. We show that if

with Several Poles

Z BIGNIEW B ŁOCKI

A

. We show that if

is a

smooth, strictly pseu- doconvex domain in

, then the pluricomplex Green function for

with several fixed poles and positive weights is

.

1. I

If

is a bounded domain in

,

,

,

are distinct, and

,

,

0, then the corresponding pluricomplex Green function is given by

g=

sup

where

B =

v∈ PSH(Ω) | v <

0

lim sup

z→pⁱ

(u(z)− µi

log

1

One can show that

,

is a maximal plurisubharmonic (psh) function in

, and

Mg= πⁿ n

!2

X

i

µiδ_pi

(see [Le]), where

is the complex Monge-Amp`ere operator. For smooth

det

∂z_i∂z

¯

! ,

and by [De]

can be well defined as a nonegative Borel measure if

and

is locally bounded near

.

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

335

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

Theorem 1.1. Assume that

is

smooth and strictly pseudoconvex. Then

, and

|∇²g(z)| ≤ C

min

where

is a constant depending only on

,

,

,

,

,

,

.

One can treat it as a regularity result for the complex Monge-Amp`ere operator and indeed, this the main tool in the proof. The obtained regularity is the best possible: as shown in [Co] and [EZ], the Green function for a ball with two poles and equal weights is not

inside. In the case of one pole it is known from [BD]

that the Green function need not be

up to the boundary, but in this example it is not clear how regular the function is inside. Therefore, a full counterexample is still missing in this case.

The case

1 was treated in [Gu] and [Bł3]. In [Gu] the

regularity for

1 was claimed. However, the proof contained an error (inequality (3.6) on p.

697 in [Gu] is false). Then in [Bł3], using some results from [Gu] and a method similar to the one used in [BT1] involving holomorphic automorphisms of a ball, the

regularity was shown. Afterwards, in the correction to [Gu], a di fferent method was used to show the

regularity.

Here we adapt the methods from [Gu] and [Bł3] for

1. This yields also a slightly di fferent proof for

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

z7 -→ z + (z₁− p1¹)· · · (z1− p^k1) (a₁− p1¹)· · · (a1− p1^k)h

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

and small

fixes

and maps

to

.

To get an priori estimate for the second derivative on the boundary, we follow the method from [CKNS] and prove Theorems 4.1 and 4.2 below. In the case of Theorem 4.2 we also use a modification of this method from [Gu]. We present the full proofs of Theorems 4.1 and 4.2 for two reasons: firstly, since given functions are constant on the boundary and their complex Monge-Amp`ere measure is also constant, the proofs are simpler than in the general setting, and secondly, we get a precise dependence of the a priori constants which was stated neither in [CKNS]

nor in [Gu]. In fact, all quantitative estimates necessary to obtain the constant from Theorem 1.1 are included here. We only make use of the existence result – [Gu, Theorem 1.1] (it would even be enough to use [CKNS, Theorem 1] and Theorem 4.1 and 4.2 below instead).

By the way, we are also able to show the following regularity of

.

Theorem 1.2. If

is hyperconvex, then

is continuous as a function defined on the set

{(z, p¹, . . . , p^k, µ₁, . . . , µk)∈_{Ω × Ω}

¯

if

(1.1)

where for

we set

:

0.

(Recall that

is called hyperconvex if there exists

with

0 and lim

0.)

Theorem 1.3. Assume that lim sup

z→∂Ω

|g(z)|

dist

Then

|∇g(z)| ≤ C

min

where

is a constant depending only on

,

,

,

,

,

,

.

Notation. If

, then

Re

,

Im

. If

,

|ζ| =

1, then by

we will denote the

-th derivative of

in direction

at

. For the partial derivatives we will use the notation

ux_i = ∂u

∂xi

, uy_i= ∂u

∂yi

, ui= ∂u

∂zi

, u_i¯= ∂u

∂z

¯

.

If we write

|∇u| ≤ f

in an open

where

is locally bounded, nonnegative in

, then we mean that

is locally Lipschitz and the inequality holds almost everywhere (

makes then sense by the Rademacher theorem). If we write

, in fact it means exactly that

is psh. When proving the existence of a constant depending only on given quantities, by

,

,

we will denote positive constants depending only on those quantities and call them under control.

2. B

Given a bounded domain

in

, distinct poles

,

,

and weights

,

,

0 fix positive

,

,

, and

so that for

,

1,

,

Ω ⊂ B(pⁱ, R),

B(p

¯

and

¯

¯

One can easily check the following estimates for

:

i

µi

log

i|

R ≤ g(z) <

0

log

i|

R − (k −

1

log

r ≤ g(z) ≤ µi

log

i|

r , z∈_B(p

¯

For

with 0

, define

Ω^ε

:

i

B(p

¯

and

g^ε

:

sup



v∈ PSH(Ω) v <

0

B(pⁱ,ε)≤ µi

log

r, i=

1

 .

One can easily check that

g^ε(z)≤ µi

log max

r , z∈_B(p

¯

(2.1)

g^ε ∈ PSH(Ω), g≤ g^ε≤

log

log

1

log

in

(2.2)

g^ε ↓ g⁰

:

as

0, and the convergence is locally uniform in

. Proposition 2.1. Assume that

is

smooth and strictly pseudoconvex. Then there exists

depending only on

,

,

,

, and

, 0

, such that for

with 0

we can find

¯ with

in

,

0 on

, and for

1,

,

µi

log

r ≤ v(z) ≤ µi

log

i|

r

if

Proof. Set

w(z)

:

i

µi

log

i|

R + |z − p¹|²− R²,

so that

0 on ¯

,

, and

log

on

. On the other hand, for

we have

w(z)≥ kM

log

R + r²− R²> µi

log

r + |z − pⁱ|²− ε²,

provided that

is such that

log

r − ε²< kM

log

Similarly as in [Bł2], let

:

be

smooth and such that

0

1

χ(t)= t, t ≥

1

0

1

0

For

,

set

fj(x, y)

:

1

jχ(j(y− x)),

so that

fj(x, y)=

max

if

1

If

,

are psh functions with

,

, then

dd^cf_j(u, v)≥ (

1

Let

be a defining function for

. If we choose

,

su fficiently big, then the function

v(z)=





 fj



w(z), µi

log

r + |z − pⁱ|²− ε²



, z∈S

i_B(p

¯

¯

¯

has all the required properties.

Note that if

1, then we may choose

in Proposition 2.1.

Proof of Theorem 1.2. By (2.2)

locally uniformly on the set (1.1) as

0. It is thus enough to show that for a fixed small

,

is continuous as a function defined on

Ω × {(p

¯

dist

2

if

Let

,

as

,

1,

,

, and

g^ε_j

:

sup



v∈ PSH(Ω) v <

0

log

 .

Note that if 0

and

is big enough, then by Proposition 2.1 applied to a ball containing

we have lim

log

. Moreover, lim

0, since

is hyperconvex. Therefore, by a result from [Wa] (see also [Bł1, Theorem 1.5]),

is continuous on ¯

.

To finish the proof it is enough to show that

uniformly as

in

¯ . Fix

0. For

¯

and

big enough, by (2.1) we have

g_j^ε(z)≤ µi,j

log max

r ≤ µi,j

log

i− p^i,j|

r ≤ µi

log

whereas for

¯

g^ε(z)≤ µi

log max

r ≤ µi

log

i− p^i,j|

r ≤ µi,j

log

Thus for those

g^ε− c ≤ g^ε_j ≤ g^ε+ c

on ¯

and the theorem follows.

In the proof of Theorem 1.1 we will also need to approximate

. If 0

and 0

1, define

g^ε,δ

:

sup

in

Note that

is increasing in

and decreasing in

. We also have

(2.3)

Proposition 2.2.

,

in

. If

is hyperconvex and 0

, then

is continuous on ¯

. If

is

smooth and strictly pseudoconvex, 0

and 0

1, then

.

Proof. We use standard procedures. Let

B = {v ∈ PSH(Ω) | v ≤ g^ε, Mv≥ δ

in

By the Choquet lemma there exists a sequence

such that

sup

. (

denotes the upper semicontinuous regularization of

.) If

max

, then

in

(see e.g. [Bł2]) and thus

. There-

fore

almost everywhere, and by the approximation theorem from

[BT2]

in

. We conclude that

and

in

. The balayage procedure gives

in

.

Now assume that

is hyperconvex and 0

. By [Bł1] there exists

¯ with

0 on

and

1 in

. For

big enough

Aψ≤ g^ε,δ≤

0 in

(2.4)

Let

be given by Proposition 2.1 applied to a ball containing

. Then

log

i|

r

if

(2.5)

For small

and

with

dist

2

we have

|g^ε,δ(z+ h) − g^ε,δ(z)| ≤ C(|h|).

By the comparison principle (see [BT2]) applied to

and

, the above inequality holds for all

with dist

. By (2.4) and (2.5)

h

lim

0

which means that

is continuous.

The last part of the proposition follows from Proposition 2.1 and [Gu, Theo-

rem 1.1].

3. G

Theorem 1.3 will follow immediately from the next result applied to

0.

Theorem 3.1. Fix 0

1. Assume that lim sup

z→∂Ω

|g^0,δ(z)|

dist

Then for

satisfying Proposition 2.1 we have

|∇g^ε,δ(z)| ≤ C

min

where

is a constant depending only on

,

,

,

,

,

, and

.

The assumption of Theorem 3.1 is satisfied uniformly for

1 for example, if

is smooth and strictly pseudoconvex.

Proof of Theorem 3.1. Let

0 be such that

−g^ε,δ(z)≤ −g^0,δ(z)≤

2

dist

if dist

For

su fficiently small

g^ε,δ(z+ h) − g^ε,δ(z)≤

2

if dist

and, since by Proposition 2.1

µi

log

r ≤ g^ε,δ(z)≤ µi

log

i|

r

if

we have

g^ε,δ(z+ h) − g^ε,δ(z)≤ µi

log

i+ h|

ε ≤

2

if

1

From the comparison principle we get

2 max

 B,M

ε



|h|

if

min

dist

and thus

|∇g^ε,δ| ≤C₁

ε

in

(3.1)

We will need a lemma.

Lemma 3.2. There exists a constant

such that for given

,

,

,

we can orthonormally change variables in

so that

|a − pⁱ| ≤ eC|a1− p₁ⁱ|, i=

1

Proof. By

denote the unit sphere in

. We have to show that there exists

such that

|a − pⁱ| ≤ eC|ha − pⁱ, bi|, i=

1

that is,

* a− pⁱ

|a − pⁱ|, b+  ≥

1

Ce.

Define

Ce

:

1

min

where

f (ζ¹, . . . , ζ^k)

:

max

b∈S

min

i |hζⁱ, bi|

is a continuous function on

. It remains to show that

0 on

. Fix

,

,

and define

:

0

,

1,

,

. Then

, and thus for

iKi

we have

f (ζ¹, . . . , ζ^k)≥

min

i |hζⁱ, bi| >

0

❐

End of proof of Theorem 3.1. Fix

and choose variables as in Lemma 3.2. Set

P (λ)

:

so that

|P(z1)|

|P(a1)| ≤ C2

max

min

min

For

su fficiently small let

:



z∈ Ω z + P (z₁) P (a₁)h∈ Ω



and

Ω⁰

:

i

B(p

¯

where

ε⁰=

min

dist

Set

v(z)

:



z+ P (z₁) P (a1)h



+ C₄

min

so that if

is big enough, then

Mv≥

1

²δ+ C₄

min

For

we have

v(z)− g^ε,δ(z)≤

2

dist

2

min

whereas for

v(z)− g^ε(z)≤ C₁ ε

|P(z1)|

|P(a1)||h| ≤ C1C₂ ε+ ε⁰

ε

min

min

Therefore, the comparison principle gives

g^ε(a+ h) − g^ε(a)≤ C6

min

if

min

and the theorem follows.

4. E

Our goal will be to estimate

for small

,

. First, we need such an estimate on

. We will follow the method from [CKNS] (see also [Gu]). We shall prove two theorems.

Theorem 4.1. Let

be a bounded strictly pseudoconvex domain in

and

a

psh defining function for

. Assume that

and that there are positive constants

,

such that

|ψ|, |∇ψ|, |∇²ψ|, |∇³ψ| ≤ A

on ¯

|∇ψ| ≥ a

on

For

0 denote

dist

. Let

¯ be such that

0 on

and

0,

in

, where 0

. Assume also that there are positive constants

,

such that

|∇u| ≥ b

on

|∇u| ≤ B

on ¯

Then there is a constant

such that

|∇²u| ≤ C

on

Theorem 4.2. Fix

1 and let

1

. Assume

that

¯ is such that

0 on

(

0

),

0,

Mu= δ >

0 in

. Suppose, moreover, that there are positive constants

,

,

such that

u≥ β

on

|∇u| ≥ b

on

|∇u| ≤ B

on ¯

Then there exist positive constants

and

such that if 0

, we have

|∇²u| ≤ C

on

Proof of Theorem 4.1. Fix

. We may assume that

0

0

1

, so that

. Since both

and

are

defining functions for

, there exists a

function

, defined in a neighborhood of

, such that

and

0 on ¯

. Therefore, if

, then

uts(z₀)= ux_n(z₀)ψts(z₀) ψ_xn(z₀)

(4.1)

and thus

|uts(z₀)| ≤ C1.

(4.2)

Suppose now that we know that

|utxn(z₀)| ≤ C2,

(4.3)

and we want to estimate

. We have

4

and by (4.1), (4.2), (4.3), and since

,

δ₀≥ δ =

det

a A

_n−1

− C3.

It thus remains to show (4.3). For

¯ we have

Re

*

∇ψ(z), ∇ψ(z0)

|∇ψ(z0)| +

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

On ¯

¯

˜ define

T

:

ψxn

u_xn,

so that

T =

0 on

¯

˜ (4.4)

We have

Tx_n(z₀)= utx_n(z₀)−ψtxn(z₀)

ψx_n(z₀)ux_n(z₀),

and thus it is enough to prove that

|Txn(z0)| ≤ C4.

Set

:

; then

|∇f |, |∇²f| ≤ C5

in ¯

¯

˜ (4.5)

Since det

is constant, one can show that

n=

0

(Here

denotes the inverse transposed matrix of

.) Hence, we can com- pute

u^ij¯T_ij¯= −uxnu^ij¯f_ij¯−

2 Re

2

2 Im

Since

u^ij¯(u²_yn)_ij¯=

2

n,

the Schwarz inequality and (4.5) give

u^ij¯



±T +

1 2

2y_n



i_j¯≥ ∓uxnu^ij¯f_ij¯∓

2

 X

i

u^ii¯+

1

On

we have

, and thus by (4.4)

±T +

1 2

2yn

 ≤ C⁷|z − z0|², z∈ ∂Ω ∩_B(z

¯

˜

Moreover,

±T +

1 2

2y_n

 ≤ C⁸

in ¯

¯

˜

and we obtain that if

, where

is big enough, then

0 on

˜ , and

u^ij¯w_ij¯≥ −C10

 X

i

u^ii¯+

1

Therefore, if

and

are big enough, then

0 on

∂(Ω ∩ B(z0,ρ))

˜ and

0 in

˜ . By the maximum principle

w+ C11ψ+ C12u≤

0 in

˜ , and thus

|Tx_n(z₀)| ≤ C11A+ C12B. ❐

Proof of Theorem 4.2. Set

ψ(z)= λ(|z|²−

1

where

1

, so that

in

for

su fficiently small. We now follow the proof of Theorem 4.1. Fix

, we may assume that

0

0

1

. We may reduce the problem to the estimate

|utx_n(z₀)| ≤ C1.

Similarly as before we get that if

, where

is big enough, then

u^ij¯w_ij¯≥ −C3

 X

i

u^ii¯+

1

in

1

and

0 on

1

.

Now by the inequality between arithmetic and geometric means we have

i

u^ii¯− n ≥λ

2

X

i

u^ii¯+ n

 λ

2

1



≥ λ

2

 X

i

u^ii¯+

1

for

small enough. Thus

u^ij¯(w+ C4(ψ− u))_ij¯≥

0 in

1

if

is su fficiently big, and by the maximum principle we conclude that

|Tx_n(z₀)| ≤ C4B. ❐

Proof of Theorem 1.1. Let

be a

defining function for

with

in

and

|ψ|, |∇ψ|, |∇²ψ|, |∇³ψ| ≤ A

on ¯

|∇ψ| > a

on

for some positive

and

. We can find ˜

0 such that for every

there exists a ball

2 ˜

, contained in

and tangent to

at

. Then

g(z)≤ − γ

log 2 log

2 ˜

if ˜

2 ˜

where

γ=

max

dist(z,∂Ω)≥r˜g(z).

Therefore we can find

with lim inf

z→∂Ω

|g(z)|

dist

0

Let

be the standard regularization of

and let

0

. If

is big enough, then the constants

,

, and

are good also for

and

. Thus, we may assume that

(and thus

) is

, provided that we prove that the constant in Theorem 1.1 depends only on

,

,

,

,

,

,

,

, and

.

By Proposition 2.2,

if 0

, 0

1. It is enough to show that for small positive

and

we have

|∇²g^ε,δ(z)| ≤ C₁

min

Since

on

, by Theorems 3.1 and 4.1 we have

|∇²g^ε,δ| ≤ C2

on

(4.6)

For

1 and fixed

1,

,

set

u(w)

:

log

By (2.2) and (2.3)

u(w)≥ µi

log

Thus, if

is so big that

:

log

0, then for su fficiently small

,

on

. Moreover,

on

. Thus by the comparison principle, for su fficiently small

we have

µi

2 log

i| r +µi

2 log

r + |z − pⁱ|²− ε²≤ g^ε,δ(z)

if

Therefore

|∇g^ε,δ| ≥ µi

2

on

and

2 on

. From Theorem 4.2 it follows that for

small enough

|∇²u| ≤ C5

on

which means that

|∇²g^ε,δ| ≤ C₅

ε²

on

(4.7)

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

. From the fact that

is psh it follows that

|∇²g^ε,δ(a)| =

lim sup

h→0

g^ε,δ(a+ h) + g^ε,δ(a− h) −

2

|h|² .

(4.8)

Let

be as in the proof of Theorem 3.1 and let

,

0. For

iB(pⁱ, ε+ ε⁰)

and small

set

:



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,

so that

D(z,

0

v

is psh in

, and

v(a, h)≥ g^ε,δ(a+ h) + g^ε,δ(a− h) − C₆R²

|P(a1)|²|h|².

(4.9)

If

is su fficiently big and

su fficiently small, then

(Mv(·, h))^1/n≥

1

^2/n+

1

^2/n

! δ^1/n

+ C6

|P(a1)|²|h|²≥ (

2

The Taylor expansion of

about the origin gives

v(z, h)≤ D(z, h) + D(z, −h) ≤

2

Since

|∇²(D(z,·))(_h)

˜

|P(a1)|²

∇²g^ε,δ



z+ P (z1) P (a1)_h

˜

we get

v(z, h)≤

2

v(z, h)≤

2

where

C⁰= C7

k∇²g^ε,δk_Ω\Ω00

|P(a1)|² , C_i⁰= C8

(ε+ ε⁰)²k∇²g^ε,δkB(pⁱ,ε+2ε⁰)∩Ω^ε

|P(a1)|²

for

small enough. Now we can apply the comparison principle to

and 2

. We obtain

v(a, h)≤

2

max

By (4.8) and (4.9)

|∇²g^ε,δ(a)| ≤

max

|P(a1)|².

If we let

,

0, and use (4.6), (4.7), then the desired estimate follows.

R

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[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.