1. Introduction. Let Ω be an open and connected set in R

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INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES

WARSZAWA 1995

THE SYMMETRIC PLURICOMPLEX GREEN FUNCTION

U R B A N C E G R E L L

Department of Mathematics, University of Ume˚ a S-90187 Ume˚ a, Sweden

1. Introduction. Let Ω be an open and connected set in R

ⁿ

; x

0

∈ Ω. Then the classical Green function G

Ω

(x, x

0

) is the solution to the Dirichlet problem

G

Ω

(x, x

0

) = 0, ∀x ∈ ∂Ω

∆

x

G(x, x

0

) = δ

x0

.

In [7], Klimek introduced the pluricomplex Green function g

Ω

, that can be defined as solution to







g

Ω

(z, z

0

) ∈ PSH (Ω) g

Ω

(z, z

0

) = 0, ∀z ∈ ∂Ω (dd

^c_z

g

Ω

(z, z

0

))

ⁿ

= (2π)

ⁿ

δ

z0

where Ω is a domain (open, bounded, and connected set) in C

ⁿ

; z

0

∈ Ω.

An alternative definition of g

Ω

for any domain Ω in C

ⁿ

, z

0

∈ Ω is g

Ω

(z, z

0

) = sup{ϕ(z); ϕ ∈ PSH (Ω), ϕ ≤ 0, ϕ(z) − log |z − z

0

|

bounded above near z = z

0

}.

It is well known that the classical Green function is symmetric: G

Ω

(x, x

0

)=

G

Ω

(x

0

, x). However, the pluricomplex Green function need not be symmetric.

It was shown by Bedford and Demailly [2] that there exists a strictly pseudoconvex smooth Ω such that g

Ω

(z, z

0

) 6= g

Ω

(z

0

, z).

2. The symmetric pluricomplex Green function. In [3], we introduced the symmetric pluricomplex Green function W

Ω

(z, ω),

W

Ω

(z, ω) = sup{ϕ(z, ω) ∈ 2 − PSH (Ω × Ω), ϕ ≤ 0, ϕ(z, ω) ≤ log |z − ω| − log max[d(z, {Ω), d(ω, {Ω)]}.

1991 Mathematics Subject Classification: Primary 32F05.

The paper is in final form and no version of it will be published elsewhere.

[135]

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Here, 2 − PSH (Ω × Ω) denotes the subharmonic functions that are also sep- arately plurisubharmonic. The purpose of this note is to consider some basic properties of W

Ω

.

Definition. A domain Ω is said to be strongly hyperconvex if W

Ω

(z, ω) is an exhaustion function for Ω for each fixed ω ∈ Ω.

R e m a r k. Every strictly pseudoconvex set is strongly hyperconvex.

Lemma 1. Suppose ϕ is plurisubharmonic near zero and that |ϕ(z) − log |z|| <

K near zero for some constant K. Then µ(0) ≥ (2π)

ⁿ

where µ is the weak

^∗

-limit of (dd

^c

max[ϕ, t])

ⁿ

, t → −∞.

P r o o f. Let 1 > r > 0 so that ϕ is plurisubharmonic and so that

|ϕ(z) − log(z)| ≤ K on B(0, r).

Given 0 < < 1, then

|z| > e

⁻¹

⇒ log |z| ≥ −1

⇒ − log |z| < 1 ⇒ (ϕ − log |z| > −K) ⇒

ϕ − log |z| > −(K + 1) − log |z| ⇒ ϕ(z) > (1 − ) log |z| − (K + 1).

Thus

Ω

= {z ∈ B(0, r); ϕ(z) < (1 − ) log |z| − (K + 1)}

is a neighborhood of zero and relatively compact in B(0, r) if e

⁻¹

< r.

Let t < inf

z∈∂Ω

(1 − ) log |z| − (K + 1) = δ < 0 and define ϕ

2t

= max[ϕ, 2t].

Then Ω

^t

= {z ∈ B(0, r); ϕ

2t

< max[(1 − ) log |z| − (K + 1), t]} is a neighborhood of zero and relatively compact in B(0, r). Thus

R

Ω^t

(dd

^c

ϕ

t

) ≥ R

Ω^t

(dd

^c

max[(1 − ) log |z| − K + 1, t])

ⁿ

= (2π)

ⁿ

(1 − )

ⁿ

and since Ω

^t

⊂ Ω

⊂ B(0, e

⁻¹

),

R

B(0,e^{− 1})

(dd

^c

ϕ

2t

)

ⁿ

≥ (2π)

ⁿ

(1 − )

ⁿ

.

So if µ is the weak

^∗

-limit of (dd

^c

ϕ

2t

)

ⁿ

, t → −∞, then µ(0) ≥ (2π)

ⁿ

which proves the lemma.

Theorem 1. Suppose Ω is strongly hyperconvex. Then g

Ω

≥ W

_Ω

with equality if and only if

τ (z) = R

Ω

(dd

^c_ξ

max[W

Ω

(z, ξ), −1])

ⁿ

= (2π)

ⁿ

, ∀z ∈ Ω.

P r o o f (cf. [3, Prop. VII:2). Note first that R

Ω

(dd

^c_z

max[g

Ω

(z, ξ), t])

ⁿ

=

(2π)

ⁿ

,

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∀t < 0, ∀z ∈ Ω and that

τ (z) = R

Ω

(dd

^c_ξ

max[W

Ω

(z, ξ), t])

ⁿ

is independent of t for all negative t. Also, it follows from definitions that g

Ω

≥ W

Ω

. It follows from Lemma 1 that τ (z) ≥ (2π)

ⁿ

with equality if W

Ω

≡ g

Ω

.

On the other hand, assume τ (z) ≡ (2π)

ⁿ

. Again, by Lemma 1, (dd

^c

W

Ω

(z, ξ))

ⁿ

= 0 on z 6= ξ. Let ξ ∈ Ω be given and consider for 0 < < 1, (1 − )W

Ω

(z, ξ).

Then (1 − )W

Ω

(z, ξ) = 0 on ∂Ω, (1 − )W

Ω

(z, ξ) ≥ g

Ω

(z, ξ) for z near ξ. Since (dd

^c_z

(1 − )W

Ω

(z, ξ))

ⁿ

= 0 outside ξ, (1 − )W

Ω

(z, ξ) ≥ g

Ω

(z, ξ) on Ω. Letting

& 0, we find that W

Ω

= g

Ω

.

Lemma 2. Let Ω

1

⊂ C

ⁿ

, Ω

2

⊂ C

ⁿ

be two open and connected sets. Then max(W

Ω1

, W

Ω2

) ≤ W

Ω1×Ω2

.

P r o o f.

0 ≥ max[W

Ω1

(z

1

, ω

1

), W

Ω2

(z

2

, ω

2

)]

≤ max(log |z

₁

− ω

₁

| − log max[d(z

₁

, {Ω

1

), d(ω

1

, {Ω

1

)],

log |z

2

− ω

₂

| − log max[d(z

₂

, {Ω

2

), d(ω

2

, {Ω

2

)])

≤ log |(z

1

, z

2

)−(ω

1

, ω

2

)|− log min[d(z

1

, {Ω

¹

), d(ω

1

, {Ω

¹

), d(z

2

, {Ω

¹

), d(ω

2

, {Ω

²

)]

so the inequality now follows from the definition of W

Ω1×Ω2

via [3, Cor. VII:1].

Example. Denote by C

Ω

the Carath´ eodory pseudodistance on Ω. We give an example of a bounded pseudoconvex set Ω, such that

log tanh C

Ω

6= W

Ω

6= g

Ω

.

Let Ω

1

= {z ∈ C;

¹₂

< |z| < 1} and let Ω

2

be any strictly pseudoconvex domain where W

Ω2

(z

₂⁰

, ω

₂⁰

) < g

Ω2

(z

₂⁰

, ω

₂⁰

) for a point (z

₂⁰

, ω

₂⁰

) ∈ Ω

2

× Ω

₂

(by [2], such a set exists). Note first that W

Ω1

= g

Ω1

and that

W

Ω1

(z

1

, ω

1

) > log tanh C

Ω

(z

1

, w

1

), ∀z

1

6= ω

1

∈ Ω

1

(cf. Klimek [7], p. 234–235]).

Then log tanh C

Ω1×Ω2

((z

1

, z

2

), (ω

1

, z

2

)) < W

Ω1

(z

1

, ω

1

) ≤ max[W

Ω1

(z

1

, w

1

), W

Ω2

(z

2

, z

2

)] ≤ W

Ω1×Ω2

((z

1

, z

2

), (ω

1

, z

2

)) by Lemma 2. Thus log tanh C

Ω1×Ω2

6=

W

Ω1×Ω2

; it remains to prove that W

Ω1×Ω2

6= g

Ω1×Ω2

. Suppose W

Ω1×Ω2

≡ g

Ω1×Ω2

. Since Ω

1

and Ω

2

are pseudoconvex, it follows from Theorem 9. 6 in [6] that g

Ω1×Ω2

= max[g

Ω1

, g

Ω2

] so W

Ω1×Ω2

((z

1

, z

2

), (z

1

, ω

2

)) = g

Ω2

(z

2

, ω

2

) is plurisubharmonic in ω

2

which is a contradiction to the assumption

W

Ω2

(z

₂⁰

, ω

⁰₂

) < g

Ω2

(z

₂⁰

, ω

⁰₂

)

by Proposition VII:2 in [3].

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3. Some estimates. If Ω is a domain in R

ⁿ

, regular for the classical Dirich- let problem, then for every function ϕ, subharmonic near Ω we have the Riesz representation formula:

ϕ(ω) = R

Ω

G(ξ, ω)∆ϕ(ξ) + R

∂Ω

ϕ(ξ)dσ

ω

(ξ), ω ∈ Ω

where G is the Green function for Ω and dσ

ω

is the harmonic measure relatively Ω and ω.

Stokes theorem gives a similar formula for plurisubharmonic functions (cf.

Demailly [4], [5] and Ko lodziej [10]). Suppose V, ϕ and ψ ∈ PSH (Ω) ∩ L

^∞

(Ω) and define

s(r) = {z ∈ Ω; ϕ(z) = r}; B(r) = {z; ϕ(z) < r}.

We assume that B(r) ⊂⊂ Ω ∀r < 0. Consider

R

S(r)

V d

^c

ϕ ∧ (dd

^c

ψ)

^k−1

= (Stokes)

= R

B(r)

dV ∧ d

^c

ϕ ∧ (dd

^c

ψ)

ⁿ⁻¹

+ R

V dd

^c

ϕ ∧ (dd

^c

ψ)

ⁿ⁻¹

= R

B(r)

d(ϕ − r) ∧ d

^c

V ∧ (dd

^c

ψ)

ⁿ⁻¹

+ R

B(r)

V (dd

^c

ϕ) ∧ (dd

^c

ψ

ⁿ⁻¹

= (Stokes)

= − R

B(r)

(ϕ − r)dd

^c

V ∧ (dd

^c

ψ)

ⁿ⁻¹

+ R

B(r)

V dd

^c

ϕ ∧ (dd

^c

ψ)

^k−1

. Hence

R

B(r)

V (dd

^c

ϕ) ∧ (dd

^c

ψ)

ⁿ⁻¹

= R

B(r)

(ϕ − r)dd

^c

V ∧ (dd

^c

ψ)

ⁿ⁻¹

(1)

+ R

S(r)

V d

^c

ϕ ∧ (dd

^c

ψ)

ⁿ⁻¹

.

We now claim that d

^c

ϕ ∧ (dd

^c

ψ)

ⁿ⁻¹

is a positive measure on S(r).

For let 0 ≤ h ∈ C

^∞

be given. Let > 0 and define ϕ

= max{ϕ, r − }. Then

R

S(r)

hd

^c

ϕ ∧ (dd

^c

ψ)

ⁿ⁻¹

= R

S(r)

h(d

^c

ϕ

) ∧ (dd

^c

ψ)

ⁿ⁻¹

= R

S(r)

dϕ

∧ d

^c

h ∧ (dd

^c

ψ)

ⁿ⁻¹

+ R

B(r)

h(dd

^c

ϕ

) ∧ (dd

^c

ψ)

ⁿ⁻¹

= R

r−≤ϕ<r

dϕ

∧ d

^c

h ∧ (dd

^c

ψ)

ⁿ⁻¹

+ R

r−≤ϕ<r

hdd

^c

ϕ

∧ (dd

^c

ψ)

ⁿ⁻¹

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= R

r−≤ϕ<r

d(ϕ

− (r − )) ∧ d

^c

h ∧ (dd

^c

ψ)

ⁿ⁻¹

+ R

r−≤ϕ<r

hdd

^c

ϕ

∧ (dd

^c

ψ)

ⁿ⁻¹

= R

S(r)

(ϕ

− (r − ))d

^c

h ∧ (dd

^c

ψ)

ⁿ⁻¹

− R

r−≤ϕ<r

(ϕ

− (r − ))dd

^c

h ∧ (dd

^c

ψ)

ⁿ⁻¹

+ R

r−≤ϕ<r

hdd

^c

ϕ

∧ (dd

^c

ψ)

ⁿ⁻¹

. Here, the last term is nonnegative so

R

S(r)

hd

^c

ϕ ∧ (dd

^c

ψ)

ⁿ⁻¹

≥ −

R

B(r)

dd

^c

h ∧ (dd

^c

ψ)

ⁿ⁻¹

− R

|dd

^c

h ∧ (dd

^c

ψ)

ⁿ⁻¹

| → 0, → 0 which proves the claim.

Example. Let 0 ≤ V ∈ PSH (Ω), ψ ∈ PSH ∩L

^∞

(Ω) and ϕ

t

= max[W (z, ξ), −t].

Then (1) gives

R

B(r)

V (dd

^c

ϕ

t

) ∧ (dd

^c

ψ)

ⁿ⁻¹

= R

B(r)

(ϕ

t

− r)dd

^c

V ∧ (dd

^c

ψ)

ⁿ⁻¹

+ R

s(r)

V d

^c

ϕ

t

∧ (dd

^c

ψ)

ⁿ⁻¹

. Letting r → 0 we get

R

Ω

− ϕ

_t

dd

^c

V ∧ (dd

^c

ψ)

ⁿ⁻¹

≤ R

Ω

V dd

^c

ϕ

t

∧ (dd

^c

ψ)

ⁿ⁻¹

≤ sup

Ω

V R

Ω

dd

^c

ϕ

t

∧ (dd

^c

ψ)

ⁿ⁻¹

so if we choose ψ = P

n

j=1

|z

_i

|

²

then

(i) R

Ω

− W (z, ξ)∆V ≤ kV k

_L^∞

R

Ω

∆

z

W (z, ξ), if we choose ψ = V , then

(ii) R

Ω

− W (z, ξ)(dd

^c

V (z))

ⁿ

≤ kV k

_L∞

R

Ω

dd

^c_z

W (z, ξ)(dd

^c

V )

ⁿ⁻¹

, and finally if ψ = ϕ

t

,

R − ϕ

_t

dd

^c

V ∧ (dd

^c

ϕ

t

)

ⁿ⁻¹

≤ R

V (dd

^c

ϕ

t

)

ⁿ

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and so

(iii) R

− W (z, ξ)dd

^c

V ∧ (dd

^c

W )

ⁿ⁻¹

≤ R

V (dd

^c_ξ

W (z, ξ))

ⁿ

.

4. Integrability of plurisubharmonic functions. Suppose µ is a positive measure on Ω. How do we know there is a ϕ ∈ PSH ∩L

^∞

loc (Ω) with (dd

^c

ϕ)

ⁿ

= µ?

Here is a necessary condition.

Proposition 1. Let R > 1 fixed , B the unit ball. Then there exists a constant c such that

R

B¯

− ϕ(dd

^c

u)

ⁿ

≤ c R

B

− ϕdV for all 0 ≥ ϕ ∈ PSH (RB ) and −1 ≤ u ≤ 0, u ∈ PSH (RB ).

P r o o f. See [3, Prop. VI:2].

Let now Ω be hyperconvex with exhaustion function ψ. Let µ be a positive measure and assume 0 ≥ V ∈ PSH ∩L

^∞

(Ω), (dd

^c

V )

ⁿ

= µ. For m > 0, define V

m

= max(V, mψ). Then, by (1),

R V

m

(dd

^c

ψ)

ⁿ

= R

ψdd

^c

V

m

∧ (dd

^c

ψ)

ⁿ⁻¹

≤ 1 m

R V

m

dd

^c

V

m

∧ (dd

^c

ψ)

ⁿ⁻¹

≤ . . . ≤ 1 m

ⁿ⁻¹

R ψ(dd

^c

V

m

)

ⁿ

. so

0 ≤ R

− ψ(dd

^c

V

m

)

ⁿ

≤ m

ⁿ⁻¹

R

− V

_m

(dd

^c

ψ)

ⁿ

≤ m

ⁿ⁻¹

(sup

z∈Ω

−V (z)) R

(dd

^c

ψ)

ⁿ

. If τ (z) ≡ (2π)

ⁿ

, we take ψ(ξ) = W (z, ξ) and get

(2π)

ⁿ

m

ⁿ⁻¹

V

m

(z) ≤ R

W (z, ξ)(dd

^c

V

m

ξ)

ⁿ

≤ 0.

If supp µ is compact, then V

m

= V near the support of µ for m large enough and therefore

0 ≤ R

−W (z, ξ)dµ(ξ) ≤ m

ⁿ⁻¹

R

−V (ξ)(dd

^c

W (z, ξ))

ⁿ

≤ m

ⁿ⁻¹

(sup

ξ∈Ω

−W )τ (z)).

We are thus led to consider the pluricomplex potential Ω3z7→ R W (z, ξ)dµ(ξ) for positive measures µ. We have just proved

Theorem 2. Suppose −1 ≤ u ≤ 0, u ∈ PSH (Ω) and that Ω is strongly hyperconvex. Then

0≤ − R

W

Ω

(z, ξ)(dd

^c

max[u(ξ), mW

Ω

(η, ξ)])

ⁿ

≤−m

ⁿ⁻¹

R

max[u(ξ), mW

Ω

(η, ξ)](dd

^c_ξ

W

Ω

(z, ξ))

ⁿ

≤ m

ⁿ⁻¹

τ (z), z ∈ Ω, η ∈ Ω.

5. A metric defined by W . It is known that g

Ω

gives rise to an infinitesimal

Finsler pseudometric, cf. [1], [8] and [9]. We show here that W

Ω

also defines an

infinitesimal Finsler pseudometric.

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Definition. Let w ∈ Ω, ξ ∈ C

ⁿ

. We define T (ω, ξ) = lim

|l|→0 l∈C

W

Ω

(w + lξ, ω) − log |l|.

Proposition 2. T (ω, ξ) is upper semicontinuous on Ω × C

ⁿ

.

P r o o f. Note that (ω, ξ, l) 7→ W

Ω

(ω +lξ, ω)−log |l|, l 6= 0 is upper semicontinuous and subharmonic in l, for ω, ω + lξ ∈ Ω. Also, for ω, ξ fixed W

Ω

(ω + lξ, ω) ≤ c+log |l|. Therefore W

Ω

(ω +lξ, ω)−log |l| has a uniquely determined subharmonic extension over l = 0. Also

T (ω, ξ) = lim

|l|→0

W

Ω

(ω + lξ, ω) − log |l| = lim

r&0

W

Ω

(ω + rξ, ω) − log r.

By the mean value property for subharmonic functions, 1

2π

R

0

[W

Ω

(ω + re

^iθ

ξ, ω) − log r]dθ & T (ω, ξ), r & 0.

Note that for r > 0 fixed the left hand side is upper semicontinuous in (ω, ξ) and since it decreases in r, the proposition follows.

Furthermore, exp T (ω, tξ) = |t| exp T (ω, ξ), t ∈ C so exp T (ω, ξ) defines an infinitesimal Finsler pseudometric.

References

[1] K. A z u k a w a, The invariant pseudometric related to negative plurisubharmonic function, Kodai Math. J. 10 (1987), 83–92.

[2] E. B e d f o r d and J. P. D e m a i l l y, Two counterexamples concerning the pluri-complex Green function in C

ⁿ

, Indiana Univ. Math. J. 37 (1988), 865–867.

[3] U. C e g r e l l, Capacities in Complex Analysis. Aspects of Mathematics, 14, Vieweg, 1988.

[4] J. P. D e m a i l l y, Mesures de Monge-Amp` ere et mesures pluri-sousharmoniques, Math.

Z. 194 (1987), 519–564.

[5] —, Mesures de Monge-Amp` ere et caract´ erisation g´ eom´ etrique des vari´ et´ es alg´ ebriques affines, M´ em. Soc. Math. France 19 (1985), 1–125.

[6] M. J a r n i c k i and P. P f l u g, Invariant Distances and Metrics in Complex Analysis, Walter de Gruyter & Co., 1993.

[7] M. K l i m e k, Extremal plurisubharmonic functions and invariant pseudodistances, Bull.

Soc. Math. France 113 (1985), 231–240.

[8] —, Infinitesimal pseudometrics and the Schwarz lemma, Proc. Amer. Math. Soc. 105 (1989), 134–140.

[9] —, Pluripotential Theory , Oxford Science Publications, 1991.

[10] S. K o l o d z i e j, The logarithmic capacity in C

ⁿ

, Ann. Polon. Math. 48 (1988), 253–267.

1. Introduction. Let Ω be an open and connected set in R

THE SYMMETRIC PLURICOMPLEX GREEN FUNCTION

U R B A N C E G R E L L

Department of Mathematics, University of Ume˚ a S-90187 Ume˚ a, Sweden