Approximation of Plurisubharmonic Functions

Approdmation Theorg and Functional AnaZysds J.B. Prolla l e d . ) 0iVort.h-Holland PA Zishing Company, 1979

APPROXIMATION OF PLURISUBHARMONIC FUNCTIONS

PHILIPPE NOVERRAZ U n i v e r s i t g d e Nancy I Ma t h g m a t i q u e s 5 4 0 3 7 NANCY CEDEX, F r a n c e

If

U

i s a n open and c o n n e c t e d s u b s e t o f

a , an a p p l i c a t i o n

s i o n a l l o c a l l y convex v e c t o r s p a c e E o n (resp. [ -

a,

+ 1

an i n f i n i t e

dimen

-

f :U+a

i s s a i d t o b e h o t o m o h p h i c ( r e s p . p.&.U&ubhahmonLc) i f

)

a)

f

b)

t h e r e s t r i c t i o n of

i s continuous ( r e s p . upper semicontinuous)

f

t o any f i n i t e d i m e n s i o n a l

subspace

i s holomorphic ( r e s p . plurisubharmonic). L e t us d e n o t e by

(resp. P ( U )

H(U)

,

P,(U))

t h e s e t o f holomor-

p h i c ( r e s p . p l u r i s u b h a r m o n i c , p l u r i s u b h a r m o n i c a n d c o n t i n u o u s ) funct i o n s on

U.

If

K i s a compact , s u b s e t of

= Ix E

(U)

In

an,

n

2

2,

1) Any v i n

u,

U

,

l e t u s d e n o t e by

v ( x ) 5 s u p v , wv E P ( U ) ) . K

t h e f o l l o w i n g r e s u l t s are w e l l known ( 3 ) : P(U) i s t h e p o i n t w i s e d e c r e a s i n g l i m i t

of

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 i n a s t r i c t l y smaller o p e n

2)

(ie

U'

of

If

U i s pseudo-convex

compact

U

K of

Cm

set

d(U', C U ) > 0).

U)

then

(ie Kp(u)

343

Kp(U)

-

i s compact i n

Kpc(U)

.

U

f o r any

344

NOVERRAZ

If

U i s pseudo-convex,

compact s u b s e t of al,

..., a j

Iv

If

then f o r v i n

U there e x i s t

fll..

Pc(U)

. , f 7.

,

in

E >

0 and K

and

H(U)

p o s i t i v e numbers such t h a t

-

K = KH(U)

sup ai l o g I f i

i

i s compact i--a pseudo-convex open set

U,

then any holomorphic f u n c t i o n i n a neighborhoodof Kcan be a p p r o x i m a t d u n i f o r m l y on K by elements If

u

H(U).

-

A

and U' are pseudo-convex, U C U' t h e n K H ( U ) , = K H ( " , )

f o r any compact s u b s e t of in

of

U i f and o n l y i f

H(U') i s d e n s e

H(U) f o r t h e compact open topology.

P r o p e r t i e s 31, 4 ) and 5 ) have been g e n e r a l i s e d t o l a r g e r c l a s ses of l o c a l l y convex s p a c e s w i t h Schauder b a s i s i n c l u d i n g

Banach

spaces ( 6 ) .

8, c o n d i t i o n

W e s h a l l i n v e s t i g a t e c o n d i t i o n s 1 and 2 . I n

i s o b t a i n e d by r e g u l a r i s a t i o n ( i e c o n v o l u t i o n ) of

se-

v by a D i r a c

quencer so it i s n a t u r a l t o c o n s i d e r s o m e measure.

1)

For t h e sake

of

s i m p l i c i t y w e s h a l l c o n s i d e r h e r e only ( i n f i n i t e dimensional) Banach spaces and Gaussian measures f o l l o w i n g Gross ( 5 ) . I t i s w e l l known t h a t i n a Banach s p a c e E there are no

s t i t u t e t o t h e Lebesgue measure t h a t means t h e r e does n o t e x i s t

sub-

a

measure i n v a r i a n t by t r a n s l a t i o n s o r r o t a t i o n s . A Gaussian measure l.~ on E can be c h a r a c t e r i z e d as follows: there e x i s t s an H i l b e r t space H

v

d e n s e l y and c o n t i n u o u s l y imbeded i n E such t h a t

u

t h e c y l i n d r i c a l Gauss measure on t h e c y l i n d r i c a l s e t s of

arises H

1-I

.

from The

t r i p l e t ( H p , i , E ) is c a l l e d an a b s t r a c t Wiener space. The f o l l o w i n g p r o p e r t i e s hold:

1)

L e t be

T in

P(E,E), i f

and i s u n i t a r y t h e n

p

T restricted t o H

i s i n v a r i a n t by

T

i s i n P(H H ) I-r P I !J ( i e pT-' = 11).

APPROXIMATION OF PLURISUBHARMONIC FUNCTIONS

L e t be

2)

clx(A)

= p(x

+

,A

A)

346

u

Bore1 i n E l t h e n

and p x are

e i t h e r e q u i v a l e n t o r o r t h o g o n a l , t h e y a r e e q u i v a l e n t if and only i f

x belongs t o

H

P

.

W e have t h e f o l l o w i n g Lemma:

16

LEMMA 1:

i n a Gaubbian meabutre on E and

p

hatrmonic 6uncLion i n an open n u b s e t U

doh

E we h a v e

Suppose t h a t v i s bounded from above i n t h e b a l l B ( x , r ) , t h e

mapping

x

+

eiex

- invariant,

but

a pLuhisub-

r bmale enough.

PROOF:

Te

06

i d v in

V(X)

induce& a

u n i t a r y mapping T e on H

P

I

so

u

is

and w e have

5

v(x

+

y e i e )do

.

The r e s u l t f o l l o w s from

Fubini

t h e o r em. L e t us note

P R O P O S I T I O N 1: 1)

A(v,x

A(x,v r )

p(r)A

2)

v(x) = l i m r =O

3)

A(x,v,r)

LA apLutribubhatrmonic d u n c t i o n 06

a c nwex and inctreasing dunc-tion

i n in6initely

any x i n E t h e f u n c t i o n y = 0.

and

0 6 Log r .

A(x,V,r).

L e t us r e c a l l t h a t a f u n c t i o n

entiable a t

x

y

+

cp

H

u - di66etrenZiable. 9 is H -differentiable

u

( x + y ) , d e f i n e d on

Hcl

I

if

for

is differ

-

NOVERRAZ

346

PROOF:

1) i s a consequence of t h e f a c t t h a t p l u r i s u b h a r m o n i c

func-

t i o n s depending o n l y from I1 x I) are l o g a r i t h m i c a l l y convex. S i n c e v i s upper s e m i c o n t i n u o u s , f o r any

2)

5 v(x) +

v ( x + y)

w e have

II y II 5 r X f E hence

for

E

> 0

E

Is a consequence of a r e s u l t of Gross (5).

3)

L e t us n o t i c e t h a t , u n l e s s

v i s continuous, A(v,x,r)

i n g e n e r a l a continuous f u n c t i o n of

is

not

x.

A s a consequence of 2 ) and 3 ) w e have:

A p l u t i b u b h a h m a n i c dunc-tion v

PROPOSITION 2:

wine l i m i t

a nequence

06

06

i-6

L a c a l l y ,the p o i n t -

i n , 5 i n i t d y H - d i d , 5 e h e n t i a b l e @~L5ubha/unonic

iuncztio nA . T h e r e i s a n o t h e r way t o a p p r o x i m a t e bounded f u n c t i o n s : l e t p be a Gaussian measure o f p a r a m e t e r

vt{ll x 11 2

c1

> 01

+

f u n c t i o n Ptf ( x ) = f Ptf

0

i,

if f (x

t

+

+

t > 0 , then

t h e n Gross ( 5 ) h a s proved t h a t

0

1.1

/f(x)

-

f

is uniformly continuous

f u n i f o r m l y on E .

tends t o

For

the

y)ut(dy) is i n f i n i t e l y H -differentiable if

i s bounded and m e a s u r a b l e . Moreover i f

PROOF:

t and

= 1

vt(E)

E

f(y)I 5

< 0, t h e r e i s E

.

<

2 E

n

such t h a t

if

Ix

- yi 5

t < t E .

rl

i m p 1i e s

APPROXIMATION

If

f

OF PLURISUBHARMONIC FUNCTIONS

347

i s only continuous, t h e n t h e convergence of

Ptf

to

f

i s u n i f o r m on e v e r y compact s u b s e t . I t is a l s o w e l l known (1) t h a t t h e r e e x i s t

ceding r e s u l t gives a

separable

f u n c t i o n s are n o t

Banach s p a c e s s u c h t h a t t h e bounded and 'C i n t h e space o f uniformly

several

dense

c o n t i n u o u s and bounded f u n c t i o n s . T h e p r e

uniform a p p r o x i m a t i o n by H-inf i n i t e l y differen-

t i a b l e f u n c t i o n . F o r 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 t h i s k i n d of approxim a t i o n g i v e s more o r less t h e same r e s u l t as p r o p o s i t i o n 2 . Now w e s h a l l s t a t e t h e f o l l o w i n g p r o p o s i t i o n :

PROPOSITION 3:

Let

U

be a pheudo-convex open b u b b e t o d aBanach bpace

v be a pLuhinubhahmonic

E and L e t

pointwide l i m i t

06 a

d e c h e a n i n g oequence a d con,Chow ( i n

p ~ u h i n u b h a h m o n i c 6unct i onh i n U

Let

COROLLARY: E,

60%

v

6unctian on U , t h e n

i n

the

duct L i p b C c k i t Z )

.

U b e a pbeudo-convex o p e n b u b n e t 0 6 a Bunach

Apace

then

K ad

any compact Aubbet

U.

F o r t h e p r o o f w e s h a l l follow a n u n p u b l i s e d p a p e r o f

C.Herves

and M. E s t e v e z ( 2 ) . They f i r s t g e n e r a l i z e i n t h e Banach case a n i d e a of

( 3 ) : L e t f be a l o w e r s e m i - c o n t i n u o u s f u n c t i o n bounded from

above,

t h e n for any i n t e g e r k d e f i n e

f

k

( x ) = i n f [ kll x

I t i s e a s y t o show t h a t

Moreover

f

Y

fk-l- < fk ( f

-

yll

and

+

f ( y )1

.

1 f k ( x ) - f k ( y ) I -<

i s t h e p o i n t w i s e l i m i t of t h e s e q u e n c e

fk

.

k I1 x

-

yll.

NOVERRAZ

348

If U

U i s pseudo-convex and v i s a 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 n

w e t a k e t h i s approximation sequence of t h e f u n c t i o n f d e f i n e d by

e x p ( - v)

i n U and z e r o o u t s i d e

and i f w e c o n s i d e r t h e norm

U

If we s t a t e

+ Iw 1

kII z 1 I

on

E x 4

, we

subharmonic i n

U,

moreover

v = l i m [-log f k 1 k

proved.

xo

B

cp(u)

.

there i s v i n

P ( U ) such t h a t

u(xo) > a

P r o p o s i t i o n 3 i m p l i e s t h a t t h e r e i s a d e c r e a s i n g sequence

> sup

K

(vn)

v. in

, hence: K C { X E U, v ( x ) < a ) =

U

n

{x E U, vn(x) < a ] .

vn+l < v

S i n c e K i s compact and

t h e r e i s an index

p

that

We have v (x 1

P

hence

is

Proposition 3

It i s s u f f i c i e n t toprove t h a t

PROOF OF THE COROLLARY:

Pc (U)

have

i s a pseudo-convex domain it follows t h a t -log fk i s p l u r i -

Since

If

.

O

2 v(xo)

xo does n o t belong t o

*

>

c1

sup v K P The c o r o l l a r y i s proved.

such

APPROXIMATION OF PLURISUBHARMONIC FUNCTIONS

340

REFERENCES

11

R. BONIC and J. FRAMPTON, Smooth functions on Banach manifold, J. Math. Mech. 15(1966) , 877 - 898.

2

1

M. ESTEVEZ and C. HERVES, Sur une proprigts de l'enveloppeplurisousharmonique dans les espaces normds, preprint.

3

1

L. HORMANDER, An i n t h o d u c t i o n t o campeex a n a l y d i b , VanNostrand 1966.

14 1

J. P. FERRIER and N. SIBONY, Approximation ponddrde sur une sous-vari6tG totalement r6elle de an, Ann. Inst.Fourier 26 (1976)I 101 - 115.

5 1

H. H. KUO, Gaudhian meabuhe i n Banach d p a c e b , Springer Lecture Notes 464.

61

Ph. NOVERRAZ, Approximation of holomorphic or plurisubharmonic functions in certain Banach spaces. Phoc. on ' I n d i n i A e Dimen&Lonad Holomohphy, Springer Lecture Notes 364,p. 178-185.

1

Ph. NOVERRAZ, P n e u d o - c o n v e x i t e , c o n v e x i t e p o l y n o m i a l e etdomainen d ' h o l o m o h p h i e , North-Holland Publishing Cn., Amsterdam, 1972.

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