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