1.Itiswellknownthatanyproperholorphicmappingfromaunit卯一balltoaunit 犯 一 b a l l i s b i h o l o m o r p h i c i f 〃 ≧ 2 . F o r t h e n o n ‑ e q u i d i m e n s i o n a l c a s e , t h i s i s n o t t r u e ( [ 1 ] ) ． B u t a n y p r o p e r h o l o m o

Anoteonaproperholomorphicmappingfrom an"‑balltoa"‑ball(">").

DedicatedtoDr・EiichiSakaiforhis70th.birthday.

ChikaraWatanabe*)

1.Itiswellknownthatanyproperholorphicmappingfromaunit卯一balltoaunit 犯 一 b a l l i s b i h o l o m o r p h i c i f 〃 ≧ 2 . F o r t h e n o n ‑ e q u i d i m e n s i o n a l c a s e , t h i s i s n o t t r u e ( [ 1 ] ) ． B u t a n y p r o p e r h o l o m o r p h i c m a p p i n g f r o m a u n i t " ‑ b a l l t o a u n i t ( " + 1 ) ‑ b a l l w h i c h i s O f c l a s s C 2 o n i t s b o u n d a r y i s e s s e n t i a l l y l i n e a r i s o m e t r y ( [ 1 ] , T h e o r e m 4 ) i f

" ≧ 3 . I n t h i s n o t e , u s i n g t h e s a m e i d e a o f [ 1 ] , w e s h a l l s h o w t h a t a n y p r o p e r h o l o m o r ‑

p h i c m a p p i n g f r o m a u n i t " ‑ b a l l t o a u n i t だ − b a l l ( " > " ) w h i c h i s o f c l a s S C 2 o n i t s

b o i m d a r y i s l i n e a r i s o m e t r y i f i t h a s s o m e l i n e a r a l g e b r a i c c o n d i t i o n s .

2.LetB"beamitballofC",thatis,

B " = { Z E C " ; ￨ Z , 1 2 + … + ￨ z " ￨ 2 < 1 } .

Forapoint@zofB",aninvolutionjaisdefinedby

j ｡ ( z ) = a − = 1‑<z,"> @ ( = = g 2 ｡ ( z ) , w h e r e

語 長 ≧ α , ， = ､ / I 二 『 & ( z ) = 息 一 P 画 ( 屋 ）

P"(Z)=

a n d < z , C z > i s a n h e r m i t i a n i n n e r p r o d u c t , ￨ z l 2 = < z , z > .

I t i s e a s i l y s e e n t h a t の α ･ j a i s a n i d e n t i t y m a p P i n g a n d の α ( L z ) = 0 , t h e o r i g i n o f C " . Remarkl.Forpointszz,zofB",put"=(Lz,o')EB",2=(z,o')EBh.

Thenitholdsthat"(2)=(のα(z),o').

LetUbeadomainofC",/beaholomorphicmappingfromUintoB".

D e f i n i t i ･ n 2 . D A / ( " ) ( " 偽 ) = , 愚 磁 鵠 鈴 z ) i ] ル " " w h e r … U , "

＝(〃1,〃2,…,〃兎)andIII=j,+j2+…+3".

（磁鶚論

+ T z ]

Defmition3.DRf(")(て庫‑')=

爾 川 ￥

× , 星( 皇 ) … ( 堂 ) n i ‑ s : " f ､ … 甥 ‑ 鋤 ' 芽 ）

*)DepartmentofMathematics

12 ChikaraWATANABE

（ ; ) = ( 俺 二 等 w r a n d r = ( 画 曲 … , Z y @ ) , " = = ( " , , " 2 , … ル )

where

Remark4.Byaneasycalculation,wehave

( : ) w ( " ) ( r " "

（ , ) w ( " ) ( r 偽 ‑ 璽 , ")

た

りγ(")((r+")")=Z

〃＝I

Remark5.TheFrechetderivativesofaninvolution｡"sftisfy

［ , ≦ ぞ " 完 ] "' D ' ｡ ( z ) ( "

D々α(z)(zﾉ鹿)="!

Let/beaproperholomorphicmappingfromaunitballB"toaunitballBA(">") whichisofclassC2onB"・Wemayassume/(o)=o.;Wefixanorthonormalbasis { " , で ( 2 ) , … , z ･ ( n ) } o f C " . N o w s u p p o s e t h a t t h e r e e x i s t s a v e c t o r " E C 泥 一 { o } s u c h t h a t f o r anychoiceofanautomorphismdofBnandajofB"withjo/･の(o)=o,thefollowing

condition(*)issatisfied:

( * ) D 2 ( j o f o の ) ( z ' ) ( " 2 ) i s i n t h e s p a c e s p a n n e d b y { 妙 ｡ f ･ の ( " ) , D ( ' ｡ / ･ の ) ( " ) ( r ( 2 ) ) ,

･ ･ ･ , D ( ' ｡ / ･ の ) ( " ) ( r ( " ) ) } .

T h e n i t h o l d s t h a t / i s l i n e a r a n d i s o m e t r y , t h a t i s l / ( z ) ￨ = ￨ z l . W e s h a l l p r o v e t h i s . SincetherankofthelinearmappingD/(zd)isequalto"([1],page492),wecan chooseapoint4zEB"suchthattherankofDf(Lz)isequalto〃・Takeinvolutionsj ofB"andajofB"suchthatd(o)=",'(/("))=oandputg='｡/･の．ThenD29(zz) ( 2 ﾉ 2 ) i s i n t h e s p a c e s p a n n e d b y { g ( " ) , D g ( " ) ( r ( 2 ) ) , … , D g ( z z ) ( r ( M ) ) } . S i n c e " v e c t o r s { g ( " ) , D 9 ( " ) ( z ． ( 2 ) ) , … , D g ( " ) ( r ( " ) ) } a r e l i n e a r y i n d e p e n d e n t , t h e r a n k o f t h e ( " + 1 , " )

matrix

I 鮴≦ ^l

Then

isequalto〃・Letl≦j,≦た…<jiz+,≦た andputG=(9f,,…,9宛.,).

臓 : : ≦ ， ≦ " ）

det

F i x a c o m p l e x n u m b e r ﾉ l w i t h l ｣ ￨ = 1 . S i n c e g ( / I z ) s a t i s f i e s t h e c o n d i t i o n ( * ) , b y a n e a s y

c a l c u l a t i o n ,

（ ; 漁≦ , ≦ " ■ ） a

det

ThisholdsforalllECwithlAI=1andsincethelefthandsideoftheaboveisholomor‑

phicinlll<1,theaboveequalityholdsforalllECwithl｣￨≦1.Thereforewehave

、

〜 、

A n o t e o n a p r o p e r h o l o m o r p h i c m a p p i n g f r o m a n " ‑ b a l l t o a " ‑ b a l l ( " > " ) ¹³

G(")

嶬 仙 , … ‑ α 仇 1

limdet

A→O

D2G(A")("2)￨

（ 職≦ ' ≦ " o 1 a

det

Wenowfixanarbitrarynon‑zerovector77EC"andchooseaunitarytransformationV ofC"suchV'"isinthespacespannedby2ﾉ．Putj=.･VandC=G｡V.Sincethe

codition(*)holdsforj,

（ ; , ： ≦ , ≦ " 」 1

det

Since{V",Vr(i),2≦j≦"}isanorthonormalbasisofC",itholdsthat

（ 蝋≦ ｡ ≦ " l

det

F r o m t h i s e q u a l i t y a n d f r o m t h e f a c t t h a t t h e r a n k o f D g ( o ) i s e q u a l t o " , i t h o l d s t h a t D 2 9 ( o ) ( " 2 ) i s i n t h e s p a n n e d b y { D 9 ( o ) ( " ) , D g ( o ) ( r ( 2 ) ) , 2 ≦ j ≦ " } , t h a t i s D 2 9 ( o ) ( " 2 ) E D 9 ( o ) ( C " ) f o r a n y 7 7 E C " 3 T a k e a p o s i t i v e n u m b e r l ' < 1 s u c h t h a t f o r z E B " w i t h l z l

〜 〜

<",therankofDg(z)isequalto"andfixanyofthisz.Let'EAut(B"),jEAut

〜 〜 〜 〜

(B")beinvolutionssuchthatd(o)=zandthatjogoの(o)=o.Put"=jog｡j.Since

t h e c o n d i t i o n ( * ) h o l d s f o r t h e m a p p i n g " , b y t h e s a m e m e t h o d , i t h o l d s t h a t D 2 / 2 ( o ) ( ' 7 2 )

〜 o

isinthespaceD"(o)(C")forany'7EC".SmceZand｡areinvolutions,byRemark 5 o f s e c t i O n l , i t h o l d s t h a t D Z ( 9 ( z ) ) ( D 2 9 ( z ) ( ( D Z ( o ) ( ' 7 % ) ) i s i n t h e s p a c e s p a n n e d b y 5 o f s e c t i o n l , i t h o l d s t h a t D 妙 (

g ( z ) ) ( D 2 g ( z ) ( ( D の ( o ) ( 〃 ) 2 ) ） i s i n t h e s p a c e s p a n n e d b y { D " ( 9 ( z ) ) ( D 9 ( z ) ( D a ( o ) ( " ) ) , D " ( 9 ( z ) ) ( D g ( z ) ( D a ( o ) ( r ( i ) ) ) ) , 2

{ D(

g ( z ) ) ( D g ( z ) ( D の

( o ) ( 〃 ) ) , D 妙

( g ( z ) ) ( D g ( z ) ( D( o ) ( r ( ) ) ) ) , 2 ≦ j ≦ " } ̲

S i n c e D "

( 9 ( z ) ) a n d D a ( o

) a r e n o n ‑ s i n g u l a r a n d s i n c e { D . ( o ) ( " ) , D j ( o ) ( r ( i ) ) 2 ≦ j ≦

" } a r e l i n e a r y i n d e p e n d e n t , D 2 9 ( z ) ( 7 7 2 ) i s i n t h e s p a c e s p a n n e d b y { D g ( z ) ( " ) , D g ( z )

(z･(i))2≦j≦"}forany"EC".Then

｛ 職 " ≦ , ≦ " 、 ）

det

foranyりEC施．Byconsideringthevectorg+"insteadof77,wehave

（ ; ≦ , ≦ " ， ）

det

foranyfand〃．Bythesamemanner,

14 ChikaraWATANABE

職 掌 ≦≦ " 」 1

det

foranyr,の,andり.Thisequalityholdsforallzwithlzl<".Forafixednon‑zero

Vectorg,let

( i 蝋 ､ ≦ ＃ ≦ 』

ん(4)=det

T h e n " ( ﾉ l ) i s h o l o m o r p h i c i n l ﾉ l 1 < 1 ; ｢ a n d v a n i s h e s i n l l l < f ｢ , ｡ o 伽 t v a n i s h e s i n l ﾉ l 1 <

T # r B y d i f f e r e n & 鮒 ｡ n o f " ( 1 ) ,

（ ; 職薑≦ 1 小 ｡ Ⅷ

det

（ 雛 ． ≦ ; ≦ 繩 )

det

Fromthis,itholdsthatD39(o)(f3)EDg(o)(C")．Bythesamewayitholdsthat D4g(o)(f')EDg(o)(C")

foranygEC"andforanypositiveinteger".Therefore,

9 ( e ) ̲ Z D ' g ( g ( E ' ) m g ( o ) ( C " ) ,

sothatg(B")cDg(o)(C"),alinearsubspaceofC"ofdimension".

LetUbeaunitarytransformationofC"suchthatU(D9(o))(C"))={z",,…,"")EC";

z""+'=0,…,2""=0}.PutU｡9(z)=(j,(z),…，§鹿(z)),then'"+,(z)=0,…,，庫(z)=0for anyzEB".HenceV(z)=(j,(z),…,""(z))isabiholomorphicmappingfromB"onto B"sendingtheorigintotheorigin,sothatV(z)isaunitarytransformationofC".Put x(z)=U｡9｡V‑'(z)=U｡"｡/｡V '(z),thenx(z)=(z,o').LetV(a)=6,U'(o)=",then since/(o)=o,X(6)=U'(o)=",sothat"=(b,o').Thenitiseasilyseenthatthere

existsunitarytransfOrmationsVofC",tiofC"andinvolutionSgofB"andZof B"withj(6)=o,'(")=osuchthat/=面｡夕。X･缶｡V(seeRudin[2]page28).By Remarklofsection2,夕。X･乞・V(z)=(V(z),o'),sothat/(z)=fj((V(z),o'))andthis

isalinearisometrymappin9.

Byaneasycalculation,itfollowsthat

LenZWZZZ.D2(妙｡/･の)(Z)=D2jU｡の(Z))((D/('(Z))(D.(z)(")))2)

+Dj(/･の(z))(D2f(j(z))((Dj(z)("))2))

+D'(/･の(Z))(DJ('(z))(D2'(z)("2))).

I

I t i s c l e a r f r o m t h i s l e m m a t h a t i f / i s l i n e a r a n d i s o m e t r y , t h e n t h e c o n d i t i o n ( * ) m e n t i o n e d p r e v i o u s l y i s s a t i s f i e d ・ C o n s e q u e n t l y , w e o b t a i n e d t h e f o l l o w i n g

T " e o " e w z . L e t f b e t z p 7 o p e γ 肋 ん 畑 o " " j C " α "

" g / う ℃ 班 B 通 加 B 々 （ ん > " ) " " た 〃 i s Q f c ﾉ t z S s C 2 o " B 〃 α " d / ( o ) = o . T " e " / i S 〃 " e α γ " o l " e t " が α " α o " " が 肋 e だ

〃 魔 " a " o " ‑ z e γ り ひ e c 加 γ 2 ノ Q f C " s z I c 〃 鋤 a t / b γ α " y C 加 允 e Q f α 〃 α 〃 わ " @ o " Z S 班 の Q f B " z z " d t z ' Q f B " z " 肋｡ / ･ の ( o ) = 0 , " e " " " " g c o " 或 加 〃 ( * ) f s s α " s / J e d :

( * ) D 2 ( j o f o j ) ( " ) ( " 2 ) f S 加 肋 e S m c e 幼 α " " e d b y { " ｡ / ° の ( " ) , D ( j ｡ / ･ の ) ( " ) ( r ( 2 ) ) ,

･ ･ ･ , D ( ' ｡ / ･) ( " ) ( r ( " ) ) } .

1 ．

2

Reference.

C i m a , J . A ､ , S u f f r i d g e . T . J . : A R e f l e c t i o n p r i n c i p l e w i t h H o l o m o r P h i c M a p p i n g s . M a t h . A m . 2 6 5 , 4 8 9 ‑ 5 0 0 ( 1 9 8 3 ) .

，..A;詞、x7．両､,n戸十inn+hPnTvintheunitballofC".Berlin,

R u d i n , W ､ : F u n c t i o n t h e o r y i n t h e S p r i n g e r l 9 8 0 .

ApplicationstoProper

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