Lecture notes in mathematics

Lecture Notes in Mathematics Edited by A.. Eckmann Series: California Institute of Technology, Pasadena... All rights are reserved, whether the whole or part of the material is concer

Lecture Notes in

Mathematics

Edited by A Dold and B Eckmann

Series: California Institute of Technology, Pasadena

AMS Subject Classifications (1970): 46J 10, 46J 15

ISBN 3-540-08252-2 Springer-Verlag Berlin • Heidelberg • New York ISBN 0-38?-08252-2 Springer-Verlag New York • Heidelberg • Berlin

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The p r e s e n t w o r k w a n t s to be the s y s t e m a t i c p r e s e n t a t i o n of a f u n c -

t i o n a l - a n a l y t i c t h e o r y It is an a b s t r a c t v e r s i o n of t h o s e p a r t s of

c l a s s i c a l a n a l y t i c f u n c t i o n t h e o r y w h i c h c a n be c i r c u m s c r i b e d b y b o u n d a r y

v a l u e t h e o r y a n d H a r d y s p a c e s H p T h e f a s c i n a t i o n of t h e f i e l d c o m e s f r o m the fact t h a t f a m o u s c l a s s i c a l t h e o r e m s of t y p i c a l c o m p l e x - a n a l y t i c fla-

C h a p t e r I B o u n d a r y V a l u e T h e o r y f o r H a r m o n i c a n d H o l o m o r p h i c

F u n c t i o n s i n t h e U n i t D i s k

I H a r m o n i c F u n c t i o n s I 2 P o i n t w i s e C o n v e r g e n c e : T h e F a t o u T h e o r e m a n d i t s C o n v e r s e 6 3 H o l o m o r p h i c F u n c t i o n s 12

4 T h e F u n c t i o n C l a s s e s H o I # ( D ) a n d H # ( D ) 16

N o t e s 21

C h a p t e r II F u n c t i o n A l g e b r a s : T h e B o u n d e d - M e a s u r a b l e S i t u a t i o n 2 2 I S z e g 6 F u n c t i o n a l a n d F u n d a m e n t a l L e m m a 2 2 2 M e a s u r e T h e o r y : P r e b a n d s a n d B a n d s 26

3 T h e a b s t r a c t F a n d M R i e s z T h e o r e m 31

4 G l e a s o n P a r t s 34

5 T h e a b s t r a c t S z e g ~ - K o l m o g o r o v - K r e i n T h e o r e m 36

N o t e s 42

C h a p t e r I I I F u n c t i o n A l g e b r a s : T h e C o m p a c t - C o n t i n u o u s S i t u a t i o n 44 I R e p r e s e n t a t i v e M e a s u r e s a n d J e n s e n M e a s u r e s 44

2 R e t u r n t o t h e a b s t r a c t F a n d M R i e s z T h e o r e m 4 7 3 T h e G l e a s o n a n d H a r n a c k M e t r i c s 4 8 4 C o m p a r i s o n o f t h e t w o G l e a s o n P a r t D e c o m p o s i t i o n s 54

N o t e s 5 8 C h a p t e r IV T h e A b s t r a c t H a r d y A l g e b r a S i t u a t i o n 59

I B a s i c N o t i o n s a n d C o n n e c t i o n s w i t h t h e F u n c t i o n A l g e b r a S i t u a t i o n 6 0 2 T h e F u n c t i o n a l ~ 66

3 T h e F u n c t i o n C l a s s e s H # a n d L # 69

4 T h e S z e g ~ S i t u a t i o n 76

N o t e s 79

C h a p t e r V E l e m e n t s o f A b s t r a c t H a r d y A l g e b r a T h e o r y 81

w e see t h a t Harm~(D) c HarmP(D) c H a r m l ( D ) We f o r m u l a t e the b o u n d a r y

b e h a v i o u r of the f u n c t i o n s in HarmP(D) in the s u b s e q u e n t p r o p o s i t i o n s

~(c(s)~c(s) ~

ii) L e t I ~ < ~ F o r F£LP(1) c o n s i d e r the f u n c t i o n

f = <Fl>:f(z) = / P ( z , s ) F ( s ) d l ( s ) V z6D

S I~ F u r t h e r m o r e for R+I w e h a v e c o n v e r -

Proof: I) < 8 > 6 H a r m ( D ) for 86ca(S) is o b v i o u s s i n c e for r e a l - v a l u e d

8 the d e f i n i t i o n r e p r e s e n t s <8> as the real p a r t of a f u n c t i o n in HoI(D) 2) In o r d e r to p r o v e iii) it s u f f i c e s to s h o w the u n i f o r m c o n v e r g e n c e fR÷F for R+I F o r O~R<I and z6S w e h a v e

S S

- 2~ f P ( R ' e i t ) ( F ( z e l t ) - F ( z ) ) d t '

and hence for O<~<z after s u b d i v i s i o n into !tl~6 and d ~ I t I ~

IfR(Z)-F(z) i ~2~_~6 p(R, e i t ) ~ ( 6 ) d t + 2 ]IFII P(R,e i6) =< ~(~) + 2 IIFII P(R,e i6) ,

where ~ is the modulus of c o n t i n u i t y of the function F6C(S) T h e r e f o r e

lim sup Hf R- F H ~ ~(6> for each o<6<~,

R+I

SO that l{fR-F{l+o for R+I

3) We next prove i) For f=<e>6Harm(D) and O<R<I we have

as in the proof of 1 2 Thus fEHarmP(D) and N p f ~ IIFI P(%)

case 1< p<~ we use the fact that C(S) is dense in LP(I) Thus for HEC(S)

I/S HfRd%l = < IIHIILI(%) [IfRII L ~(l) =< IIHII L 1 (l)N~f for O<R<I=

i m p l i e s t h a t IfS NFdll % IIHII L I (l) N ~ f for e a c h H 6 L I (X) T h i s m e a n s

[IFII ~ ~ N f, so t h a t w e o b t a i n N f= IIF!I ~ QED

S ii) F o r 1 ~ q < ~ t h e c o n j u g a t e e x p o n e n t w e h a v e L P ( x ) = L q ( I ) " In v i e w o f IIfRI[L p <N f < ~ f o r O < R < I t h e r e e x i s t s a w e a k • l i m i t p o i n t F 6 L P ( 1 ) o f

2t + 2~

T h e p r o o f s h a v e t o b e b a s e d o n t h e r e l a t i o n

e +z d f(z) = P ( z , e l t ) d @ ( t ) = / R e it @(t) V z6D

f(z) = CS + S - ~ - ~ d ~ ( x ) = C S + S - - ~ - - - ~ ( X ) ,

w i t h ~ : ~ ( x ) = ~ ( x ) - ~ ( - x ) for x~O a f u n c t i o n of b o u n d e d v a r i a t i o n w i t h the

n o r m a l i z a t i o n ~ ( x ) = ½ ( ~ ( x + ) + ~ ( x - ) ) Vx>O and ~(O)=0, and ~ r e a l - v a l u e d and

m o n o t o n e i n c r e a s i n g if ~ is so It follows that f ( z ) ÷ A for z+1 is e q u i v a - lent to

2 ~ ~ d~(x) for s>O

F: F(s) = ~ s2+x 2

T h e n ~(x) ÷ a for x+O implies that F ( s ) ÷ a for s+O

x

2.5 L O O M I S THEOREM: Let ~ : [ O , ~ [ ÷ ~ be monotone i n c r e a s i n g and b o u n d e d

w i t h ~ ( 0 ) = 0 (and F as above) T h e n F ( s ) ÷ a for s%O i m p l i e s that ~(x) ÷ a

2 S u p { i ~ ( X ) _ a ] : O < x < 6 } + 2 s

w h i c h m e a n s t h a t for e a c h z6D t h e m e a s u r e P ( z , - ) l (and in p a r t i c u l a r for z=O the m e a s u r e I itself) is m u l t i p l i c a t i v e on A ( D ) T h e same is

t r u e for H~(D)

3.1 REMARK: L e t 86an(D) a n d F 6 A ( D ) T h e n F86an(D) a n d < F S > = < F I > < 8 >

Proof: L e t h = < e > and f=<Fl> F o r O~R<I t h e n FhR6A(D) and

t a i n s the p o l y n o m i a l s i i i ) ~ i i ) is t r i v i a l , i i ) ~ i ) a n d the last a s s e r -

ii) L e t 1~p<~ T h e n H P ( D ) c L P ( 1 ) is the L P ( 1 ) - n o r m c l o s u r e of A(D) (and h e n c e of t h e s u b a l g e b r a of the p o l y n o m i a l s )

iii) H ~ ( D ) c L ~ ( I ) is the w e a k * c l o s u r e of A(D) (and h e n c e of the

of 3.2 the m e a s u r e s 06ca(S) w h i c h a n n i h i l a t e A(D) are the same as t h o s e

w h i c h a n n i h i l a t e the p o l y n o m i a l s T h e d e n s i t y of the s u b a l g e b r a of the

p o l y n o m i a l s a l s o has a s i m p l e d i r e c t proof: For F6A(D) and f = < F l > w e

h a v e llfR-Fll+Ofor R+I, and in v i e w of the T a y l o r s e r i e s e x p a n s i o n e a c h

fR for O~R<I is the u n i f o r m l i m i t of p o l y n o m i a l s In o r d e r to p r o v e t h e

D i r i c h l e t p r o p e r t y let @6ca(S) a n n i h i l a t e ReA(D) a n d h e n c e F a n d F for

F £ A ( D ) It f o l l o w s t h a t / s U d S ( s ) = 0 V n 6 $ a n d h e n c e e=O in v i e w

S the W e i e r s t r a S t h e o r e m

ii)iii) In v i e w of 3.2 HP(D) c o n s i s t s of the F6LP(I) w i t h /snF(s)dl(s)

3.6 L E M M A : L e t 0 6 an(D) be ~ O T h e n t h e r e e x i s t s a n n ( n = O , 1 , 2 )

s u c h t h a t

1 Z n e 6 an(D) and Sf Z ~ d n e + O,

w h e r e Z : Z ( s ) = s is the i d e n t i t y f u n c t i o n

P r o o f of 3.6: F r o m 3.2 we k n o w that ~ j s n d @ ( s ) = O Vn~1 Thus in v i e w

S

of the W e i e r s t r a B t h e o r e m we have / s - n d Q ( s ) 4 0 for some n>=O and hence

S for a s m a l l e s t n>=O T h e n 3.2 shows that this n~O fulfills the a s s e r - tion QED°

P r o o f of 3 5 ~ 3.4: i) L e t @£an(D) and @=~+~ w i t h l - c o n t i n u o u s ~ and

l - s i n g u l a r ~ T h e n @ - @ ( S ) I = ( e - @ ( S ) I ) + ~ a n n i h i l a t e s A(D) so that f r o m 3.5 we see that ~ l i k e w i s e a n n i h i l a t e s A(D) ii) In p a r t i c u l a r i) shows that a l - s i n g u l a r a n a l y t i c m e a s u r e m u s t a n n i h i l a t e A(D), that is has an integral=O But if n o w the a b o v e B w e r e 40, then 3.6 w o u l d lead to a

l - s i n g u l a r a n a l y t i c m e a s u r e w i t h integral 40 This c o n t r a d i c t i o n shows that 8 m u s t be = O QED

3.7 J E N S E N INEQUALITY: Let F6A(D) and f=<Fl> T h e n

log]f(z) I ~ / P ( z , s ) l o g I F ( s ) Idl(s)

s

H e n c e the same is true for F6HI(D)

3.8 COROLLARY: If F6HI(D) is 4 ° then l o g I F I 6 L 1 ( 1 )

N o w IIfR-FII L 1 ( 1 ) ÷ O for R+I so that for a s u i t a b l e s e q u e n c e R(n)+1 we

have f R ( n ) + F p o i n t w i s e and u n d e r an L l ( 1 ) - m a j o r a n t T h e n the same is true for log(IfR(n) I+c) ÷ log(IFl+e) It f o l l o w s that

loglf(z) I ~ f P ( z , s ) l o g ( I F ( s ) l+e)dl(s) V z6D and e>O,

S and for s%O the r e s u l t follows from B e p p o Levi Proof of 3.8: If F40 then f ( z ) 4 0 for some z6D Thus the a s s e r t i o n is obvious QED

For the last t h e o r e m we i n t r o d u c e the f u n c t i o n a l s

DP:DP(o) = I n f { / I F I P d ~ : F £ A ( D ) w i t h f(O)=1} V ~ 6 Pos(S),

S

w h e r e 1<p< ~ Thus o < D P ( o ) < a ( S )

3.9 S Z E G 0 - K O L M O G O R O V - K R E I N THEOREM: For 1<=p<~ we h a v e

d~ dl) DP(o) = exp(f ( l o g ~ )

4 The F u n c t i o n C l a s s e s HoI#(D) and H#(D)

In the future a b s t r a c t t h e o r y the f u n c t i o n class w h i c h c o r r e s p o n d s

to the class H#(D) to be d e f i n e d in the p r e s e n t s e c t i o n w i l l be far m o r e

i m p o r t a n t than the f u n c t i o n c l a s s e s w h i c h c o r r e s p o n d to the HP(D) for finite p~1 As b e f o r e our m a i n c o n c e r n w i l l be the t r a n s i t i o n f r o m D

4.1 P R O P O S I T I O N : For f6Hol#(D) the r a d i a l limit

F(s) := lim fR(s) exists for l - a l m o s t all s6S,

R+I

and p r o d u c e s an e l e m e n t F6H#(D) The map f ~ F thus d e f i n e d is a b i j e c - tion H O I # ( D ) ÷ H # ( D )

Proof:i) Let f£Hol#(D), and take f u n c t i o n s f 6HoI~(D) as r e q u i r e d in

n the d e f i n i t i o n w i t h g n : = f n f 6 H o l ~ ( D ) Let Fn,Gn6H~(D) the r e s p e c t i v e boun-

d a r y functions T h e n IF <I and

n =

f IFn-I 12dl < 2-2Re / Fnd~ = 2 { 1 - R e F n ( O ) ~ ÷ O

T h e r e f o r e after t r a n s i t i o n to a s u i t a b l e s u b s e q u e n c e we can a s s u m e that

F +I p o i n t w i s e , ii) We c h o o s e a B a i r e set NcS w i t h ~ (N)=O such that in

n

each p o i n t s6S-N I) r a d i a l c o n v e r g e n c e fn (Rs)÷Fn(s) and g n ( R S ) ÷ G n ( S ) for R+I takes p l a c e for each n>1, and 2) the r e p r e s e n t a t i v e s S÷Fn(S) of the Fn6H~(D) thus o b t a i n e d on S-N f u l f i l l s Fn(S)÷1 for n÷ ~ C o n s i d e r now the e q u a t i o n f n ( R s ) f ( R s ) = g n ( R S ) for s6S-N and O~R<I For fixed s6S-N

c h o o s e an n> 9 w i t h Fn(S) t O T h e n f n ( R S ) + O for R s u f f i c i e n t l y close to I Thus the limit F ( s ) : = l i m f ( R s ) exists in each p o i n t s6S-N The e l e m e n t

R+I F6L(I) thus p r o d u c e d f u l f i l l s F n F = G n for all n> 1 T h e r e f o r e F6H#(D) iii) In case F=O we have G n = O and hence g n = O for all n>1 In v i e w of fn÷1 this i m p l i e s that f=O T h e r e f o r e the a b o v e m a p f~F is injective iv) L e t us n o w start w i t h a f u n c t i o n F6H #(D), and take f u n c t i o n s F 6H~(D) n

4.2 C O R O L L A R Y : Let f6Hol#(D) T h e n for l - a l m o s t all s6S the a n g u l a r limit

+I and h ef=exp(fn)6Hol~(D) Thus h n : = e x p ( f n - f ) 6 H o l ~ ( D ) w i t h lhnI~1 and h n n

Vn~1 T h e r e f o r e e f 6 HoI#(D) ii) Let @=~+8 w i t h l-continuous ~ and l-sin- gular 8 T h e n ~ 8>=0 And f=u+v w i t h

and let Fn,Gn6H~(D) be the respective b o u n d a r y functions N o w Re v(Rs)÷O

for Rfl for l-almost all s6S after 2.2 Thus JgnJ=Jfnle Rev implies that

JGnJ=JFnJ<l and hence IgnJ=IfnleReV<1 and hence eReV<1 since fn "

fore Re v = O and hence B=O QED

4.4 COROLLARY: Let F6Re L1(1) and

4.6 J E N S E N INEQUALITY: Let f£Hol#(D) w i t h b o u n d a r y f u n c t i o n F£H#(D)

In case f#O we have logjFl6L1(1) and

logJf(z) j £ ] P(z,s)logjF(s)ldl(s) v zeD

S

= +I and gn:=fnf6 Proof: Take functions fn£HOl (D) with JfnJ<1 and fn

6HoI~(D) Vn~1, and let Fn,Gn6H~(D) be the r e s p e c t i v e b o u n d a r y functions

T h e n Gn=FnF We can of c o u r s e a s s u m e fn#O and hence gn~O Thus loglFnl, logIGnlfL1(~) from 3.8 and hence l o g l F l 6 L l ( ~ ) A n d from 3.7 we have for z6D

l°glgn(Z) i ~ / P ( z , s ) l o g I G n ( S ) Idl(s) ~ f P ( z , s ) l o g l F ( s ) Idl(s)

in v i e w of IFnl~1, from w h i c h the result follows for n÷~ QED

4 7 P R O P O S I T I O N : • (HoI#(D)) x := the set of i n v e r t i b l e e l e m e n t s of the

a l g e b r a HoI#(D) c o n s i s t s of the f u n c t i o n s h=ce f, w h e r e

s+z f:f(z) = / ~ L ~ F(s)dl(s) V z6D w i t h F £ R e L I ( I ) ,

= Vz6D is in HoI+(D) and its b o u n d a r y f u n c t i o n F=I-~-~6H (D) has ReF=O

S o - F 6 H # ( D ) has R e ( - F ) > O as well, but it c o r r e s p o n d s to the f u n c t i o n -f6Hol#(D) w h i c h is ~HoI+(D) so that -F~H+(D)

To e x p l a i n the p h e n o m e n o n recall from 4.3 that to f6Hol+(D) t h e r e exists a m e a s u r e ~£Pos(S) such that Re f =<8> and hence

o n t h e s u b s p a c e A m o d ~ c L P ( o ) a n d is L P ( o ) - n o r m c o n t i n u o u s t h e r e , t h a t

o f A m o d o O t h e r w i s e t h e r e w e r e a n f E L q ( o ) w i t h ~ u f d o = O V u E A a n d

w i t h ~ P f d d * O B u t t h e n f E T a n d O = [ f d d = e(f) = c ~ f P d o w h i c h i s a

c o n t r a d i c t i o n

~n V u 6 A T h u s f r o m H a h n - B a n a c h a n d 3) W e k n o w t h a t l~(u) I ~ u I L p ( o )

P r o o f : C o m b i n e 1.4 w i t h 1.2 QED

1.6 C O R O L L A R Y : L e t 8 £ c a ( X , Z ) w i t h ~(f) = f f d 8 V f 6 A T h e n t h e r e

e x i s t s an m 6 M ( ~ ) w h i c h is << 9 In p a r t i c u l a r M ( ~ ) ~

P r o o f : W e h a v e f l f l d l S l _>_ IffdSl > I V f 6 A w i t h <0(f) = I a n d h e n c e DI(IsI) > 0 QED

e: = (oT-e m) + e~ , II e,~il = It e~-%lI + It o.~ il •

tl e~ I1 - It emil = tl emil - II e~ II = II e~-emll •

iii) I n t h e a l g e b r a s i t u a t i o n c o n s i d e r e d i n t h e p r e s e n t c h a p t e r f o r

e a c h ~ E Z ( A ) t h e s e t M(~) = M ( A , ~ ) c P r o b ( X , E ) is a p r e b e n d In f a c t ,

f o r m l £ M ( ~ ) ( 1 = 1 , 2 , ) w e h a v e m : = ~ 2 - 1 m 1 6 M ( ~ ) a n d m l < < m V I ~ I

1=I

~ £ Z ( A ( D ) ) is r e p r e s e n t e d b y a u n i q u e p r o b a b i l i t y m e a s u r e : M(~) = { m ( ~ ) }