69 R e vista de la Uni6n M atem ática Argentina V o lum e n 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 A B S T RAC T . L e t G� = G a ( P ± io , m , n) b e the c au s a l ( an t i causal ) d i s tri bu t i o n de fined by 1 l( a - n) '2 2 2 . G a ( P ± o l , m , n ) = H a ( m , n ) ( P ± l' o ) K n _ a [m ( p ± io ) ] , -' 2 whe r e m i s a po s i t ive r e a l numb er , a E e , K � d e s i gna t e s the mo d i f i e d B e s s e l func t i o n of the third kind and H a (m , n) i s the cons tant d e f i ned by 'Ir 1 n-a 'Ir Q ± l. '2 1 -� ( 1'2 a ) 2 . 2 m ( ) 2" ""2 e 2 e H a ( m , n) n • ( 2 '1r) "2 r ( �) 2 The d i s tr ibu t i on s G 2 k ( P ± i o , m , n) , wher e n = inte ger � 2 and k = 1 , 2, . . , are e l ementary c aus a l ( an t i c aus a l ) s o lut ions o f t h e ul trahyp erb o l ic Kl e i n - Gordon operator , i t era ted k - t ime s : . + • • • + L e t B a f b e the u l tr ahyp erb o l i c B e s s e l operator de fined by the fo rmu l a f e S. 70 Our To. prob 1 em eóns l s ts i n the o b t a inmen t · o f an ope rator = {B o,) - 1 su e h tha t if then In th is. No te we prove ( Theor em I I I . 1 , formu l a ( I 1 I , 7) ) that fo r a 1 1 a E C. We o b s erve tha t the d i s tr ibu t i o n G o. ( P ± i o , m , n) i s a cau s a l ( an t ie aus a 1 ) ana 1 0 gu e o f t h e kerne l due to N . Ar ons z a j n - K . T . Smi th and A . P . Ca lderón ( e f . [ 1 ] and [ 2 ] , r e sp e e t ive1y) . The p ar t icu lar rad i a l c a s e o f our prob 1 em was s o l ved by No gin , for a t= 1 , 2 ; 3 , . . . ( e f . [3 ] ) . l. I N TRO DUCT I O N L e t x = ( x 1 , x Z " " ' x ) b e a po int o f the n - d imen s iona l Eue 1 i ­ n dean spae e R n • Cons id er a non - d e gener a t e quadr a t ie form i n n var iab l e s o f the form P = P ( x) = x Zl + - p + q . The d i s tr ibut ion ( P whe r e n (P i o ) ;' ± Xl The d i s tr ibu t ions ( P = - I - k , ( e f . [4 ] , p . 2 7 S ) . E�O Z + wher e E > O , I x l Z at A 1 im { P k = . . • ± ± 10) • A e Z p +q . (I , 1 ) A i s d e f in e d by iE j x l Z }A + xZ . ' n x ( I , 2) C. i O ) A ar e ana 1 y t i e in A everywh e re exe ept 0 , 1 , . . . ; where they have s imp l e po 1 e s ± 71 Q ± i O ) A a r e d e f in e d i n an ana l o gu e " manner a s the d i s tr ib u t i o n s ( P ± i o ) A . L e t u s p u t ( c f . [4 ] , The d i s tr ibu t i o n s ( m 2 + . p . 28 9) (m 2 + Q ± = iO) A l im ( m e:-+O 2 + ' 2 A Q ± i e: l y l ) ( I , 3) whe r e m i s a p o s i t iv e r e a l numb e r , A E e , e: i s a n arb i t r ary p o s i t iv e numb er . I n the formu l a ( 1 , 3 ) we have wr i t t e n p + Q Q ( y) q n , 2 + YI ••• y + 2 p - y and ••• + 2 p q + ( I , 4) 2 yn I t i s us e fu l t o s t a t e an e qu iva l e n t de f in i t i o n o f the d i s tr i ­ b u t i o n s ( m2 + Q ' ± i o ) A. . I n th i s d e f i n i t i o n app e a r the d i s tr ibu t i o n s (m (m 2 2 + + Q) � 2 Q) A if m O if m O if m if m (m Q) : ( -m We can p r o v e , w ':' �:. + 2 Q) A _ _. " ,.l � 2 2 2 2 + Q ;;;. O + Q < O + Q + Q .;;;; O > ( I , 5) O ( I , 6) r f i c u l ty , tha t the fo l l ow i n g f o r mu l a i s va l i d ( c f . [ 7 ] , p . 5 6 6 ) ( 1 , 7) F r o m t h i s formu l a we c o nc lude ílmmed i a t e l y t h a t (m whe n 2 A = k + Q + . io) Let Ga ( P f ined b y ± = (m 2 + Q - 10 ) • A = (m 2 + ( 1 , 8) p o s i t ive i n t e g e r . We o b s e rve tha t ( m t ions o f A . A 2 + Q ± i o ) A a r e e n t i r e d i s t r ib u t i o n a l func ­ i o , m , n) b e the caus a l ( an t i c au s a l ) d i s tr ibu t i o n d e ­ 72 io ) ] , (I , 9) wher e m i s a po s it iv e r e a l number , a E e , K� d e s ignat e s the we l l - known mod i f i ed B � s s e l fun c t ion of the t h ird k ind ( c f . [ 5 ] , p . 7 8 , formu l a e ( 6 ) and ( 7 ) ) : K v (z) lv ( z ) = ( 1 , 1 0) 00 L m= O m ! (7) r 2m+v (i , 1 1 ) (m+v + l ) and Ha (m , n ) i s the con s t ant d e f ined by H (m , n) a . 1T len - a) +- 1T Q 1. 1 -� 1z a z e e 2 2 (m 2 ) 2 2 n Z 1T 2 r (�) ) ( 2 (1, 1 2) The fol l owing formu l a i s val id ( c f . [ 6 ] , p . 3 5 , formu l a ( I I , 1 . 8 ) ) : n e Z ( 2 1T ) . a 11T - a -- 2 ( m 2 + Q ± io ) 2 (I , 1 3) Here A d eno t e s the Four ier t r an s fo rm o f a d i s t r ibut ion . We o b s erv e t hat the r ight - hand memb er o f ( 1 , 1 3 ) is an ent i r e d i s t r ibut ion ' o f a ; t her e fo r e Ga i s a l so a n ent ir e d i s t r ibut io ­ nal funct ion o f a . 1 1 . T H E P RO P E RT I E S O F G (P ± a io , m , n ) The B e s s e l pot ent ial o f o r der a ( a b e ing any c ompl ex numb er ) o f a t emp erat e d i s t r ibut ion f , deno ted by J a f i s def ined by a -Z 2 2 Q A J X J ) 1T 4 + (Il , l ) fA (J f ) = ( 1 73 wa s int r o du e ed by N . Ar o n s z a j n - K . T . Sm i t h and A . P . C a l d e r 6 n ( e í . [1 ] and [ 2 ] , r e s p e e t iv e l y ) . A . P . C a l d er 6 n pr o v e s in [ 2 ] , Theo r em 1 , t ha t (ll , 2) wh er e y (a) G (x ) a J: x e- I l e- I xl t � (t + ) n-a - l -----2 dt (ll , 3) Re a < n + 1 . , and [y ( a ) ] -1 ( 2 1T ) n- l -2- r a ( ) 2" r . ( n-a+1 ) . 2 (ll,4) We s t a r t by o b s erv ing t ha t t h e d i s t r ibut i o n a l fun e t ion G a ( P ± io , m , n ) (ef . 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 k e r n e l d e f in e d by t h e fo rmu l a ( 1 1 , 3 ) . The d i s t r i bu t i o n s G a = G a ( P ± io , m , n ) s ha r e many p r o p er t i e s w i t h t h e B e s s e l k e rn e l o f wh i e h t h ey ar e ( c au s a l , ant i e au s a l ) ana l o gu e s . T h e ío l l o w ing t h eo r ems ho l d : THEO REM 1 1 . 1 . L e t us p u t a E {G * G a - 2k } A .= C, k ( 2 1T ) n 2" = {G } a 0, 1 , A . . . , then . {G - 2k} A . ( l l , S) H e r e * d e s i g n a t e s , a s u s u a l , t h e e o nvo l ut i o n . THE O REM 1 1 . 2 . T h e fo Z Z o w ing fopmu Z a i s v a Z i d ( l l , 6) when a E C, k = 0, 1 ,2, . . , . Mo p e g e n e p a H y,- t h e 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 ± io ,m,n) o (ll , 7) 74 ( I r" 8 ) and ( I r , 9) Let us define t he n - d imens ional ul t r ahyp erbo l ic Kl e in - Gordon operat o r , it erated 2 - t imes : 2 2 2 2 a + . . . + a - -a 2- 2 2 a K = m 2 aX 2 aX ax aX 2 1 p+ 1 p+q � 2 2 = {O - m } where p + q f � ( I r , 1 0) n , m E R+ , 2 = 1,2, . . . From t h e pr e c ed ing r e sul t s we d educ e the expl i c it expr e s s ion of a fam i l y of el emen t ary ( c aus a l , ant icausa l ) so lut ion of t he u l t r ahyperbo l ic Kl e in - Go rdon operator , i t er a t e d k - t ime s . I n fact , t h e fo l l owing propo s it ion i s val id o THEOREM 1 1 . 3 . The di s t r i b ut i o n a l fun c t i o n s G 2 k ( P ± io , m , n ) w h e r e n = i n t e g e r > 2 an d k = 1 , 2 , . . . , a r e e l im e n t ary c a u s a l (an t i c a u s a l ) s o l u t i o n s o f t h e u l tra hyp e r b o l i c Kl e i n - Go r don o p e r a t o r, i t e r a t e d k- t i m e s : K k { G 2 k ( P ± io , m , n ) } = 15 . (Ir , 1 1 ) The pro o f s o f t he formulae ( I r , S ) , ( I r , 6 ) , ( I r , 7 ) , ( I r , 8 ) , ( I r , 9 ) and ( l l , 1 1 ) app ear in [ 6 ] . I t may b e o b s erved t hat the el ement ary so lut ion s G Z k ( P ± io , m , n ) have t he sam e form fo r a l l n > 2 . T h i s do e s no t happen for o t her e l ement ary so lut ion , who s e fo rm depends e s s ent ial l y on t he par ity o f n ( c f . [ 7 ] , p . S 8 0 and [ 8 ] , p. 4 0 3 ) . We o b s erve that t he part icular c a s e o f Theor em 1 1 . 3 co r r e spon ­ d in g t o n = 4 , k = t , q = 1 i s espec ia l l y impo r t ant . The c o r r e spond ing el ement ary so lut ion s c an b e wr it t en 75 K 1 lm ( P + io ) 1 / Z ] m i 1I 41f Z ( P + io ) Z i G Z ( P - io , m , � ) = m � 41f " ) 1 /Z] 10. 1 7Z ( P - 1" 0 ) K 1 lm ( P . - (II , 1 Z) (I I , 1 3) The fo�mul a ( I I , 1 Z ) i s a us eful expr e s s ion o f the famo u s "ma ­ g ic funct ion" or " c au s a l propaga t o r " o f Feynman . Fo r t h i s r e a s on we have d e c ided t o c a l l " c ausa l " ( "ant i c au sal " ) t he d i s t r ibut ions G ( P ± i o , m , n ) . a. f l l . THE I NVERSE ULTRAH Y P E RBOL I C B E S S E L KERN EL L e t Ba. f be t he u l t r �hyperbo l ic B é s s el o perator def ined b y t he fo rmula (III , 1 ) f E S. Our obj �ct iv e i s t h e a t t a inment o f T a. = ' ( B a. ) - 1 such t hat if � = Ba.f , t hen T a.� = f . , We not e t hat t he inver s e u l t r ahyperbo l ic Be s s el kernel (B a. ) - 1 i s , forma l l y , by v irtue o f ( 1 , 1 3 ) and ( 1 1 , 1 0 ) , a fract ional power o f t he d ifferent i a l Kl e in - Go rdon operat o r : ( B a. ) - 1 = ( O _ a. mZ ) 2 • ( I II , Z ) Ther efo r e , her e we ar e s e ek ing an expl ic it expr e s s ion for ( B a. ) - 1 . The fo l l owing t h e o r em expr e s s e s t hat i f we put , by d e f in i t ion_, B a. = G (I I I , 3) a. t hen ( Ba. ) fo r a l l comp l ex a. . -1 = (G ) a. -1 .. 'G -a (III , 4) 76 Now we s ha l l s t a t e our ma in t h e o r em . THEOREM I I T . 1 . If ( I I I , 5) w h e r e B OL f i s de fin e d by ( 1 1 1 , 1 ) , f E S ; t h e n ( I I I , 6) where G OL E C . (III , 7) - OL Here G OL i s defin e d by ( 1 , 9 ) a n d OL b e ing any a o mp Z e x n um b e r . Pro o f. From t h e d e f in i t o ry formu l a ( 1 I T , 1 ) we hav e ( I I I , 8) wher e G OL i s g iven by ( 1 , 9 ) , OL E C and f E S . Then , in v i ew o f ( 1 1 , 9 ) and ( 1 1 , 7 ) , we o bt a in G - OL * ( G OL * f ) (G_ * G ) * f OL ct Go * f Ther efore G - OL = = 15 * f ( B ct ) Formu l a ( 1 1 1 , 1 0 ) i s t h e de s i r e d t h e proo f of Theo r em 1 1 1 . 1 • -1 G - OL +ct *. f f ( II I , 9 ) ( I I I , 1 0) r e su l t and t h i s f in i s hed 77 REFEREN C E S [1] N . A RO N S Z A J N a n d K . T . S M I T H , T h e o � y 0 6 B e � � ef p o t e ntiaf� , P a r t 1 , A n n . 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 hu s e t t s I n s t i t u t e o f T e c hno 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 u n et i o n � a n d di� t�i b utio n � , S y m p o s . P u r e M a t h . , 4 , 3 3 - 4 9 , 1 9 6 1 . [3] V . 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