(1)TheC1,1Regularity of the Pluricomplex Green Function Z b i g n i e w B ł o c k i If is a domain in Cnandζ

TheC^1,1Regularity of the Pluricomplex Green Function

Z b i g n i e w B ł o c k i

If is a domain in Cⁿandζ ∈ , then the pluricomplex Green function in  with pole atζ is defined as

g = sup{u ∈ PSH() : u < 0, lim sup

z→ζ (u(z) − log|z − ζ|) < ∞}

(see [5] for details). The main goal of this note is to prove the following result.

Theorem 1. Let be a C^∞strictly pseudoconvex domain inCⁿ, and let g be the pluricomplex Green function of  with pole at some ζ ∈ . Then g is C^1,1 in ¯ \ {ζ} (that is, g is C^1,1in \ {ζ } and the second derivative of g is bounded near∂).

An example given in [1] shows thatg need not be C²smooth up to the boundary.

It remains an open problem if, in that example,g /∈ C²( \ {p}).

In [4], Guan claimed to prove theC^1,α regularity for everyα < 1. However, the proof was incomplete because the inequality (3.6) in [4] is false. In a correc- tion to [4], written after I had sent him a preliminary version of this paper (with the proof of Theorem 1), Guan has given a new proof of theC^1,αregularity.

Our proof will be based on a construction from [4] of an approximating se- quence forg and an idea from [2] used to show C^1,1regularity for the solutions of the complex Monge–Ampère equation in a ball (see also [3]).

Using similar methods, one can also characterize domains where the Green function is Lipschitz up to the boundary. We recall that a domain inCⁿis called hyperconvex if it admits a bounded PSH exhaustion function.

Theorem 2. Let  be a bounded hyperconvex domain in Cⁿ, and let g be the Green function of  with a pole at ζ ∈ . Then g ∈ C^0,1( ¯ \ {ζ}) if and only if there existsψ ∈ PSH() with

−C dist(z, ∂) ≤ ψ(z) < 0, z ∈ , for someC > 0.

Proof of Theorem 1. We may assume that ζ = 0. Choose ε > 0 such that Bε b , and set ε =  \ ¯Bε. By [4], there is a sequence of functions u^ε ∈ PSH(ε)∩C^∞( ¯ε) which increase locally uniformly to g on ¯\{0} as ε ↓ 0 and

Received June 15, 1999. Revision received September 29, 1999.

Partially supported by KBN Grant no. 2 PO3A 003 13.

211

which satisfyu^ε = 0 on ∂ and u^ε = log|z| + ψ on ∂B_ε, where ψ is smooth in

¯ and det(u_{i ¯}^ε) = ε. It follows that the tangential derivatives of the second order ofu^εwith respect to∂B_εare bounded; that is,

k∇²(u^ε|_∂Bε)k ≤ C1. (1)

In addition, it was shown in [4] that theu^εsatisfy

k∇u^εk∂, k∇²u^εk∂≤ C2. (2) HereC1andC2are constants depending only on.

FixK b  \ {0}. By C3, C4, ... we will denote positive constants depending only on and K. We need to show that

k∇²u^εk_K ≤ C3. (3)

Forζ ∈ Cⁿ\ {0} with |ζ| = 1, let ∂ζ denote the directional derivative in the direc- tionζ. Since u^εis plurisubharmonic, we have

∂_ζ²u^ε+ ∂_iζ²u^ε≥ 0.

This easily gives

|∇²u^ε(a)| = sup

|ζ|=1∂_ζ²u^ε(a) = lim sup

h→0

u^ε(a + h) + u^ε(a − h) − 2u^ε(a)

|h|² (4)

fora ∈ K.

We will need a lemma as follows.

Lemma. Let 0< ε0 < r1 < r2andR > 0. Then there exist δ > 0 and a C^∞ smooth mapping

T : [0, ε0]× ( ¯Br2\ Br1) × ¯Bδ× ¯BR 7−→ Cⁿ

(Brstands for an open ball centered at the origin with radiusr) such that T (ε, a, h, ·) is holomorphic in B_R,

T (ε, a, h, ·) maps ∂B_ε onto∂B_ε, (5) T (ε, a, h, a) = a + h,

T (ε, a, 0, z) = z.

Proof. LetT (ε, a, h, ·) be a holomorphic automorphism of Bε (defined, in fact, onBR) of the form U B P, where

P(z) = ε hz, bi

|b|² b +p 1− |b|²



z −hz, bi

|b|² b



− εb

ε − hz, bi ,

|b| < R/ε (see [6]), and U is a linear orthogonal mapping with P(a) =|a + h|

|a| a, U

|a + h|

|a| a



= a + h.

C Regularity of the Pluricomplex Green Function 213

One can check that the first condition is satisfied ifb = εαa, where α = |a + h| − |a|

|a|(|a + h||a| − ε²). This gives

P(z) = hz, ai

|a|² a +p

1− ε²α²|a|²



z −hz, ai

|a|² a



− ε²αa

1− αhz, ai .

The existence of an appriopriateU, depending smoothly on a and h and in fact in- dependent ofε, is clear.

Proof of Theorem 1 (cont.). Let⁰and⁰⁰be domains such thatK b ⁰b ⁰⁰b

. We will use the foregoing lemma with r1, r2 andR such that K ⊂ ¯B_r2\ B_r1

and ⊂ BR. For z ∈ ⁰⁰andh, ε small enough, set

v(z) := u^ε(T (ε, a, h, z)) + u^ε(T (ε, a, −h, z)) so that it is well-defined andv(a) = u^ε(a + h) + u^ε(a − h).

A Taylor expansion about the origin of an arbitrary smooth functionf gives f(h) + f(−h) = 2f(0) + ¹₂(∇²f(h⁰) + ∇²f(h⁰⁰)) · h²

for someh⁰∈ [0, h] and h⁰⁰∈ [0, −h]. Therefore, by (1) and (2),

v(z) ≤ 2u^ε(z) + C4|h|², z ∈ ∂B_ε. (6) On the other hand,

v(z) ≤ 2u^ε(z) + ˜C|h|², z ∈ ∂⁰⁰, (7) where

˜C = sup

|h⁰|≤|h|,z∈∂⁰⁰|∇_h²(u^εB T )(ε, a, h⁰, z)|.

It follows that

˜C ≤ C5(k∇²u^εk_¯\0+ k∇u^εk²_¯\0) (8) forh small enough. Since the mapping A 7→ (det A)^1/nis superadditive on the set of positive hermitian matrices, we have

(det(v_{i ¯}))^1/n ≥ ε^1/n¡

|JacT (ε, a, h, ·)|^2/n+ |JacT (ε, a, −h, ·)|^2/n

≥ ε^1/n(2 − C6|h|²). (9)

LetM > 0 be such that |z|²− M ≤ 0 for z ∈ , and define

w(z) = v(z) − max{C4, ˜C}|h|²+ ε^1/nC6|h|²(|z|²− M ).

Thenw is PSH in ⁰⁰, w ≤ 2u^εon∂B_ε∪ ∂⁰⁰by (6) and (7), and det(w_{i ¯}) ≥ 2ⁿε in⁰⁰by (9). The comparison principle (see e.g. [2]) now implies thatw ≤ 2u^ε in⁰⁰. In particular, w(a) ≤ 2u^ε(a), and this coupled with (4) and (8) gives

|∇²u^ε(a)| ≤ C7(k∇²u^εk_¯\0+ k∇u^εk²_¯\0) + C8.

Since⁰can be chosen to be arbitrarily close to, (3) follows thanks to (2).

Proof of Theorem 2. The “only if ” part is obvious. Assume again thatζ = 0 and fixK b  \ {0}. Let r > 0 be such that Br b . For 0 < ε < r, define

u^ε:= sup{v ∈ PSH() : v < 0, v|Bε≤ log(ε/r)}.

Then one can easily show thatu^ε∈ PSH()∩C( ¯), u^ε= 0 on ∂, u^ε= log(ε/r) on ¯Bε, and u^ε↓ g as ε ↓ 0 (see e.g. [5]). Since g is a maximal PSH function near

∂, we may assume that

u^ε≥ g ≥ ψ near ∂. (10)

Fora ∈ K, ε as before, and h small enough, define

⁰= {z ∈  : T (ε, a, h, z) ∈ }.

By (10) and the assumption onψ we have

u^ε(z) ≥ ψ(z) ≥ −C dist(z, ∂) ≥ −C⁰|h|, z ∈ ∂⁰, whereC⁰depends only onK and . Hence, for z ∈ ∂⁰we have

u^ε(T (ε, a, h, z)) ≤ 0 ≤ u^ε(z) + C⁰|h|.

Sinceu^εis maximal on⁰\ ¯Bε, (1) gives

u^ε(T (ε, a, h, z)) ≤ u^ε(z) + C⁰|h|, z ∈ ⁰.

Thus, ifz = a for a ∈ K and |h| < δ, where δ depends only on K and , we have u^ε(a + h) ≤ u^ε(a) + C⁰|h|

and the theorem follows.

Acknowledgment. This paper was written during my Fulbright Fellowship at the Indiana University in Bloomington and the University of Michigan in Ann Arbor. I would also like to thank Professor E. Bedford for calling my attention to [4].

References

[1] E. Bedford and J.-P. Demailly, Two counterexamples concerning the pluri-complex Green function inCⁿ, Indiana Univ. Math. J. 37 (1988), 865–867.

[2] E. Bedford and B. A. Taylor, The Dirichlet problem for a complex Monge–Ampère equation, Invent. Math. 37 (1976), 1–44.

[3] A. Dufresnoy, Sur l’équation de Monge–Ampère complexe dans la boule deCⁿ, Ann. Inst. Fourier 39 (1989), 773–775.

[4] B. Guan, The Dirichlet problem for complex Monge–Ampère equations and regularity of the pluri-complex Green function, Comm. Anal. Geom. 6 (1998), 687–703.

C Regularity of the Pluricomplex Green Function 215

[5] M. Klimek, Pluripotential theory, Clarendon Press, Oxford, 1991.

[6] W. Rudin, Function theory in the unit ball of Cⁿ, Grundlehren Math. Wiss., 241, Springer-Verlag, New York, 1980.

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