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## with Several Poles

BSTRACT

C2,1

Cn

C1,1

NTRODUCTION

Cn

p1

. . .

pk∈ Ω

µ1

. . .

µk>

g=

B,

B =

v∈ PSH(Ω) | v <

,

z→pi

(u(z)− µi

### log

|z − pi|) < ∞, i =

, . . . , k .

g ∈ B

g

### is a maximal plurisubharmonic (psh) function in

Ω \ {p1, . . . , pk}

Mg= πn n

n

X

i

µiδpi

M

u Mu=

2u

∂zi∂z

j

! ,

Mu

u∈ PSH(Ω)

u

### In this paper we want to show the following regularity result.

335

Indiana University Mathematics Journal c , Vol. 50, No. 1 (2001)

(2)

C2,1

### smooth and strictly pseudoconvex. Then

g∈ C1,1(Ω \ {p1, . . . , pk})

|∇2g(z)| ≤ C

### min

i|z − pi|2, z∈ Ω \ {p1, . . . , pk},

C

p1

. . .

pk

µ1

. . .

µk

C2

C2

k=

C1

α <

C1,1

C1

k≥

k=

### 1, as instead of the lemma from [Bł3] we use a holomorphic mapping

z7 -→ z + (z1− p11)· · · (z1− pk1) (a1− p11)· · · (a1− p1k)h

### (in appriopriate variables given by Lemma 3.2 below), which for

a∉ {p1, . . . , pk}

h∈ Cn

pi

a

a+ h

g

(3)

g

### is continuous as a function defined onthe set

{(z, p1, . . . , pk, µ1, . . . , µk)∈Ω × Ω

### ¯

k× (R+)k| z ≠ pi≠ pj

i≠ j},

z∈ ∂Ω

g

=

ψ∈ PSH(Ω)

ψ <

z→∂Ωψ(z)=

z→∂Ω

|g(z)|

(z, ∂Ω) <∞.

|∇g(z)| ≤ C

### min

i|z − pi|, z∈ Ω \ {p1, . . . , pk},

C

p1

. . .

pk

µ1

. . .

µk

### Notation. If

z= (z1, . . . , zn)∈ Cn

xi=

zi

yi=

zi

ζ∈ Cn

|ζ| =

ζmu(z)

m

u

ζ

z

uxi = ∂u

∂xi

, uyi= ∂u

∂yi

, ui= ∂u

∂zi

, ui¯= ∂u

∂z

i

.

|∇u| ≤ f

D⊂ Cn,

f

D

u

|∇u|

ddcu≥ ddc|z|2

u− |z|2

C1

C2

. . .

ASIC ESTIMATES

Cn

p1

. . .

pk∈ Ω

µ1

. . .

µk>

R

r

m

M

i

j=

. . .

k

Ω ⊂ B(pi, R),

B(p

i, r )⊂ Ω

B(p

i, r )B(p

### ¯

j, r )= ∅, m≤ µi≤ M.

(4)

g

X

i

µi

|z − p

i|

R ≤ g(z) <

, z∈ Ω, µi

|z − p

i|

R − (k −

)M

R

r ≤ g(z) ≤ µi

|z − p

i|

r , z∈B(p

i, r ).

ε

< ε < r

ε

= Ω \[

i

B(p

i, ε),

gε

=



v∈ PSH(Ω) v <

, v ¯

B(pi,ε)≤ µi

ε

r, i=

, . . . , k,

 .

gε(z)≤ µi

{|z − pi|, ε}

r , z∈B(p

i, r ),

### (2.1)

gε ∈ PSH(Ω), g≤ gε

(r /ε)

(R/ε)+ (k −

)(M/m)

(R/r )g

ε,

gε ↓ g0

= g

ε↓

### 0, and the convergence is locally uniform in

Ω \ {p1, . . . , pk}

C

r0

k

r

R

m

M

< r0 ≤ r

ε

< ε < r0

### we can find

v ∈ PSH(Ω) ∩ C(Ω)

ddcv≥ ddc|z|2

v=

i=

. . .

k

µi

ε

r ≤ v(z) ≤ µi

|z − p

i|

r

### if

ε≤ |z − pi| ≤ r .

w(z)

=X

i

µi

### log

|z − p

i|

R + |z − p1|2− R2,

w <

ddcv ≥ ddc|z|2

w < µi

(ε/r )

∂B(pi, ε)

z∈ ∂B(pi, r )

w(z)≥ kM

r

R + r2− R2> µi

### log

ε

r + |z − pi|2− ε2,

(5)

ε

m

ε

r − ε2< kM

r R − R2.

χ

R → R

C

χ(t)=

, t≤ −

,

χ(t)= t, t ≥

,

≤ χ0(t)≤

, t∈ R, χ00(t)≥

, t∈ R.

x

y∈ R

fj(x, y)

= x +

jχ(j(y− x)),

fj(x, y)=

{x, y}

|x − y| ≥

j.

u

v

ddcu

ddcv≥ ddc|z|2

ddcfj(u, v)≥ (

### 1

− χ0(j(v− u)))ddcu+ χ0(j(v− u))ddcv≥ ddc|z|2.

ψ

j

A

v(z)=



 fj



w(z), µi

### log

ε

r + |z − pi|2− ε2



, z∈S

iB(p

### ¯

i, r ), fj(w(z), Aψ(z)), z∈Ω \

SiB(p

i, r )

k=

r0= r

gε → g

ε

ε

gε

Ω × {(p

### ¯

1, . . . , pk)∈ Ωk|

### dist

(pi, ∂Ω) > ε, |pi− pj| >

ε

i≠ j} × (R+)k.

pi,j→ pi

µi,j→ µi

j→ ∞

i=

. . .

k

gεj

=



v∈ PSH(Ω) v <

### 0

, v B(p¯ i,j,ε)≤ µi,j

ε r

 .

(6)

< ε < r0

j

### we have lim

z→∂B(pi,ε)gjε(z) = µi

(ε/r )

z→∂Ωgjε(z)=

gjε

gjε→ gε

j→ ∞

c >

z∈B(p

i, ε)

j

gjε(z)≤ µi,j

{|z − pi,j|, ε}

r ≤ µi,j

ε+ |p

i− pi,j|

r ≤ µi

ε r + c,

z∈B(p

i,j, ε)

gε(z)≤ µi

{|z − pi|, ε}

r ≤ µi

ε+ |p

i− pi,j|

r ≤ µi,j

ε r + c.

### Thus for those

j

gε− c ≤ gεj ≤ gε+ c

Ω,

gε

≤ ε < r

≤ δ ≤

gε,δ

=

### sup

{v ∈ PSH ∩ L(Ω) | v ≤ gε, Mv≥ δ

ε}.

gε,δ

ε

δ

### . We also have

gε+ δ(|z − p1|2− R2)≤ gε,δ≤ gε.

gε,δ ∈ PSH(Ω)

Mgε,δ = δ

ε

< ε < r0

gε,δ

C

< ε < r0

< δ≤

gε,δ∈ C(ε)

### Proof. We use standard procedures. Let

B = {v ∈ PSH(Ω) | v ≤ gε, Mv≥ δ

ε}.

vj ∈ B

(gε,δ) = (

jvj)

u

u

wj =

{v1, . . . , vj}

Mwj ≥ δ

ε

wj ∈ B

wj ↑ (gε,δ)

M(gε,δ)≥ δ

ε

gε,δ∈ PSH(Ω)

Mgε,δ ≥ δ

ε

Mgε,δ= δ

ε

(7)

< ε < r0

ψ∈ PSH(Ω) ∩ C(Ω)

ψ=

Mψ≥

A

Aψ≤ gε,δ

Ω.

v

### . Then

v(z)≤ gε,δ(z)≤ µi

|z − p

i|

r

### if

ε≤ |z − pi| ≤ r .

h∈ Cn

z∈ Ωε

|h| <

(z, ∂ε) <

|h|

### we have

|gε,δ(z+ h) − gε,δ(z)| ≤ C(|h|).

gε,δ

gε,δ(·+h)

z

(z, ∂Ωε) >|h|

h

0C(|h|) =

,

gε,δ

δ=

≤ δ ≤

z→∂Ω

|g0(z)|

(z, ∂Ω) ≤ B < ∞.

ε

|∇gε,δ(z)| ≤ C

### min

i|z − pi|, z∈ Ωε,

C

n

k

R

r

m

M

B

δ≤

ρ >

### 0 be such that

−gε,δ(z)≤ −g0(z)≤

B

(z, ∂Ω)

(z, ∂Ω) ≤ ρ.

(8)

h

### su fficiently small

gε,δ(z+ h) − gε,δ(z)≤

B|h|

(z, ∂Ω) = |h|,

µi

ε

r ≤ gε,δ(z)≤ µi

|z − p

i|

r

### if

ε≤ |z − pi| ≤ r ,

### we have

gε,δ(z+ h) − gε,δ(z)≤ µi

|z − p

i+ h|

ε

µi ε |h|,

### if

z∈ ∂B(pi, ε+ |h|), i =

, . . . , k.

### From the comparison principle we get

gε,δ(z+ h) − gε,δ(z)≤

 B,M

ε



|h|

|h| ≤

{ρ,

(z, ∂ε)},

|∇gε,δ| ≤C1

ε

ε.

Ce = eC(k, n)

p1

. . .

pk∈ Cn

### ,

a∈ Cn\ {p1, . . . , pk}

Cn

### sothat

|a − pi| ≤ eC|a1− p1i|, i=

, . . . , k.

S

Cn

b∈ S

### such that

|a − pi| ≤ eC|ha − pi, bi|, i=

, . . . , k,

* a− pi

|a − pi|, b+

Ce.

Ce

=

Skf,

(9)

### where

f (ζ1, . . . , ζk)

=

b∈S

i |hζi, bi|

Sk

f >

Sk

ζ1

. . .

ζk∈ S

Ki

### :

= {b ∈ S | hb, ζii =

}

i=

. . .

k

SiKi ≠ S

b∈ S \S

iKi

### we have

f (ζ1, . . . , ζk)≥

i |hζi, bi| >

.

a ∈ Ωε

P (λ)

### :

= (λ − p11)· · · (λ − p1k),

|P(z1)|

|P(a1)| ≤ C2

i|z − pi|

i|a − pi| C3

i|a − pi|, z∈ Ω.

h

00

### :

=



z∈ Ω z + P (z1) P (a1)h∈ Ω



0

= Ω00\[

i

B(p

i, ε+ ε0),

ε0=

{ε, r − ε,

(a, ∂Ωε), ρ}.

v(z)

### :

= gε,δ



z+ P (z1) P (a1)h



+ C4

### min

i|a − pi|(|z − p1|2− R2)|h|,

C4

Mv≥

+P0(z1) P (a1)h1

2δ+ C4

### min

i|a − pi||h| ≥ δ.

(10)

z∈ ∂Ω00

v(z)− gε,δ(z)≤

B

(z, ∂Ω) ≤

B C3

i|a − pi||h|,

### whereas for

z∈ ∂B(pi, ε+ ε0)

v(z)− gε(z)≤ C1 ε

|P(z1)|

|P(a1)||h| ≤ C1C2 ε+ ε0

ε

i|a − pi| C5

i|a − pi||h|.

### Therefore, the comparison principle gives

gε(a+ h) − gε(a)≤ C6

i|a − pi||h|

|h| ≤ ε0

i|a − pi| C3 ,

### 4. E

STIMATES OF THE SECOND DERIVATIVE

|∇2gε,δ|

ε

δ

∂Ωε

Cn

ψ

C

ddcψ≥ ddc|z|2

A

a

### such that

|ψ|, |∇ψ|, |∇2ψ|, |∇3ψ| ≤ A

Ω,

|∇ψ| ≥ a

∂Ω.

ρ >

U = {z ∈ Cn |

(z, ∂Ω) < ρ}

### . Let

u ∈ PSH(Ω ∩ U) ∩ C(Ω ∩ U)

u =

u <

Mu = δ

Ω ∩ U

< δ≤ δ0

b

B

|∇u| ≥ b

∂Ω,

|∇u| ≤ B

Ω ∩ U.

### Then there is a constant

C= C(n, ρ, a, A, b, B, δ0)

|∇2u| ≤ C

∂Ω.

α >

Ω = {z ∈ Cn |

< |z| < α}

### that

u ∈ PSH(Ω) ∩ C(Ω)

u =

∂B1

Bα = B(

, α)

u >

(11)

Mu= δ >

β

b

B

u≥ β

∂Bα,

|∇u| ≥ b

∂B1,

|∇u| ≤ B

Ω.

δ0 = δ0(n, α, β)

### and

C = C(n, α, β, b, B)

< δ≤ δ0

|∇2u| ≤ C

∂B1.

z0∈ ∂Ω

Nz0 = (

, . . . ,

,

)

Nz0 = ∂/∂xn

ψ

u

C

C

v

∂Ω

u= vψ

v >

Ω ∩ U

### . Therefore, if

t, s∈ {x1, y1, . . . , xn−1, yn−1, yn}

### , then

uts(z0)= uxn(z0ts(z0) ψxn(z0)

|uts(z0)| ≤ C1.

|utxn(z0)| ≤ C2,

|uxnxn(z0)|

uxnxn =

unn¯ − uynyn,

ddcψ≥ ddc|z|2

δ0≥ δ =

### det

(uij¯(z0))≥ unn¯(z0)

a A

n−1

− C3.

z∈

ψxn(z)=

### Re

*

∇ψ(z), ∇ψ(z0)

|∇ψ(z0)| +

≥ |∇ψ(z0)| − A|z − z0| ≥ a − A|z − z0|.

(12)

Ω ∩B(z

0,ρ)

T

= ut ψt

ψxn

uxn,

T =

∂Ω ∩B(z

0,ρ).

### We have

Txn(z0)= utxn(z0)−ψtxn(z0)

ψxn(z0)uxn(z0),

|Txn(z0)| ≤ C4.

f

= ψtxn

### ; then

|∇f |, |∇2f| ≤ C5

Ω ∩B(z

0,ρ).

(uij¯)

### is constant, one can show that

uij¯uijt¯ = uij¯uijx¯

n=

.

(uij¯)

(uij¯)

### .) Hence, we can com- pute

uij¯Tij¯= −uxnuij¯fij¯

### 2 Re

uij¯uixnfj¯= −uxnuij¯fij¯

fxn

uij¯uiynfj¯.

uij¯(u2yn)ij¯=

uij¯uiynujy¯

n,

uij¯



±T +

### 1 2

u

2yn



ij¯≥ ∓uxnuij¯fij¯

### 2

fxn− uij¯fifj¯≥ −C6

 X

i

uii¯+

.

∂Ω

uyn= uxnψynxn

±T +

### 1 2

u

2yn

≤ C7|z − z0|2, z∈ ∂Ω ∩B(z

0,ρ).

(13)

±T +

u

2yn

≤ C8

Ω ∩B(z

0,ρ),

### and we obtain that if

w = ±T + 12u2yn− C9|z − z0|2

C9

w≤

∂(Ω ∩ B(z0,ρ))

uij¯wij¯≥ −C10

 X

i

uii¯+

.

C11

C12

w+ C11ψ+ C12u

∂(Ω ∩ B(z0,ρ))

### ˜ and

uij¯(w + C11ψ+ C12u)ij¯

Ω ∩ B(z0,ρ)

w+ C11ψ+ C12u≤

Ω ∩ B(z0,ρ)

### ˜ , and thus

|Txn(z0)| ≤ C11A+ C12B.

ψ(z)= λ(|z|2

),

λ= β/(α2

)

ψ≤ u

δ

z0∈ ∂B1

z0= (

, . . . ,

,

)

|utxn(z0)| ≤ C1.

### Similarly as before we get that if

w= ±T + 12u2yn− C2|z − z0|2

C2

uij¯wij¯≥ −C3

 X

i

uii¯+



Ω ∩ B(z0,

),

w≤

∂(Ω ∩ B(z0,

))

### Now by the inequality between arithmetic and geometric means we have

uij¯(ψ− u)ij¯≥ λX

i

uii¯− n ≥λ

X

i

uii¯+ n

 λ

δ1/n



λ

 X

i

uii¯+

,

δ

### small enough. Thus

uij¯(w+ C4(ψ− u))ij¯

Ω ∩ B(z0,

)

(14)

C4

|Txn(z0)| ≤ C4B.

ψ

C2,1

ddcψ≥ ddc|z|2

### and

|ψ|, |∇ψ|, |∇2ψ|, |∇3ψ| ≤ A

Ω,

|∇ψ| > a

∂Ω,

a

A

r >

z0∈ ∂Ω

B(z1,

r )

z0

g(z)≤ − γ

|z − z1|

r

r ≤ |z − z1| ≤

r ,

γ=

### max

dist(z,∂Ω)≥r˜g(z).

b

z→∂Ω

|g(z)|

(z, ∂Ω) > b >

.

ψj = ψ ∗ ρ1/j

ψ

j = {ψj <

}

j

A

a

b

ψj

j

ψ

C

n

k

r

R

m

M

A

a

b

gε,δ ∈ C(ε)

< ε < r0

< δ≤

ε

δ

|∇2gε,δ(z)| ≤ C1

### min

i|z − pi|2, z∈ Ωε.

|∇gε,δ| ≥ b

|∇2gε,δ| ≤ C2

∂Ω.

|w| ≥

i=

. . .

k

u(w)

### :

= gε,δ(pi+ εw) − µi

ε r.

(15)

u(w)≥ µi

|w| − C3.

α

β

= m

α−C3>

ε

u≥ β

∂Bα

gε,δ ≥ −C4

∂B(pi, r )

ε

µi

|z − p

i| r i

### 2 log

ε

r + |z − pi|2− ε2≤ gε,δ(z)

### if

ε≤ |z − pi| ≤ r .

|∇gε,δ| ≥ µi

ε

∂B(pi, ε),

|∇u| ≥ µi/

∂B1

δ

|∇2u| ≤ C5

∂B1,

|∇2gε,δ| ≤ C5

ε2

∂B(pi, ε).

### The rest of the proof will be a compilation of the methods from [Bł3] and from the proof of Theorem 3.1. Fix

a∈ Ω \ {p1, . . . , pk}

gε,δ

|∇2gε,δ(a)| =

### lim sup

h→0

gε,δ(a+ h) + gε,δ(a− h) −

gε,δ(a)

|h|2 .

P

00 b Ω0 b Ω

ε0 >

z∈ Ω0\S

iB(pi, ε+ ε0)

h

D(z, h)

### :

= gε,δ



z+ P (z1) P (a1)h



### and

v(z, h)= D(z, h) + D(z, −h) + C6

|P(a1)|2(|z − p1|2− R2)|h|2,

D(z,

### 0

)= gε,δ(z), D(a, h)= gε,δ(a+ h),

(16)

v

z

### , and

v(a, h)≥ gε,δ(a+ h) + gε,δ(a− h) − C6R2

|P(a1)|2|h|2.

C6

h

(Mv(·, h))1/n

### 1

+ P0(z1) P (a1)h1

2/n+

−P0(z1) P (a1)h1

2/n

! δ1/n

+ C6

|P(a1)|2|h|2≥ (

δ)1/n.

### The Taylor expansion of

D(z,·)

v(z, h)≤ D(z, h) + D(z, −h) ≤

### 2

gε,δ(z)+ k∇2(D(z,·))kB(¯ 0,|h|)|h|2.

|∇2(D(z,·))(h)

### ˜

| = |P(z1)|2

|P(a1)|2

2gε,δ



z+ P (z1) P (a1)h

 ,

v(z, h)≤

### 2

gε,δ(z)+ C0|h|2, z∈ ∂Ω0,

v(z, h)≤

### 2

gε,δ(z)+ Ci0|h|2, z∈ ∂B(pi, ε+ ε0),

### where

C0= C7

k∇2gε,δkΩ\Ω00

|P(a1)|2 , Ci0= C8

(ε+ ε0)2k∇2gε,δkB(pi,ε+2ε0)∩Ωε

|P(a1)|2

h

v

gε,δ

v(a, h)≤

gε,δ(a)+

### max

{C0, C10, . . . , Ck0}|h|2.

|∇2gε,δ(a)| ≤

### max

{C0, C10, . . . , Ck0} + C6R2

|P(a1)|2.

00 ↑ Ω

ε0

(17)

### R

EFERENCES

[BD] E. BEDFORD& J.-P. DEMAILLY, Two counterexamples concerning the pluri-complex Green function inCn, Indiana Univ. Math. J. 37 (1988), 865-867.

[BT1] E. BEDFORD&B.A. TAYLOR, The Dirichlet problem for a complex Monge-Amp`ere equation, Invent. Math. 37 (1976), 1-44.

[BT2] , A new capacity for plurisubharmonic functions, Acta Math. 149 (1982), 1-41.

[Bł1] Z. BŁOCKI, The complex Monge-Amp`ere operator in hyperconvex domains, Ann. Scuola Norm.

Sup. Pisa 23 (1996), 721-747.

[Bł2] , Equlibrium measure of a product subset ofCn, Proc. Amer. Math. Soc. 128 (2000), 3595-3599.

[Bł3] , TheC1,1regularity of the pluricomplex Green function, Michigan Math. J. 47 (2000), 211-215.

[CKNS] L. CAFFARELLI, J.J. KOHN, L. NIRENBERG& J. SPRUCK, The Dirichlet problem for non-linear second order elliptic equations II: Complex Monge-Amp`ere, and uniformly elliptic equations, Comm. Pure Appl. Math. 38 (1985), 209-252.

[Co] D. COMAN, The pluricomplex Green function with two poles of the unit ball ofCn, Pacific J.

Math. 194 (2000), 257-283.

[De] J.-P. DEMAILLY, Mesures de Monge-Amp`ere et mesures plurisousharmoniques, Math. Z. 194 (1987), 519-564.

[EZ] A. EDIGARIAN&W. ZWONEK, Invariance of the pluricomplex Green function under proper mappings with applications, Complex Variables Theory Appl. 35 (1998), 367-380.

[Gu] BOGUAN, The Dirichlet problem for complex Monge-Amp`ere equations and regularity of the pluri-complex Green function, Comm. Anal. Geom. 6 (1998), 687-703, correction: ibid. 8 (2000), 213-218.

[Le] P. LELONG, Fonction de Green pluricomplexe et lemmes de Schwarz dans les espaces de Banach, J. Math. Pures Appl. 68 (1989), 319-347.

[Wa] J.B. WALSH, Continuity of envelopes of plurisubharmonic functions, J. Math. Mech. 18 (1968), 143-148.

Jagiellonian University Institute of Mathematics Reymonta 4, 30-059 Krak´ow POLAND

E-MAIL: blocki@im.uj.edu.pl

ACKNOWLEDGMENT: Partially supported by KBN Grant #2 PO3A 003 13.

KEY WORDS AND PHRASES:

pluricomplex Green function with several poles, complex Monge-Amp`ere operator.

1991MATHEMATICSSUBJECTCLASSIFICATION: Primary 32U35; Secondary 32W20 Received : March 14th, 2000; revised: September 8th, 2000.

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