Comparison of Two Capacities

8/8/2019 Comparison of Two Capacities

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M ath, Z . 1 86, 407-4 17 (1984) M a th e m a t is c h eZ e i t s c h r i f t9 Springer-Verlag 1984

C o m p a r i s o n o f T w o C a p a c i t i e s i n l l ; "H e r b e r t J. A l e x a n d e r 1 a n d B .A . T a y l o r 21 Depa rtment of M athematics, Unive rsity of Illinois at Chicago, Box 4348, Chicago,Illinois 60680, US A2 Dep artment o f M athematics, University of M ichigan,Ann Arbo r, Michigan 48104, USA

1 . I n t r o d u c t i o nI n t h e p a s t f e w y e a r s , s e v e r a l c a p a c i t a r y s e t f u n c t i o n s h a v e b e e n i n t r o d u c e di n c o n n e c t i o n w i t h t h e s t u d y o f a n a ly t ic a n d p l u r i s u b h a r m o n i c f u n c ti o n s o fs e v e r a l v a r i a b l e s ( 1" 1-5 , 9 - 1 3 ] ) . T h e r e a r e m a n y d i f f e r e n t s u c h f u n c t i o n s a n d t h er e l a t i o n s h i p b e t w e e n t h e m i s n o t a t a l l c l e a r . W e s h a l l c o n s i d e r h e r e t h erelative capacity o f B e d f o r d a n d T a y l o r , a n d t h e c a p a c i t y d e f in e d in t e r m s o fc e r t a i n T c h e b y c h e f f c o n s t a n t s a s s tu d i e d b y Z a h a r j a t a [ 1 3 ] a n d A l e x a n d e r [ 1 ] .( S e e S e c t . 2 f o r t h e d e f i n i t io n s . ) W e s h o w t h a t t h e s e c a p a c i t i e s a r e e s s e n t i a l lyt h e s a m e . O u r m a i n r e su l t, T h e o r e m 2 .1 , g i ve s th e q u a n t i t a t i v e r e l a t i o n s h i pb e t w e e n t h e m .

I t h a p p e n s t h a t t h e r e l a t i o n s h i p b e t w e e n t h e t w o c a p a c i t i e s i s c l o s e l yc o n n e c t e d w i t h a r e s u l t o f J o s e f s o n [ 8 ] o n t h e e q u i v a l e n c e o f l o c a l ly a n dg l o b a l l y p l u r i p o l a r s et s. T h e q u a n t i t a t i v e e s t i m a t e s o f T h e o r e m 2 .1 a l lo w u s t og i v e i n S ec t. 4 a n e w p r o o f o f J o s e f s o n 's l e m m a a b o u t n o r m a l i z e d p o l y n o m i a l sw h i c h a r e v e r y sm a l l o n t h e s e ts w h e r e a g i v e n p l u r i s u b h a r m o n i c f u n c t i o n i sn e a r l y - o e . J o s e f s o n 's o w n p r o o f is a d ir e c t c o n s t r u c t i o n ; o u r o r i g in a l m o t i -v a t i o n w a s t o g i v e a p r o o f b a s e d o n c a p a c i t y n o ti o n s . I t t u r n s o u t t h a t t h eT c h e b y c h e f f p o l y n o m i a l s t h e m s e lv e s a l re a d y d o t h e jo b .

E 1 M i r [ 7 ] h a s o b t a i n e d a n e x t e n s i o n o f J o s e f so n ' s t h e o r e m . G i v e n af u n c t i o n v p s h o n a n o p e n s e t h e o b t a i n s a g l o b a l p s h f u n c t i o n u o f s m a l lg r o w t h a t i n f i n i t y s u c h t h a t u i s d o m i n a t e d b y a f u n c t i o n h ( v ) o f v . H i s c h o i c eo f h is es s e n ti a ll y h ( x ) = - l o g [ x l. T h u s h e g e t s u < - l o g Iv[ a n d s o th e v a l u e o ft h e g l o b a l f u n c t io n u is - o e w h e n e v e r v = - o e. O u r m e t h o d s g i ve a sh o r tp r o o f o f t h i s ; i n f ac t , t h e y a p p l y t o f u n c t i o n s h s a t is f y i n g

i Ih(x)____~ld x < o e ._~lxll2 . S t a t e m e n t o f M a i n R e s u l tF o r Q a s m o o t h l y b o u n d e d , s tr o n g l y p s e u d o c o n v e x d o m a i n i n " a n d K ac o m p a c t s u b s e t o f g 2 , t h e capacity of K relative to g2 is d e f i n e d i n t e r m s o f th e


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408 H, Alexan derand B.A. Taylorc o m p le x M o n g e - A m p e r e o p e r a to r b y

cap (K ; ~2): = sup {~ (d d cu)": 0 < u < 1, u e P ((J)}, (2.1)K

w h e r e P ( Q ) d e n o t e s t h e c l a ss o f a l l p l u r i s u b h a r m o n i c f u n c t i o n s o n (2. T h eo p e r a t o r

[ J u 1ddCu)" = (2 iO ~ u) " = 4" n ! d e t i_0zjfk I fl"w h e r e / ~ , i s t h e u s u a l v o l u m e f o r m o n ti? ", a n d t h e p r o p e r t i e s o f t h e c a p a c i t a r yf u n c t i o n c a p ( K , f2 ) a r e d i s c u s s e d e x t e n s i v e l y in [ -3 ]. W e r e f er t h e r e a d e r t o t h a tp a p e r f o r m o r e d e t a il s . W h e n t h e o p e n s e t f2 i s u n d e r s t o o d , w e w i ll w r i t es i m p l y c a p ( K ) f o r c a p ( K ; ~ ) a n d c a ll i t t h e r e l a ti v e c a p a c i t y o f K .

T h e g l o b a l c a p a c i t y f u n c t i o n w e c o n s i d e r c a n b e g i v e n i n s e v e r a l e s s e n t i a l l ye q u i v a l e n t w a y s. I t is d e f in e d in a m a n n e r u s u a l w h e n d e a l i n g w i t h " T c h e b y -che f f co ns ta n t s" . F o r K a su bse t o f I1~", l e t

I l f l l K= s up { I f ( z) [: z ~ K }d e n o t e t h e s u p r e m u m n o r m . L e t ~ d e n o t e t h e c la s s o f a ll p o l y n o m i a l s o n C "o f d e g re e < d , n o r m a l i z e d b y r e qu i ri n g t h a t t h e m a x i m u m o f t h e p o l y n o m i a lon th e un i t ba l l i s a t le a s t 1 . Th a t i s, a po lyn om ia l Pe o f deg ree < d be long s to~ d i f a n d o n l y i f IIPdlIB_-->1, (2.2)w h e r e B = { z e C ' : [z[ < 1 }. T h e n d e f in e th e T c h e b y c h e f f c o n s t a n t s f o r K b y

M e ( K ) = inf{ IIP~ I [K: Pe~ a } ( 2 . 3 )a n d t h e c a p a c i ty o f K b y

T ( K ) = in f [Me(K)] l id = lim [M e( K )] 1/e . (2.4)d_->O d~r

V a r i o u s o t h e r d e f i n i t i o n s o f T ( K ) c a n b e g i v e n b y c h a n g i n g t h e n o r m a l i z a t i o n(2 .2 ) in de f in ing the c l a ss ~a - Fo r example , the projective capacity t r e a t e d b yt h e f ir s t a u t h o r i n [ 1 ] i s e q u i v a l e n t to r e p l a c i n g t h e n o r m a l i z a t i o n ( 2 . 2 ) b y

lo 71z .C r ( l + l z l ) =

log [zx] d2(z )whe re C =Se- ( 1 + ] Z 1 2 ) n + l i s a d i m e n s i o n a l c o n s t a n t ( d 2 d e n o t e s L e b e s g u em e a s u r e) . O t h e r n o r m a l i z a t i o n s m a y b e g iv e n b y r e q u i ri n g t h a t t h e s u m o f t h eabs o lu te va lue s o f the coe f f i c i en t s o f the p o ly no m ia l s Pe i s a t l e a s t 1 , o r them a x i m u m c o e f f ic i e n t is a t l e a s t 1 ( w i t h r e s p e c t t o t h e u s u a l b a s i s o f m o -n o m i a l s ) . T h e s e a l l g i v e c a p a c i t i e s w h i c h a r e b o u n d e d a b o v e a n d b e l o w b y ac o n s t a n t m u l t i p l e o f T ( K ) . T h u s , i t m a k e s l i tt l e d if f e re n c e w h i c h o n e i s u s e d .A n a c c o u n t o f m a n y d i f f e re n t c a p a c it i es a n d t h e i r r e l a ti o n s h i p s i s g iv e n i n


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Com parison of Two Capa cities in Ig" 409[ 1 2 ]. W e h a v e u s e d t h e p a r t i c u l a r n o r m a l i z a t i o n ( 2 .2 ) b e c a u s e o f it s c h a r a c t e r -i z a t i o n i n t e r m s o f a n e x t r e m a l f u n c t i o n , g i v e n b y S i c i a k ( T h e o r e m 9 ] o f 1-1 2]).T h i s i s s t a t e d b e l o w a s T h e o r e m 3 . 2 .

N o t e t h a t a l t h o u g h w e ca ll T ( K ) t h e g l o b a l c a p a c i t y f u n c t i o n , i t i s i n s o m es e n s e r e l a t i v e t o t h e u n i t b a l l b e c a u s e o f t h e c h o i c e o f n o r m a l i z a t i o n i n t h ed e f i n i ti o n o f ~ d- I n f a c t, i f K c o n t a i n s t h e u n i t b a l l t h e n M d ( K ) = I f o r a ll d > 0a s f o ll o w s b y c o n s i d e r i n g t h e c o n s t a n t p o l y n o m i a l P c - 1 i n ~ a . T h e f o l lo w i n g iso u r m a i n r e s u l t .T h e o r e m 2 . 1 ( C o m p a r i s o n T h e o r e m ) . Let K be a compact subset of the uni t bal lin 117". The n

T ( K ) _ < e x p [ - { c . ~l/n] (2.6)- \ c a p ( K ; B ) ] ]where c, is a constant given in (3.12). For each r < l , there i s a cons tan t A=A(r )such that for al l compact se ts K c {I z] < r}

- AT ( K ) > e X P ( c a p ~ ; B ) ) . (2.7)R e m a r k 1. B o t h t h e s e t f u n c t i o n s c a p ( K , f2 ) a n d T ( K ) a r e k n o w n t o b e " g e n e r -a l i z e d c a p a c i t i e s " . H e n c e , t h e e s t i m a t e s o f t h e t h e o r e m a l s o h o l d f o r a l l ca p a c -i t ib le s e t s - in pa r t i cu la r a l l Bore l s e t s .R e m a r k 2 . T h e i n e q u a l i t i e s a r e s h a r p , a t l e a s t a s f a r a s t h e e x p o n e n t s o nc a p ( K ; B ) a r e c o n c e r n e d . F o r i f K = { z: I zl < 5 } , t h e n T ( K ) = e a n d c a p ( K ; B )= c , . H e n c e , e q u a l i t y h o l d s i n ( 2 .6 ) . O n t h e o t h e r h a n d , i f K is a s m a l lp o l y d i s c , K = { ( z l , . . . , z , ) : ] Z l l < _ 6 , Iz a]_ c o n s t , l o g ~ . o s ee t h e l a st i n e q u a l i t y p u t

log + (Izkl~ log+ ]z l l" \ 1 / 2 ] au ( z ) = k= 2 l o g 2 - ~ - - 1 n .l o g ~

N o t e t h a t u < 0 o n B a n d u < - 1 o n K . H e n c ec a p ( K ; B ) > ! ~ " c on st \( d d u ) _ _ > 7 5 - . )l o g ~

T h u s t h e e x p o n e n t i n ( 2 .7 ) c a n n o t b e i m p r o v e d .

3 . P r o o f o f t h e C o m p a r i s o n T h e o r e mT h e p r o o f o f t h e c o m p a r i s o n t h e o r e m f o l lo w s f r o m s o m e s i m p l e q u a n t i t a t i v er e l a t io n s h i p s b e t w e e n e x t r e m a l p l u r i s u b h a r m o n i c f u n c t i o n s r e l at e d t o t h e c a p a c -


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410 H. Alexander and B.A. Tay lori t i e s . F o r t h e r e l a t i v e c a p a c i t y , t h e e x t r e m a l f u n c t i o n i s

U~ (z) = U~ (z, f2) = l im su p UK(0, (3.1)~ z

t h e u p p e r s e m i c o n t i n u o u s r e g u l a r i z a t i o n o f t h e e n v e l o p eU K ( z ) = s u p { v ( z ) : v e P ( f 2 ) , v < - 1 o n K , v < 0 o n f2 }. (3 .2 )

T h e m a i n p r o p e r t i e s o f U ~ a r eU * ~ P ( f 2 ) , - 1 < U * < 0 , l ir a U ~ (z ) = 0 . (3 .3 )

z ~Y 2( d d ~U * ) " = 0 o n f 2 \ K (3 .4 )

U * = - 1 o n K , e x c e p t o n a s e t o f ( r e la t iv e ) c a p a c i t y z e r o (3 .5 )c a p ( K , ( 2 ) = ~ (dd ~U*)" = ~ (dd ~ U~)" . (3.6)

$2 K

P r o o f s o f t h e s e f a c ts a r e g i v e n i n P r o p . 5 .3 o f [ 3 ] .T h e o t h e r e x t r e m a l f u n c t i o n i s t h e o n e i n t r o d u c e d b y S i c i a k [ 1 1 ]

u~: (z) = l im su p uK(0 (3.7)w h e r e

uK(z)=sup{v(z): v~P(C"), v(z) 0 , A ( e ) i s s o l a r g e t h a t u ( z ) < 0 o n K , a n d f 2 i s a l a r g e b a l l , t h e n w e c o n c l u d e


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Com parison of Tw o Capacities in C" 411~(ddCu~:)" ~ . (3.13 )B e c a u s e o f S ic i a k 's t h e o r e m , t h i s is e q u i v a l e n t t o (2 .6 ).

F o r t h e p r o o f o f t h e o t h e r p a r t , (2 .7 ), o f t h e c o m p a r i s o n t h e o r e m , t w oa d d i t i o n a l f a ct s a re n e e d e d . T h e f ir st is a n e s t i m a t e f o r th e m a s s o f t h e m e a s u r e(ddCu)" w h i c h is s h a r p e r t h a n t h e o r i g i n a l e s t im a t e o f C h e r n , L e v i n e , a n dN i r e n b e r g . ( S e e e .g . [ 6 ] , S e c t . 2 a n d [ 3 ] , T h e o r e m 2.4 .)L e m m a 3 .3 . L e t c o - {Iz-zol


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412 H. Atexan der and B.A. TaylorL e m m a 3.4. L et v l , v 2 . . . . G 6 P(O)c~L~176 and le t ~ be a c losed pos i t ive (s, s )current on (2 wi th s + k < n. Then there is a w el l de f ined c losed po s i t ive (k +s , k+ s ) c u r r e n t 4 ) = O /x dd C va /x . . . / x d d C G o n fJ . I f K ~ c o ~ f 2 t h e re i s a c o n s ta n t C= C(K , co, O) ( independent of O) such tha t

kI S O A P . - ( s + j - - < c S o A P . - , 9 ] 7 [ I Iv ~ l lo ~ . ( 3 . 1 4 )K co j = l

P r o o f o f L e m m a 3 .4 . D e f i n e 4 b y i n d u c t i o n o n k o n a C ~ ( f 2 ) ( n - ( s + k ) , n - ( s+ k)) fo rm )~ b y IAX=fvkAddCzw he r e qS~ = 0 A d d ~ v , / x . . . A d d C G _ , i s g i v e n b y i n d u c t i o n . T h e v e r i f i c a t i o n t h a t0 is a p os i t i ve c l o s e d ( s + k , s + k ) cu r r e n t i s j u s t a s i n [ - 4 ], P r op os i t i o n 2 .9 .

T h e e s t im a t e (3 .1 4) is o b t a i n e d b y i n d u c t i o n o n k . F o r k = l , l e t0 < o- e C ~ (co), o - - 1 on K. T he nl Y ~ ' A d d ~v 1 A fl.- (k + 1)J < I ( . ~ O / ' dd~ v l A a f i ._(k + ) l

= l y v l A ~dd~o-A f i ._ (k+ l )[(o

N o w t h e g e n e r a l c a s e fo l lo w s f r o m t h is .

K K

w h e r e q ~ = O / x d d ~ v a / x . . . / x d d ~ G _ l . B y t h e k = 1 c a s e t h i s l a s t i n t e g r a l i s d o m i -n a t e d b y

C IIG[Io,~ ~ q~ A f i , - (~+ k- 1 ) ( wh e r e K _ _ co ~ c o l ~ f 2 )C01

k--1< C [[vk[Io, ~ ~ A f t , _ , [ I I1@~, (by in du ct i on h yp oth es i s ) .

co j=lThis g ives (3 ,14) .P r o o f o f L e m m a 3 .3 . W e c a n a s s u m e Z o = 0 . I n L e m m a 3 .4 w e ta k e ~ = d d ~ ua n d k = n - 1 w i t h G = u fo r l < - k < n - 1 . Appl y i ng ( 3 .14 ) we wi l l be f i n i s hedo n c e w e s h o w t h a t 5 d d ~ u A f i , _ , < C ( - u ( O ) ) . T o s e e t h i s f i r s t a p p l y J e n s e n ' sf o r m u l a : o ,

1u (O ) + N ( r ) = ~ u ( r~ ) d a (o : )o-2n-- 1 l a[= i

w h e r e do - i s s u r f a c e a r e a m e a s u r e o n t h e u n i t s p h e r e , o - 2 . _ ~ = y d o - , a n dw h e r e h i= 1- i n ( t )N ( r ) - o ~ d t


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414 H. Alexander and B.A. TaylorI n t h e s p e c i a l c a s e o f t h e b a l l s, w e c a n t a k e

p(z) = - 1 + [ log + (Izl /P)] [ log(R/p)] - 1t o o b t a i n

6 = [ log 1/p] [ logR/p] -1W e c a n n o w p r o v e t h e o t h e r h a lf , (2 .7 ), o f t h e c o m p a r i s o n t h e o r e m .

P r o o f o f (2.7) . Let K c { l z l < r } , r < l , K c o m p a ct , a n d a = m a x { u * ( z ) : I z l < l }= u ~ ( Z o ) , Iz o[ = 1. I f a = + o% t h e n K h a s c a p a c i t y z e r o f o r b o t h c a p ( K ; ) a n dT ( K ) s o t h e r e i s n o t h i n g t o p r o v e . T h u s , w e a s s u m e a < + o o . T h e n u ~ ( z ) - a < Oo n [z[__


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Comparison of Two Capac ities in 112 415Then there ex is ts a sequence o f polynomials {pd(z )} for in f in i te ly many in tegers ds uch tha t

(i ) pa(z) has degree l,(iii) sup [pa(z)[ = 1,I z [ = l

( i v ) I p a ( z ) l < e x p ( - C A 1/" . d) for a l l z in Kw h e r e C = C(r ) i s a cons tant depending only on r< 1 and n .P r o o f . L e t U ~ b e t h e r e l a t i v e e x t r e m a l f u n c t i o n f o r K . T h e n U ~ > u / A , soU ~ ( O ) > - 1 / A . T h u s b y ( 3. 6) a n d L e m m a 3 .3 t h e r e is a c o n s t a n t C = C ( r ) s u c ht h a t c a p ( K ; ]z[ < 1 ) < C / A . (4.1)B y ( 2.7 ) o f t h e C o m p a r i s o n T h e o r e m w e h a v e

C < [ca p (K ; Iz[ < 1 3 1 / n . (4.2)

H e n c e T ( K ) < e x p ( - C A 1 / " ) . C h o o s e p o l y n o m i a l s p a e N d s u c h t h a t LlpallB=la n d ]lpalhK=Ma(K) f o r e a c h d > 0 . T h e n s i n c e i n f l l p a l k ~ : / a = r ( K ) < e x p ( - C A X / " ) ,w e h a v e I lpa l lK< exp ( - 89 fo r in f in i t e ly ma n y d ' s ; th i s g ives ( iv ). F in a l ly ,( ii ) fo l lows beca use

1 lo g IPal 1.

C o n s i d e r a c o n v e x , i n c r e a s i n g f u n c t i o n h ( x ) d e f i n ed f o r - o o < x < + oo s u c ht h a t h ( 0 ) = 0 a n d i Ih(x)]d x-oo [x] 1+1/" < + o o . (4.3)F o r e x a m p l e , i f 0 < ~ < 1/n, t h e n

x ~ 1 [ ( 1 - x ) ~ - 1] x < 0h f x ) = x > 0i s s u c h a f u n c t i o n . A n o t h e r e x a m p l e , t h e o n e u s e d b y E1 M i r , is

x < 0x > 0 .

T h e o r e m 4 .2 . Let h be as above . Le t v be psh on { I z ] < l } wi th v < 0 and v ( 0 ) =- 1 . Then there ex is ts a fun c t io n u psh on C" such that

(i ) u = O ( l o g l z [ ) a s z ~ o o ,(ii) u ( z ) < h ( v ( z ) ) f o r Iz l< 89


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4 1 6 H . A l e x a n d e r a n d B . A . T a y l o r

R e m a r k . O n p . 7 4 o f [-7 ], i t is s t a t e d t h a t t h e t h e o r e m is t r u e w i t h h ( x ) =- [ - x ] ~/". H o w e v e r , i t d o e s n o t a p p e a r t h a t t h e l i n e o f a r g u m e n t u s e d i n [ 7 ] w i lly i e l d t h is r e s u lt . I t s e e m s t o b e a n i n t e r e s t i n g p r o b l e m t o d e c i d e i f t h e " l / n " isr e a l ly n e c es s a r y . C o u l d t h e t h e o r e m h o l d w i t h h ( x ) = - ( - x ) l - ~ ? A n e x a m p l eo f E 1 M i r s h o w s t h a t s > 0 is n e e d e d ; i .e ., o n e c a n n o t a l w a y s f i n d u s u c h t h a tu(z)__< v(z) fo r [z l < 89Proo f . L e t ( 9A d e n o t e t h e o p e n s e t { z : [ z [ < 8 9 v ( z ) < A } a n d l e t u ~ d e n o t e t h ee x t r e m a l f u n c t i o n o f (9A . T h u s , u * ( z ) < l o g Iz l + O ( 1 ) , I z [ - - , o o , a n d u * ( z ) = 0 f o rz ~ (g A , b e c a u s e (9A is o p e n . S e t c t( A ) = s u p { u ] ( z ) " [ z[ = 1 } a n d VA(Z = U~-- ~(A).W e c l a i m t h a t c ~ (A ) > c o n s t . A 1/" (4 .4 )

vx( z ) < l o g + [z [ ( 4 .5 )v ~ ( z ) = - c ~ ( A ) , z ~ ( 9 A (4 .6 )

v a ( z ) d a ( z ) > c o n s t . ( 4 . 7 )h i = 2A s s u m i n g t h a t ( 4 . 4 ) - ( 4 . 7 ) h o l d , i t i s e a s y t o s e e t h a t t h e f u n c t i o n

0u ( z ) = ~ v 4 ( z ) h ' ( A ) A - 1 / ' d A

- oo

h a s t h e d e s i r e d p r o p e r t i e s . F o r , i f l z I < 8 9 a n d v ( z ) < C , t h e n0U(Z) ~ ~ t )A(Z h' (A)A- 1IndAC0= ~ a ( A ) h ' ( A ) A - 1 / ' d AC

0< - c o n s t . ~ h ' (A) dAC

= - c o n s t . [ h ( 0 ) - h ( C ) ]= c o n s t , h(C) .

B e c a u s e t h i s h o l d s w h e n e v e r v ( z ) < C a n d h i s c o n t i n u o u s , i t f o l l o w s t h a tu(z )


11/11

C o m p ar i s o n o f T w o C ap ac i t ie s i n G " 4 17References

1 . Alexander , H. : Pro jec t ive capaci ty . Ann . o f Ma th . Stud ies I00 , pp . 3 -27 . C onferenc e on S evera lComplex Var iab les . Pr inceton : Pr inceton Univ . Press (1981)2. Alexand er, H .: A note on pro ject iv e capacity. C anad . J . M ath. 31, 1319-1329 (1982)

3 . Bedford , E . , Tay lor , B .A. : A new cap aci ty fo r p lur i subharm onic funct ions . Ac ta Ma th . 1 49 , 1 -40 (1982)4 . Bedford , E . , Tay ior , B .A.: The D i r ich le t p rob lem for a com plex Mo nge -A mp ere equat ion .Invent . Math . 37 , 1 -44 (1976)5. Cegrell , U.: Con struc t ion o f capacit ies . Pre prin t6 . Chern , S .S., Lev ine , H. , Ni ren berg , L . : In t r ins ic norm s o n a c om plex mani fo ld . Glo bal Analys is ,pp . 119-139 . Toky o: U niv . o f To kyo Press 19697 . E 1 M i r , H . : Fo n c t i o n s p l u r i s o u s h a rm o n i c e t en s em b l es p o l a i r e s . Sem i n a i r e P i e r r e L e l o n g -H en r iSk o d a 1 9 78 /7 9 . L ec t u re N o t es i n M a t h . 8 2 2 . B e r l i n -H e i d e l b e rg -N ew Y o rk : Sp r i n g e r 1 9 798 . Josefson , B . : On the e qu ivalen ce betw een local ly po lar and g lobal ly po lar fo r p lur i subharm onicfunct ions on G" . Ark . Mat . 16 , t09-115 (1978)9 . M olzo n , R . , Sh if fman , B . , Sibony , N . : A verag e growth es t imates fo r hyperp lane sect ions o f

ent i re ana lyt ic sets . M ath. Ann. 257 , 43-59 (1981)10 . Sibony , N. , Wong, P . -M. : Some resu l t s on g lobal analy t i c se t s . Seminai re Pier re Lelong-Henr iSkod a 1978/79 , 221-237 . Lec ture Notes in Math . 822 . Ber l in -H eidelberg -Ne w Yo rk : Sp r inger

197911. Siciak, J . : Ex t rem al p lur i sub harm onic funct ions in C" . Ann. P o lon . M ath . 3 19 , 175-211 (1981)12 . Sic iak , J . : Ex t remal p lur i subharmonic funct ions and capaci t i es in G" . Prepr in t 198113. Za har juta, V .: Tra nsfini te dia me ter, Ceb ysev constants, and cap aci ty for com pa cta in 112". M ath.USSR-Sb. 25, 350-364 (1975)

Received October 19 , 1983

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Comparison of Two Capacities