H o l o m o r p h i c F u n c t i o n s
S S 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)
Proof: L e t h = < e > and f= F o r O~R > I = I < < b , a > , z > l F o r z=O i n p a r t i c u l a r I < a , b > l = I < b , a > l iii) l < a , b > = < l a , l b > f o r a l l 16S iv) lal-lbl < il < l a I + I b l
1+tl zl w h i c h is at o n c e t r a n s f o r m e d i n t o the a s s e r t i o n I z l ~ c t ( u , v ) To p r o v e the s e c o n d i n e q u a l i t y it s u f f i c e s to a s s u m e t h a t nfIl O } , w h i c h is a cone in R e L Y ( m ) We deduce f r o m 3.14 that S is weak* closed
In fact, if fn6S w i t h IfnlO since S c o n t a i n s all f u n c t i o n s f
/ lhj2+tp " - - a 2 + b 2 + t a V t>O lhj2+2tP+t2vcum = a 2 + b 2 + 2 t a + t 2 ii) We p r o v e 5.4 and can a s s u m e that b=O T h e n from i) we o b t a i n
~ a ~t : / l h j 2 + t p V d m > / lhJ2+tp V d m lhI2+2tP+t 2 =[JhJ>t] lhl2+2tp+t 2
Z ~I [Jh|zt]~ V d m = ẵ ( V m ) ( [ Jhlzt]) V t > O , w h i c h is the assertion, iii) For t>O the d i f f e r e n c e t[ lh12 - lhl2+tp ] = Pt 2 [hI2-t2
Jhj2+t 2 lhl2+2tP+t '2j (lhI2+t2) (lhI%2tp+t 2) is of m o d u l u s ~P and tends ÷ -P p o i n t w i s e for t+ ~ S i n c e
Ibl2-vdm=-I 2t~X2dw(xl =-S w{xl: t I l h l Z + t z 0 x - + t - 0 + x + t ÷ ~ t(I) 2 ÷ ~ 1 ÷ ~ I
T h e r e s u l t c o u l d b e d e d u c e d f r o m 5.4 a n d 5.2 w i t h a w o r s e c o n s t a n t W e p r e f e r t o b a s e t h e p r o o f o n t h e s u b s t i t u t i o n t h e o r e m 4.7 i) L e t s 6 ~ w i t h R e s >_O T h u s s=Isle it w i t h Itl~ a n d h e n c e sT=IsITe iTt f o r t h e m a i n b r a n c h I t f o l l o w s t h a t R e s T = I S l T C O S T t ~ I s I T c o s ~ ii) F o r e > O w e h a v e
Q 6 R e L(m) b e i t s c o n j u g a t e f u n c t i o n (up t o a n a d d i t i v e r e a l c o n s t a n t ) F o r t h e l a t t e r p a i r s o f f u n c t i o n s P a n d Q in t h e u n i t d i s k s i t u a t i o n t h e above r e s u l t s 5.4 a n d 5 5 a s w e l l as 5.6 a n d 5 8 a r e w e l l - k n o w n c l a s s i c a l t h e o - r e m s d u e t o K o l m o g o r o v O f c o u r s e t h e s e t h e o r e m s c a n b e e x p e c t e d t o e x - t e n d t o c o n j u g a t e f u n c t i o n s in 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 B u t i t is a s u r p r i s e to s e e t h e m v a l i d u n d e r a s s u m p t i o ~ w h i c h a r e m u c h w e a k e r e v e n i n t h e u n i t d i s k s i t u a t i o n
W e l i s t s o m e s i m p l e p r o p e r t i e s , i) ~(f) is c l o s e d , ii) T h e r e e x i s t s a r e p r e s e n t a t i v e f u n c t i o n x ~ f(x) s u c h t h a t f ( x ) 6 m ( f ) f o r a l l x 6 X iii) e ( f ) + ~ , iv) If ~ ( f ) = { c } t h e n f = c o n s t = c
~ ( m ) C h o o s e a r e p r e s e n t a t i v e f u n c t i o n x ~ h ( x ) w i t h h ( x ) # t f o r a l l x 6 X ii) L e t K a n d B d e n o t e c l o s u r e a n d b o u n d a r y o f G r e l a t i v e t o t h e R i e - m a n n s p h e r e L e t A c C ( K ) c o n s i s t o f t h o s e f u n c t i o n s w h i c h a r e h o l o m o r p h i c o n G In v i e w o f t h e m a x i m u m m o d u l u s p r i n c i p l e t h e r e s t r i c t i o n f ~ f l B is a s u p n o r m i s o m o r p h i s m A ÷ A I B L e t 8 6 P r o b ( B ) b e a J e n s e n m e a s u r e f o r f l B ~ f ( t ) o n A I B a f t e r I I I 1 1 F o r e a c h u 6 ~ t h e f u n c t i o n z ~ U - Z - 1 _ u - a is i n A since
W e c l a i m t h a t f o r a l l u E G t h e s t r i c t < r e l a t i o n h o l d s t r u e T o s e e t h i s n o ~ t h a t u ~ / l o g ~u-z d8 (z) is a c o n t i n u o u s r e a l - v a l u e d f u n c t i o n o n G w h i c h s a t i s f i e s t h e m e a n v a l u e e q u a t i o n a n d h e n c e is h a r m o n i c o n G T h e r e f o r e u ~ / l o g u - z dg (z) l o g u - t is a h a r m o n i c f u n c t i o n ~ 0 o n [ u 6 G : u ~ t } w h i c h t e n d s ÷ ~ f o r u+t In v i e w o f t h e m a x i m u m p r i n c i p l e it m u s t t h e r e f o r e b e > 0 i n a l l p o i n t s u + t i n G as w e h a v e c l a i m e d , iii) F r o m t h e a b o v e w e h a v e
P r o o f : i) L e t U , V c ~ b e d i s j o i n t o p e n s e t s w i t h & + c U u V A s s u m e t h a t a6U A f t e r 6.2 it s u f f i c e s t o p r o v e & + D V = ~ T o s e e t h i s l e t G b e a c o m - p o n e n t o f V T h e n G is a d o m a i n i n ~ a n d a i n t h e e x t e r i o r o f G N o w ~G is d i s j o i n t t o V a n d t o U a n d h e n c e ~ G A A + = ~ T h e n 6.1 i m p l i e s t h a t G D A + = ~ as w e l l It f o l l o w s t h a t V n & + = ~ ii) A s s u m e t h a t t~A T h e n t h e r e e x i s t s
1 _ 1 _ 1 1 _ 1 ~ (h_ ~)6H, h - z ( h - s ) - ( z - s ) h - s 1_z- {s h - s Z 0 h - s s o t h a t z 6 P a n d h e n c e in f a c t P is o p e n as w e l l ii) N o w f o r s 6 ¢ - ~ ( h ) w e h a v e s 6 & ( h ) ~h1-~s ~ H # ~ h 1 _ ~ H ~ s 6 Q T h e r e f o r e A ( h ) c Q U m ( h ) = ~ - P w h i c h is c l o s e d a n d h e n c e A ( h ) c Q U ~ ( h ) F u r t h e r m o r e Q c & ( h ) a n d h e n c e Q c ( A ( h ) ) °
T h e L u m e r s p e c t r u m w a s i n t r o d u c e d ( u n d e r t h e n a m e o f i n n e r s p e c t r u m ) a n d t h e e s s e n t i a l o f 6.5 w a s a n n o u n c e d in L U M E R [ 1 9 6 5 ] T h e n 6.6 a p p e a - r e d in K O N I G [ 1 9 6 7 a ] [ 1 9 6 7 c ] a n d 6.7 in K O N I G [ 1 9 6 6 b ] T h e o r e m 6.8 (re- s t r i c t e d t o L I f u n c t i o n s w h i c h is u n e s s e n t i a l ) is o n e o f t h e m a i n r e - s u l t s i n K O N I G [ 1 9 6 5 ] F o r t h e c o n n e c t e d n e s s l e m m a 6.2 w e h a v e b e e n s u p p - l i e d w i t h i n d e p e n d e n t p r o o f s f r o m o u r t o p o l o g i c a l f r i e n d s T t o m D i e c k a n d R F r i t s c h
I n t h e u n i t d i s k s i t u a t i o n t h e c l a s s i c a l c o n j u g a t i o n is t h e o p e r a - t i o n w h i c h a s s o c i a t e s w i t h e a c h p 6 R e H a r m ( D ) t h e u n i q u e f u n c t i o n q 6 R e H a r m ( D ) s u c h t h a t p + i q 6 H o l ( D ) a n d q ( O ) = O In o r d e r t o e x t e n d t h e c o n j u g a t i o n t o 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 w e h a v e t o r e - d e f i n e it as a n o p e r a t i o n w h i c h t a k e s p l a c e o n t h e u n i t c i r c l e S: t h a t is w h i c h a s s o c i a t e s w i t h e a c h P f r o m a c e r t a i n s u b c l a s s E o f R e L ( 1 ) a u n i q u e f u n c t i o n Q 6 R e L ( 1 ) T h e i m m e d i a t e i d e a to d e f i n e i t v i a P + i Q 6 H # ( D ) p l u s s o m e n o r m a l i z a t i o n o f Q is b o u n d to f a i l s i n c e t h e r e a r e l o t s o f n o n c o n s t a n t r e a l - v a l u e d f u n c t i o n s in H # ( D ) A n d t h e r e is n o o b v i o u s i d e a h o w t o r e s t r i c t E a n d H # ( D ) in o r d e r t o e s c a p e f r o m t h e s e n o n - c o n s t a n t f u n c t i o n s a n d s t i l l p r e s e r v e a not too n a r r o w d e f i n i t i o n S o l e t u s s e e k t o t r a n s p l a n t t h e i n i t i a l d e f i n i t i o n f r o m D t o S
~ o l e t ( p + i Q ) l > o f o r a l l t6~ It is c l e a r t h a t t h i s d e f i n i t i o n c a n b e e x t e n d e d to the 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 F u r t h e r m o r e w e s e e t h a t e a c h f u n c t i o n P 6 R e L ( I ) w h i c h is c o n j u g a b l e i n t h e n e w s e n s e m u s t b e i n
R e L ( m ) ÷ [ - ~ , ~ ] is f i n i t e - v a l u e d a n d l i n e a r T h e m a i n a c h i e v e m e n t is t h e n t h e c h a r a c t e r i z a t i o n o f E w i t h t h e m e a n s o f M I t r e q u i r e s t h e f u l l p o - w e r o f t h e m a i n t h e o r e m s o f C h a p t e r IV T h e p r i n c i p a l r e s u l t is t h a t a f u n c t i o n P £ R e L ( m ) w h i c h is b o u n d e d , o r a t l e a s t n o t t o o f a r r e m o t e f r o m b o u n d e d n e s s i n s o m e s e n s e o r o t h e ~ is in E i f f t h e i n t e g r a l f P V d m h a s t h e s a m e v a l u e f o r a l l t h o s e V 6 M f o r w h i c h i t e x i s t s in t h e e x t e n d e d s e n s e H e r e a f t e r w e p r o v e f o r E ~ : = E n R e L ~ ( m ) a s i m p l e b u t p o w e r f u l a p p r o x i m a t i o n t h e o r e m : t h a t R e H c E ~ is d e n s e in E ~ in a s e n s e w h i c h is m u c h s h a r p e r t h e n w e a k * d e n s i t y T h e r e s u l t h a s i m p o r t a n t i m m e d i a t e i m - p l i c a t i o n s T h i s is c o n c e i v a b l e s i n c e i n t h e S z e g ~ s i t u a t i o n M = { F } w e h a v e E = R e L I ( F m ) a n d h e n c e E ~ = R e L ~ ( m ) , s o t h a t h e r e t h e a p p r o x i m a t i o n t h e o r e m s h a r p e n s t h e f u n d a m e n t a l r e s u l t t h a t R e H is w e a k • d e n s e i n
P r o o f : T h e u n i q u e n e s s a s s e r t i o n is o b v i o u s T h e b a s i c i d e a o f t h e e x i s t e n c e p r o o f is c a r e f u l d i f f e r e n t i a t i o n o f t h e v e c t o r - v a l u e d f u n c t i o n t ~ h t W e h a v e t o f a c e t e c h n i c a l d i f f i c u l t i e s d u e t o t h e f a c t t h a t t h e s i z e o f t h e f u n c t i o n Q £ R e L(m) t o b e c o n s t r u c t e d is n o t r e s t r i c t e d a t all i) T h e f u n c t i o n ~ ÷ L I ( m ) : t ~ h t is c o n t i n u o u s in L I ( m ) - n o r m T h e r e f o r e t w e c a n f o r m t h e e l e m e n t a r y i n t e g r a l H t : = f h u d U 6 L 1 (m) Vt£1~ F r o m t h e f u n d a -
The function Q6L(m) derived earlier is real-valued, indicating that ht equals h_t, which leads to Ht being equal to -H_t Consequently, it follows that ht - 1 is equal to iQHt for all t This relationship confirms the uniqueness assertion from the previous section Furthermore, we assert that for each fixed E(r), the expression ~ (ht - 1) approaches iQ as t approaches zero.
L 1 ( m ) - n o r m In fact, from i) ii) we see that for n fixed
(ht-1) He (n) =t (He (n) +t-He (n)) - 1 H t ÷ h e (n) -I = i Q H e ( n ) for t+O in LI (m)-norm, and on E(r) the f u n c t i o n IH (n) I is ~ some 6>0 for n s u f f i c i e n t l y large vi) We n o w p r o v e that h t = e itQ for all t6~ For this p u r p o s e o b s e r v e
:=hte-itQ that the f u n c t i o n s h~ V t 6 ~ fulfill the a s s u m p t i o n s of the theo- rem so that i)-v) can be a p p l i e d to them as well Thus we o b t a i n a u n i q u e Q~6L(m) such that h [ - 1 = i O ~ H [ Vt6~, and on each fixed E~(r) we have ~(h~-1~
~ Q ~ for t÷O in LI (m)-norm But on each E(r) we have in L I
(m)-norm It follows that Q~=O Thus h~=1 or ht=e itQ for all t6% QED
2.1 THEOREM: For P 6 ReL(m) the s u b s e q u e n t p r o p e r t i e s are e q u i v a l e n t i) a ( e t m ) a ( e - t P ) = I V t 6 ~ ii) ~(P) 6 ~ and ~(tP) =t.e(P) V t 6 ~ iii) There exists a f u n c t i o n Q 6 ReL(m) such that e t ( P + i Q ) £ H # V t6~
In this case the f u n c t i o n Q 6 ReL(m) is u n i q u e up to an a d d i t i v e real constant H e n c e there e x i s t s a u n i q u e Q6ReL(m) such that in a d d i t i o n
The f u n c t i o n c l a s s E is d e f i n e d to c o n s i s t of the f u n c t i o n s P6ReL(m)
112 w h i c h possess the e q u i v a l e n t p r o p e r t i e s i)ii)iii) in 2.1 The functions
P 6 E are called conjugable For P6E the unique f u n c t i o n Q6ReL(m) such that e t ( P + i Q ) 6 H # and ~ ( e t ( P + i Q ) ) = e ta(P) V t E ~ is c a l l e d the c o n j u g a t e f u n c t i o n of P and w r i t t e n Q=:P*
2.2 LEMMA: C o n s i d e r a sequence of functions UnEH# w i t h lUnl~ some
G6L # such that lim suplUni!1 and ~(Un) ~I T h e n there exists a subse- n÷~ q u e n c e Un(Z) w h i c h tends +I for £+~
Proof of 2.2: Take f u n c t i o n s v z 6 H w i t h IviI~1, v£÷I and Iv£1G~c Z vZ~I we can assume that ~ ( v £ ) > 1 - ~ For V E M then
/lv~(Un-1) I 2vdm = ~1 {VzUn-1)+(1-v Z) ] 2volta
I n t r o d u c e now G n : = S u p { l U s I : S ~ n } and o b s e r v e that fun = n= n lim sup lUn]~1 for n Then n÷~ lim sup ~Ivz(Un-1) ]2Vdm I n +~ = V :
Thus there exists a sequence 1 ~ n ( 1 ) < < n ( i ) ~ n ( i + 1 ) < such that
It follows that v i ( U n ( z ) - 1 ) ÷ O and h e n c e that Un(1)÷1 for ~÷~ QED
Proof of 2.1:i) ~ i i ) F r o m V.I.2 we have O < a ( e t P ) < ~ V t E ~ and hence a ( t P ) 6 ~ and a ( t P ) + e ( - t P ) = O Vt6~ F u r t h e r m o r e e t P 6 L # V t6~ N o w
~((s+t)P)=O T h e n letZ-11 = I ( e t X - 1 ) e i t y + ( e i t Y - 1 ) l ~ ( e t X - 1 ) + leitY-11
Now e t X - l < t x e t X < t x e T X and x < ~ ( 1 + e x ) < ! e Cx so that e t X - 1 < ~ e (T+E)x And leity-II~21sint2~I~tly I The a s s e r t i o n follows QED°
2.8 P R O P O S I T I O N : A s s u m e that P6E is b o u n d e d b e l o w or b o u n d e d above Then P + i P ~ 6 H # a n d ~ ( P + i P ~ ) = e ( P )
Proof: We can a s s u m e that P~O T h e n V.4.1 can be applied T h e a s s e r - tion follows QED
In one d i r e c t i o n we o b t a i n an i n s t a n t final result
3.1 THEOREM: Let P6ReL(m) w i t h e ± P 6 L # A s s u m e that the i n t e g r a l /PVdm has the same v a l u e c6[-~,~] for all those V 6 M for w h i c h it e x i s t s in the e x t e n d e d sense T h e n P6E and e(P)
Proof: We a p p l y IV.3.9 to tP w i t h real t~O It follows that - ~ < ~ ( t P ) ~
~tc Thus c 6 ~ and h e n c e ~ ( t P ) 6 ~ Vt6~ T h e n f r o m ~ ( t P ) ~ t c and ~ ( - t P ) ~ - t c we o b t a i n ~ ( t P ) = t c VtE~ QED
In the o p p o s i t e d i r e c t i o n it w o u l d be m o s t p l e a s a n t to d e d u c e from P6E that
/ I P I V d m < ~ and /PVdm = e(P) for all V6M, or at least for those V 6 M for w h i c h /PVdm exists in the e x t e n d e d sense But we c a n n o t p r o v e this u n l e s s we i m p o s e an a d d i t i o n a l b o u n d e d n e s s con- d i t i o n u p o n PEE w h i c h a p p e a r s to be s h a r p e r than the i m p l i c a t e d c o n d i - tion e ± P 6 L #
P r i o r to the m a i n p o i n t w e n o t e a s i m p l e b u t i m p o r t a n t a s s e r t i o n w h i c h is in o b v i o u s r e l a t i o n to V 4 1 o n t h e f u n c t i o n c l a s s H +
3.3 P R O P O S I T I O N : o) L e t P 6 E a n d V 6 M be s u c h t h a t / e - 6 P v d m < ~ for s o m e 6>0 T h e n l l P I V d m < ~ a n d / P V d m ~ e ( P ) i) If P 6 E is b o u n d e d b e l o w t h e n e ( P ) = S u p { / P V d m : V E M } ii) If P E E ~ : = E D R e L ~ ( m ) t h e n a(P) = / P V d m for a l l VEM
I e - t ~ ( P ) ~ f e - t P v d m , e-~(P) < (fe-tPvdm) ~ r so t h a t II.5.1 i m p l i e s t h a t - ~ ( P ) ~ / ( - P ) V d m T h e a s s e r t i o n f o l l o w s , i) f o l l o w s f r o m o) a n d 3 2 i i ) ii) is t h e n o b v i o u s QED
In the n e x t l e m m a w e i n t r o d u c e the s h a r p e n e d b o u n d e d n e s s c o n d i t i o n q u o t e d a b o v e a n d p r o v e a c e r t a i n e x t e n s i o n of the b a s i c t h e o r e m I V 3 1 0 3.5 L E M M A : D e f i n e
I n f { ~ ( ( f - h ) +] : h 6 R e L = ( m ) } < ~ + ( f ) < I n f { @ ((f-h) +] : h £ R e L ~ ( m ) } ii) f bounded above ~ e + ( f ) = O ~ e f £ L # iii) We have
Proof: i) To prove the left estimation let OO] w h i l e on [f O, w h i l e t h e f i f t h e s t i m a t i o n is b a s e d on t h e i n e q u a l i t y uv < t p - - up + I v q V U , v ~ O a n d t > O for c o n j u g a t e 1 < p , q < ~,
IunI ~ luI, IPn I ~ IPI, u n ÷ u, ~ ( U n ) r e a l a n d ÷ ~ ( u ) = O, Ivnl ~ ivl, IQnl ~ QI, v n ÷ v, M ( V n ) r e a l a n d ÷ ~ ( v ) = O
O n t h e o t h e r h a n d the i n t e g r a n d c o n v e r g e s + ( h - f h F d m - u ) (u-iv) p o i n t - w i s e a n d w i t h t h e m a j o r a n t ( l h I + I f h F d m I + l u l ) ( l u l + I v l ) 6 L 1 ( F m ) in v i e w of 2) It f o l l o w s t h a t
It is a p l e a s a n t c o n s e q u e n c e t h a t the f u n c t i o n s in H a d m i t a s i m p l e c h a r a c t e r i z a t i o n v i a m u l t i p l i c a t i v i t y u n d e r i n t e g r a t i o n as f o l l o w s
4.7 C O R O L L A R Y : A s s u m e t h a t F 6 M is d o m i n a n t o v e r X T h e n for f E L ~ ( m ) t h e s u b s e q u e n t p r o p e r t i e s a r e e q u i v a l e n t i) f6H ii) f ( f + u ) 2 V d m = ( f ( f + u ) V d m ) 2 for a l l u 6 H a n d V6M iii) / f 2 V d m = (ffVdm) 2 f o r a l l V 6 M a n d f f u F d m = f f F d m / u F d m for all u6H
P r o o f : i) ~ i i ) a n d ii) ~ i i i ) a r e o b v i o u s So a s s u m e iii) T h e se- c o n d a s s u m p t i o n s h o w s t h a t f ± H~F T h u s a f t e r 4.5 it r e m a i n s to p r o v e t h a t f I N B u t f o r U , V 6 M a n d O ~ t ~ I w e h a v e ( I - t ) U + t V 6 M a n d h e n c e
P r o o f : T h e i n c l u s i o n m is o b v i o u s In o r d e r to p r o v e = t a k e a f u n c - t i o n h 6 L ~ ( m ) w h i c h a n n i h i l a t e s t h e s e c o n d m e m b e r T h i s m e a n s t h a t h ± H F a n d h I N T h e n 4.5 i m p l i e s t h a t h 6 H a n d ~ ( h ) = / h F d m = O I t f o l l o w s t h a t h a n n i h i l a t e s K S i n c e tbe s e c o n d m e m b e r is a c l o s e d l i n e a r s u b s p a c e o f L1 (m) t h e a s s e r t i o n f o l l o w s QED
I) It is c l e a r t h a t Xs6M F u r t h e r m o r e G : = ( I - I ~ P ( a , ' ) , I - - ~ ) 6 M , w h e r e w e k n o w t h a t G > O f r o m S e c t i o n I.I i n i t i a l r e m a r k i) In fact, for f= ( u , < u l > ( a ) ) 6 H w e h a v e
2) M = { ( I - t ) X s + t G : O < t < 1 } In o r d e r to see c let V c M so t h a t O 0 s u c h t h a t V ~ c F V V 6 M ii) F is a n i n t e r n a l p o i n t o f t h e c o n v e x s e t M c R e L 1 ( m ) , t h a t is t o e a c h V 6 M t h e r e e x i s t s a n e > O s u c h t h a t F - e(V-F) £ M ii') T h e r e e x i s t s a n e > O s u c h t h a t F - e(V-F) 6 M V V 6 M iii) N = {c(V-F) :V 6 M a n d c > O}
P r o o f : i) ~ i ' ) A s s u m e t h a t t h e a s s e r t i o n is n o t t r u e T h e n t h e r e e x i s t f u n c t i o n s V n £ M s u c h t h a t V n ~ n 2 n F is f a l s e ( n = I , 2 ) ° N o w V : = ~ 1 V 6 M a n d h e n c e V ~ c F f o r s o m e c > O I t f o l l o w s t h a t V < 2 n v n=1 2 n n n c 2 n F ~ n 2 n F f o r a l l s u f f i c i e n t l y l a r g e n S o w e o b t a i n a c o n t r a d i c t i o n i') ~ i i ' ) F o r V 6 M a n d e > O t h e r e l a t i o n F - e ( V - F ) 6 M is e q u i v a l e n t t o e(V-F) ~ 0 o r V ~ (I+~)F T h u s t h e i m p l i c a t i o n is c l e a r , ii') ~ i i )
F is t r i v i a l , ii) ~ i i i ) W e h a v e t o p r o v e t h e i n c l u s i o n c L e t f £ N, t h a t is f = c ( U - V ) w i t h U , V 6 M a n d c>O F r o m ii) w e h a v e a n e > O s u c h t h a t W: = F - e ( V - F ) = ( I + s ) F - e V 6 M It f o l l o w s t h a t f = c ( U - V ) = c ( U + W - ( I + E ) F ) = c 1+a ( ~ - F )
In the context of the well-defined relationship F > O on X, we observe that for ε > 0, the function approaches certain limits, leading to a contradiction based on the established inequalities Specifically, we define fn as the minimum of f and n for n ≥ 1, which allows us to express the difference f - fn in terms of a non-negative component Consequently, we conclude that the function e(f) belongs to the space L# This result reinforces the foundational principles outlined in IV.3.13 QED.
In o r d e r to i l l u s t r a t e c o n d i t i o n ~4) above we insert the next result
We shall come b a c k to this c o n t e x t in C h a p t e r viii (see also IV.4.5)
6.3 REMARK: C o n s i d e r the s u b s e q u e n t c o n d i t i o n s i) M is c o m p a c t in o ( R e L 1 ( m ) , R e L ~ ( m ) ) ii) If O ~ fn 6 ReLY(m) and fn40 then 8(f n) ÷ O iii) = o + ) If O ~ fn 6 ReLY(m) and fn40 then e(fn ) + 0
Then i) ~ i i ) ~ i i i ) (let us a n n o u n c e that also ii) ~ i ) as it w i l l be seen in V I I I 3 1 )
Proof: i) ~ i i ) The f u n c t i o n s f n : V ~ / f n V d m are 0 ( R e L 1 ( m ) , R e L ~ ( m ) ) c o n t i n u o u s r e a l - v a l u e d f u n c t i o n s on M w i t h s u p n o r m llfnll = @(fn ) Since fn + 0 the Dini t h e o r e m implies that 8(fn)÷O ii) ~ i i i ) is o b v i o u s from IV.3.9 QED
For O ~ F 6 ReL1(m) and I ~ p < ~ let us now d e f i n e
R p(Fm):=R-~-~eLp(Fm) :={f6ReL(m) :Bfn6Re H w i t h / I f - f n l P F d m ÷ O } , so that l i k e w i s e
The final r e s u l t of the first p a r t of the p r e s e n t s e c t i o n then reads as follows
6.4 P R O P O S I T I O N : For F 6 M c o n s i d e r the s u b s e q u e n t p r o p e r t i e s i) F is internal ii) F is enveloped iii) RI (Fm) c E a n d e(f) = S f F d m V f 6 R I ( F m ) iv) R I ( F m ) c E v) R I ( F m ) A R e L Y ( m ) c E ~ a n d h e n c e = E ~
~ c / f F d m f o r a l l O ~ f 6 R e L ( m ) T h u s c o n d i t i o n I) in 6 2 is o b v i o u s , ii) iii) L e t f £ R I ( F m ) S i n c e F > O o n X w e h a v e a s e q u e n c e f £ R e H s u c h t h a t n f n ÷ f a n d I f n I ~ G w i t h G 6 R e L I ( F m ) a n d h e n c e e G 6 L ° ( F m ) c L # F r o m 2 4 i ) w e s e e t h a t f £ E a n d a ( f ) = / f F d m , iii) ~ i v ) a n d iv) ~ v ) a r e t r i v i a l T h u s i t r e m a i n s t o p r o v e t h e e q u i v a l e n c e v) ~ v i ) N o w o b s e r v e t h a t b o t h v) a n d vi) i m p l y t h a t F > O o n X F o r vi) t h i s is o b v i o u s , a n d to d e d u c e i t f r o m v) n o t e t h a t t h e c h a r a c t e r i s t i c f u n c £ i o n X B o f B : = [ F = O ] is i n
T h e n w e c a n p a s s o v e r to ~ T c L P ( F P m ) a n d h e n c e a s s u m e t h a t F = I T h e d e s i - r e d r e s u l t t h e n r e a d s as f o l l o w s : If a l i n e a r s u b s p a c e T c L ~ ( m ) is L P ( m ) - n o r m c l o s e d f o r s o m e 1 ~ p < ~ t h e n d i m T < ~ W e s h a l l p r o v e t h i s v e r s i o n ii) I t is c l e a r t h a t T is L ~ ( m ) - n o r m c l o s e d T h u s f r o m t h e c l o s e d g r a p h t h e o r e m w e o b t a i n a c o n s t a n t c > O s u c h t h a t Ilfll~ ~ clIflILp v f £ T W e
= L S ( m ) llfIlP = SlflPdm < IIfIIP~ s ] I f l S d m < cP-SllfIlP~ s llfIILs(m ) ,
II < cs Iifl S(m) a n d h e n c e t h e a s s e r t i o n a s w e l l iii) N o w w e d e d u c e f r o m IIfll ~
, [ t£fi, ~ cl, ~ tifi,IL 2 = c [ ~ Iti, 2] £=I ~=I (m) Z 1
A f t e r i n t e g r a t i o n it follows that n ~ c 2 m ( X ) Thus d i m T < ~ QED
6.6 P R O P O S I T I O N : Let F 6 M Then the s u b s e q u e n t p r o p e r t i e s are e q u i - valent i) N ReLI (m) c F ( R e L = ( m ) ) ii) N Q F ( R e L = ( m ) ) is L 1 ( m ) - n o r m c l o s e d and F f u l f i l l s the e q u i v a l e n t c o n d i t i o n s i) - vi) in 6.4 iii) dim N