theor em 1 1 . 3 corre spon ­d ing to n= 4 , k= t , q= 1 i s espec ially important .

The corre spond ing el ement ary so lut ions can be writ t en

mi -41fZ

75

K 1 lm (P + io ) 1 / Z ]

( P + io ) 1I Z

K l ( P " ) 1 / Z ] mi . 1 m - 10. G Z ( P - io , m , � ) = �

1" 0 )1 7 Z 41f ( P -

( I I , 1 Z )

( I I , 1 3 )

The fo�mul a ( I I , 1 Z ) i s a us eful expres s ion o f the famous "ma ­g ic funct ion" or " causal propagator" o f Feynman .

For this reason we have dec ided to call "causal " ( "ant icausal " ) t he distr ibut ions G

a. ( P ± i o , m , n ) .

f l l . T H E I N V E R S E U L T R A H Y P E R B O L I C B E S S E L K E R N E L

L e t Ba.f be the ultr�hyperbol ic Bé s s el o perator def ined b y t he formula

( I I I , 1 )

f E S . Our obj �ct ive i s the atta inment o f Ta. = ' (Ba. ) - 1 such that if � = Ba.f , then Ta.� = f .

, a. - 1 We not e that the inver s e ultrahyperbo l ic Be s s el kernel (B )

i s , forma l l y , by v irtue o f ( 1 , 1 3 ) and ( 1 1 , 1 0 ) , a fract ional power of the different ial Kl e in - Gordon operator :

a. (Ba. ) - 1 = ( O _ mZ ) 2

• ( I II , Z )

Ther efore , here we ar e s e eking an expl ic it expr e s s ion for

( Ba. ) - 1 . The fo l l owing f in it ion_,

t hen

for all comp l ex a. .

theorem expre s s e s

Ba. = Ga.

(Ba. ) - 1 = ( G ) - 1 ..

a.

that if we put , by de -

( I I I , 3 )

' G - a ( I I I , 4 )

76

Now we shal l s tate our ma in theorem .

THEOREM I I T . 1 . If

where BOLf i s de fin e d by ( 1 1 1 , 1 ) , f E S ; t hen

where

OL E C . G

- OL

( I I I , 5 )

( I I I , 6 )

( I I I , 7 )

Here GOL i s defin e d by ( 1 , 9 ) and OL b e ing any aomp Z ex number .

Pro o f. From the def in itory formula ( 1 I T , 1 ) we have

( I I I , 8 )

wher e G i s g iven by ( 1 , 9 ) , OL E C and f E S . OL Then , in v i ew o f ( 1 1 , 9 ) and ( 1 1 , 7 ) , we obt a in

G * ( G * f ) ( G _ OL * G ) * f G *. f - OL OL ct - OL+ct Go * f = 15 * f f ( II I , 9 )

Therefore G = ( Bct ) - 1

- OL ( I I I , 1 0 )

Formula ( 1 1 1 , 1 0 ) i s the de s ired r e sult and t h i s f in i shed the proo f of Theor em 1 1 1 . 1 •

7 7

R E F E R EN C E S

[ 1 ] N . ARO N S Z AJ N an d K . T . S M I T H , T h eo � y 0 6 B e� � ef p o t e ntiaf� , P a r t 1 , Ann . I n s t . Fo u r i e r 1 1 , 3 8 5 - 4 7 5 , 1 9 6 1 .

[ 2 ] A . P . C A L D E RO N , S i n g uf a� úí.t e g �af� ( n o t e s o n a c o u r s e t a u g h t a t t h e Ma s s a c h u s e t t s I n s t i t u t e o f T e c hn o l o g y ) , 1 9 5 9 a n d L e b e� g u e � p a e e� 0 6 di 6 6 e� e ntiaf 6 un etio n � a n d di� t�i b utio n� , S ym p o s . P u r e Ma t h . , 4 , 3 3 - 4 9 , 1 9 6 1 .

[ 3 ] V . A . NO G I N , I n v e�� io n 0 6 B e � � ef Po t e ntiaf� , P l e n um P u b l i ­s h in g C o r p o r a t i o n , 9 9 7 - 1 0 0 0 , 1 9 8 3 .

[ 4 ] I . M . G E L FAND a n d G . E . S H IL O V , G e n e�afiz e d F u n etio n� , Vo l . I , A c a d em i c P r e s s . N ew Y o r k , 1 9 6 4 .

[ 5 ] G . N . WAT S O N , A t� e ati� e o n t h e t h eo � y 0 6 B e� � ef 6 u n etio n� , S e c o n d E d i t i o n , C amb r i d g e , Un i v e r s i t y P r e s s , 1 9 4 4 . _

[ 6 ] S . E . T R I O N E , -Vi�t�i b uti o n af P�o d u et � , S e r i e I I , C u r s o s d e

Ma t e m á t i c a , N ° 3 , I n s t i t u t o A r g en t i n o d e Ma t emá t i c a , I AM­C O N I C E T , 1 9 8 0 .

[ 7 ] D . W . B R E S T E R S , O n di� t�i b utio n � e o n n e et e d wit h q ua d� at i e 6 0 �m� , S . I . A . M . , J . Ap p l . Ma t h . , Vo l . 1 6 , 5 6 3 - 5 8 1 , 1 9 6 8 .

[ 8 ] J . J . B O WMAN a n d J . D . HA�RI S , G� e e n ' � di� t�i b utio n� a n d t h e C a u c h y p�o bf em 6 0 � t h e it e�at e d Kf e i n - G o � do n o p e�at o � , J . Ma t h . P h y s . , v6 l . 3 , 39 6 - 4 0 4 , 1 9 6 4 .

D e p a r t am en t o d e M a t em á t i c a , F a c u l t a d d e C i en c i a s E xa c t a s y N a t u r a l e s Un iv e r s i d a d d e B u e n o s A i r e s .

I n s t i t u t o A r g e n t i n o d e Ma t emát i c a C o n s ej o N a c i o n a l d e I n v e s t i g a c io n e s C i en t 1 f i c a s y T é c n i c a s B u eno s A i r e s , A r g en t in a .

Rec ib ido en febrero de 1 9 8 9 .