REGULARITY OF THE PLURICOMPLEX

GREEN FUNCTION WITH SEVERAL POLES

Zbigniew Błocki

IMUJ PREPRINT 2000/07

Abstract. We show that if Ω is a C^{2,1} smooth, strictly pseudoconvex domain in ℂ^{n} then the pluricomplex Green function for Ω with several fixed poles and positive weights is C^{1,1}.

1. Introduction

If Ω is a bounded domain in ℂ^{n}, p^{1}, …, p^{k} ∈ Ω are distinct and μ_{1}, …, μ_{k} > 0 then the corresponding pluricomplex Green function is given by

g = sup B,

where

B = {v ∈ P SH(Ω) : v < 0, lim sup_{z→p^{i}}(u(z) − μ_{i} log |z − p^{i}|) < ∞, i = 1, …, k}.

One can show that g ∈ B, g is a maximal plurisubharmonic (psh) function in Ω \ {p^{1}, …, p^{k}} and

Mg = \frac{π^{n}}{n!2^{n}} ∑_{i} μ_{i}δ_{p^{i}}

Typeset by AMS-TEX

2000 Mathematics Subject Classification. Primary 32U35; Secondary 32W20.

Key words and phrases. Pluricomplex Green function with several poles, complex Monge-Ampère operator.

Partially supported by KBN Grant #2 PO3A 003 13.

(see [Le]), where M is the complex Monge-Ampère operator. For smooth u

Mu = det(\frac{∂^{2}u}{∂z_{i}∂z\overline{j}})

and by [De] Mu can be well defined as a nonegative Borel measure if u ∈ P SH(Ω) and u is locally bounded near ∂Ω.

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

Theorem 1.1. Assume that Ω is C^{2,1} smooth and strictly pseudoconvex. Then g ∈ C^{1,1}(Ω \ {p^{1}, …, p^{k}}) and

|∇^{2}g(z)| ≤ \frac{C}{min_{i} |z − p^{i}|^{2}} , z ∈ Ω \ \{p^{1}, …, p^{k}\},

where C is a constant depending only on Ω, p^{1}, … p^{k}, μ_{1}, …, μ_{k}.

One can treat it as a regularity result for the complex Monge-Ampère 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 C^{2}.

The case k = 1 was treated in [Gu] and [Bł3]. In [Gu] the C^{1,α} regularity for α < 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 C^{1,1} regularity was shown. Afterwards, in the correction to [Gu], a different method was used to show the C^{1,α} regularity.

Here we adapt the methods from [Gu] and [Bł3] for k ≥ 1. This yields also a slightly different proof for k = 1, as instead of Lemma from [Bł3] we use a holomorphic mapping

z ⟼ z + \frac{(z_{1} − p_{1}^{1}) … (z_{1} − p_{1}^{k})}{(a_{1} − p_{1}^{1}) … (a_{1} − p_{1}^{k})}h

(in appriopriate variables given by Lemma 3.2 below) which for a ∉ {p^{1}, …, p^{k}} and small h ∈ ℂ^{n} fixes p^{i} and maps a to a + h.

To get a priori estimate for the second derivative on the boundary we follow the method from [CKNS] (used also in [Gu]). In fact, all quantitative estimates necessary to obtain the constant from Theorem 1.1 are included here. We only make use of existence result - Theorem 1.1 from [Gu] (it would even be enough to use Theorem 1 from [CKNS]).

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

Theorem 1.2. If Ω is hyperconvex then g is continuous as a function defined on the set

(1.1)  \{(z, p^{1}, …, p^{k}, μ_{1}, …, μ_{k}) ∈ \overline{Ω} × Ω^{k} × (ℝ_{+})^{k} : z ≠ p^{i} ≠ p^{j} if i ≠ j\},

where for z ∈ ∂Ω we set g := 0.

(Recall that Ω is called hyperconvex if there exists ψ ∈ P SH(Ω) with ψ < 0 and

lim_{z→∂Ω} ψ(z) = 0.)

Theorem 1.3. Assume that

lim sup_{z→∂Ω} \frac{|g(z)|}{dist(z, ∂Ω)} < ∞.

Then

|∇g(z)| ≤ \frac{C}{min_{i} |z − p^{i}|}, z ∈ Ω \ \{p^{1}, …, p^{k}\},

where C is a constant depending only on Ω, p^{1}, …, p^{k}, μ_{1}, …, μ_{k}.

Notation. If z = (z_{1}, …, z_{n}) ∈ ℂ^{n} then x_{i} = Re z_{i}, y_{i} = Im z_{i}. If ζ ∈ ℂ^{n}, |ζ| = 1 then by ∂_{ζ}^{m}u(z) we will denote the m-th derivative of u in direction ζ at z. For the partial derivatives we will use the notation

u_{x_{i}} = \frac{∂u}{∂x_{i}} , u_{y_{i}} = \frac{∂u}{∂y_{i}} , u_{i} = \frac{∂u}{∂z_{i}} , u\overline{i} = \frac{∂u}{∂\overline{z}_{i}} .

If we write

|∇u| ≤ f in an open D ⊂ ℂ^{n},

where f is locally bounded, nonnegative in D, then we mean that u is locally Lipschitz and the inequality holds almost everywhere (|∇u| makes then sense by the Rademacher theorem). If we write dd^{c}u ≥ dd^{c}|z|^{2}, in fact it means exactly that u − |z|^{2} is psh. When proving existence of a constant depending only on given quantities, by C_{1}, C_{2}, … we will denote positive constants depending only on those quantities and call them under control.

1. Basic estimates

Given a bounded domain Ω in ℂ^{n}, distinct poles p^{1}, …, p^{k} ∈ Ω and weights μ_{1}, …, μ_{k} > 0 fix positive R, r, m and M so that for i, j = 1, …, k

Ω ⊂ B(p^{i}, R),
\overline{B}(p^{i}, r) ⊂ Ω and \overline{B}(p^{i}, r) ∩ \overline{B}(p^{j}, r) = ∅,
m ≤ μ_{i} ≤ M.

One can easily check the following estimates for g:

∑_{i} μ_{i} log \frac{|z − p^{i}|}{R} ≤ g(z) < 0, z ∈ Ω,

μ_{i} log \frac{|z − p^{i}|}{R} − (k − 1)M log(R/r) ≤ g(z) ≤ μ_{i} log \frac{|z − p^{i}|}{r}, z ∈ \overline{B}(p^{i}, r).

For ε with 0 < ε < r define

Ω^{ε} := Ω \ ⋃_{i} \overline{B}(p^{i}, ε)

and

g^{ε} := sup\{v ∈ P SH(Ω) : v < 0, v|_{\overline{B}(p^{i},ε)} ≤ μ_{i} log(ε/r), i = 1, …, k, \}.

One can easily check that

(2.1)  g^{ε}(z) ≤ μ_{i} log \frac{max\{|z − p^{i}|ε\}}{r}, z ∈ \overline{B}(p^{i}, r),
g^{ε} ∈ P SH(Ω),

(2.2)  g ≤ g^{ε} ≤ \frac{log(r/ε)}{log(R/ε) + (k − 1)\frac{M}{m} log(R/r)}g in Ω^{ε},

g^{ε} ↓ g^{0} := g as ε ↓ 0 and the convergence is locally uniform in Ω \ {p^{1}, …, p^{k}}.

Proposition 2.1. Assume that Ω is C^{∞} smooth and strictly pseudoconvex. Then there exists r_{0} depending only on k, r, R, m and M, 0 < r_{0} ≤ r, such that for ε with 0 < ε < r_{0} we can find v ∈ P SH(Ω) ∩ C^{∞}(\overline{Ω}) with dd^{c}v ≥ dd^{c}|z|^{2} in Ω, v = 0 on ∂Ω and for i = 1, …, k

μ_{i} log(ε/r) ≤ v(z) ≤ μ_{i} log \frac{|z − p^{i}|}{r} if ε ≤ |z − p^{i}| ≤ r.

Proof. Set

w(z) := ∑_{i} μ_{i} log \frac{|z − p^{i}|}{R} + |z − p^{1}|^{2} − R^{2},

so that w < 0 on \overline{Ω}, dd^{c}v ≥ dd^{c}|z|^{2} and w < μ_{i} log(ε/r) on ∂B(p^{i}, ε). On the other hand, for z ∈ ∂B(p^{i}, r) we have

w(z) ≥ kM log(r/R) + r^{2} − R^{2} > μ_{i} log(ε/r) + |z − p^{i}|^{2} − ε^{2},

provided that ε is such that

m log(ε/r) − ε^{2} < kM log(r/R) − R^{2}.

Similarly as in [Bł2], let χ : ℝ → ℝ be C^{∞} smooth and such that

χ(t) = 0, t ≤ −1,
χ(t) = t, t ≥ 1,
0 ≤ χ′(t) ≤ 1, t ∈ ℝ,
χ′′(t) ≥ 0, t ∈ ℝ.

For x, y ∈ ℝ set

f_{j}(x, y) := x + \frac{1}{j}χ(j(y − x)),

so that

f_{j}(x, y) = max\{x, y\} if |x − y| ≥ \frac{1}{j}.

If u, v are psh functions with dd^{c}u, dd^{c}v ≥ dd^{c}|z|^{2}, then

dd^{c}f_{j}(u, v) ≥ (1 − χ′(j(v − u)))dd^{c}u + χ′(j(v − u))dd^{c}v ≥ dd^{c}|z|^{2}.

Let ψ be a defining function for Ω. If we choose j, A sufficiently big then the function

v(z) = \{ \begin{matrix}f_{j}(w(z), μ_{i} log(ε/r) + |z − p^{i}|^{2} − ε^{2}), & z ∈ ⋃_{i} \overline{B}(p^{i}, r), \\ f_{j}(w(z), Aψ(z)) & z ∈ \overline{Ω} \ ⋃_{i} \overline{B}(p^{i}, r) \end{matrix}

has all the required properties. □

Note that if k = 1 then we may choose r_{0} = r in Proposition 2.1.

Proof of Theorem 1.2. By (2.2) g^{ε} → g locally uniformly on the set (1.1) as ε → 0. It is thus enough to show that for a fixed small ε g^{ε} is continuous as a function defined on

\overline{Ω} × \{(p^{1}, …, p^{k}) ∈ Ω^{k} : dist(p^{i}, ∂Ω) > ε, |p^{i} − p^{j}| > 2ε if i ≠ j\} × (ℝ_{+})^{k}.

Let p^{i,j} → p^{i}, μ_{i,j} → μ_{i} as j → ∞, i = 1, …, k, and

g_{j}^{ε} := sup\{v ∈ P SH(Ω) : v < 0, v|_{\overline{B}(p^{i,j},ε)} ≤ μ_{i,j} log(ε/r)\}.

Note that if 0 < ε < r_{0} and j is big enough then by Proposition 2.1 applied to a ball containing Ω we have lim_{z→∂B(p^{i},ε)} g_{j}^{ε}(z) = μ_{i} log(ε/r). Moreover, lim_{z→∂Ω} g_{j}^{ε}(z) = 0,

since Ω is hyperconvex. Therefore, by a result from [Wa] (see also [Bł1], Theorem 1.5), g_{j}^{ε} is continuous on \overline{Ω}.

To finish the proof it is enough to show that g_{j}^{ε} → g^{ε} uniformly as j → ∞ in \overline{Ω}. Fix c > 0. For z ∈ \overline{B}(p^{i}, ε) and j big enough by (2.1) we have

g_{j}^{ε}(z) ≤ μ_{i,j} log \frac{max\{|z − p^{i,j}|, ε\}}{r} ≤ μ_{i,j} log \frac{ε + |p^{i} − p^{i,j}|}{r} ≤ μ_{i} log(ε/r) + c,

whereas for z ∈ \overline{B}(p^{i,j}, ε)

g^{ε}(z) ≤ μ_{i} log \frac{max\{|z − p^{i}|, ε\}}{r} ≤ μ_{i} log \frac{ε + |p^{i} − p^{i,j}|}{r} ≤ μ_{i,j} log(ε/r) + c.

Thus for those j

g^{ε} − c ≤ g_{j}^{ε} ≤ g^{ε} + c on \overline{Ω}

and the theorem follows. □

In the proof of Theorem 1.1 we will also need to approximate g^{ε}. If 0 ≤ ε < r and 0 ≤ δ ≤ 1 define

g^{ε,δ} := sup{v ∈ P SH ∩ L^{∞}(Ω) : v ≤ g^{ε}, Mv ≥ δ in Ω^{ε}}.

Note that g^{ε,δ} is increasing in ε and decreasing in δ. We also have

(2.3)  g^{ε} + δ(|z − p^{1}|^{2} − R^{2}) ≤ g^{ε,δ} ≤ g^{ε}.

Proposition 2.2. g^{ε,δ} ∈ P SH(Ω), Mg^{ε,δ} = δ in Ω^{ε}. If Ω is hyperconvex and 0 < ε < r_{0} then g^{ε,δ} is continuous on \overline{Ω}. If Ω is C^{∞} smooth and strictly pseudoconvex, 0 < ε < r_{0} and 0 < δ ≤ 1 then g^{ε,δ} ∈ C^{∞}(\overline{Ω^{ε}}).

Proof. We use standard procedures. Let

B = {v ∈ P SH(Ω) : v ≤ g^{ε}, Mv ≥ δ in Ω^{ε}}.

By the Choquet lemma there exists a sequence v_{j} ∈ B such that (g^{ε,δ})^{∗} = (sup_{j} v_{j})^{∗}. (u^{∗} denotes the upper semicontinuous regularization of u.) If w_{j} = max{v_{1}, …, v_{j}} then Mw_{j} ≥ δ in Ω^{ε} (see e.g. [Bł2]) and thus w_{j} ∈ B. Therefore w_{j} ↑ (g^{ε,δ})^{∗} almost everywhere and by the approximation theorem from [BT2] M(g^{ε,δ})^{∗} ≥ δ in Ω^{ε}. We conclude that g^{ε,δ} ∈ P SH(Ω) and Mg^{ε,δ} ≥ δ in Ωε. The balayage procedure gives Mg^{ε,δ} = δ in Ω^{ε}.

Now assume that Ω is hyperconvex and 0 < ε < r_{0}. By [Bł1] there exists ψ ∈ P SH(Ω) ∩ C(\overline{Ω}) with ψ = 0 on ∂Ω and Mψ ≥ 1 in Ω. For A big enough

(2.4)  Aψ ≤ g^{ε,δ} ≤ 0 in Ω.

Let v be given by Proposition 2.1 applied to a ball containing Ω. Then

(2.5)  v(z) ≤ g^{ε,δ}(z) ≤ μ_{i} log \frac{|z − p^{i}|}{r} if ε ≤ |z − p^{i}| ≤ r.

For small h ∈ ℂ^{n} and z ∈ Ω^{ε} with |h| < dist(z, ∂Ω^{ε}) < 2|h| we have

|g^{ε,δ}(z + h) − g^{ε,δ}(z)| ≤ C(|h|).

By the comparison principle (see [BT2]) applied to g^{ε,δ} and g^{ε,δ}(· + h) the above inequality holds for all z with dist(z, ∂Ω^{ε}) > |h|. By (2.4) and (2.5)

lim_{h→0} C(|h|) = 0

which means that g^{ε,δ} is continuous.

The last part of the proposition follows from Proposition 2.1 and Theorem 1.1 from [Gu]. Instead of [Gu] one can use Theorem 1 from [CKNS] (p. 211) and Theorems 4.1, 4.2 below. □

3. Gradient estimates

Theorem 1.3 will follow immediately from the next result applied to δ = 0.

Theorem 3.1. Fix 0 ≤ δ ≤ 1. Assume that

lim sup_{z→∂Ω} \frac{|g^{0,δ}(z)|}{dist(z, ∂Ω)} ≤ B < ∞.

Then for ε satisfying Proposition 2.1 we have

|∇g^{ε,δ}(z)| ≤ \frac{C}{min_{i} |z − p^{i}|}, z ∈ Ω^{ε},

where C is a constant depending only on n, k, R, r, m, M and B.

The assumption of Theorem 3.1 is satisfied uniformly for δ ≤ 1 for example if Ω is smooth and strictly pseudoconvex.

Proof of Theorem 3.1. Let ρ > 0 be such that

−g^{ε,δ}(z) ≤ −g^{0,δ}(z) ≤ 2B dist(z, ∂Ω) if dist(z, ∂Ω) ≤ ρ.

For h sufficiently small

g^{ε,δ}(z + h) − g^{ε,δ}(z) ≤ 2B|h| if dist(z, ∂Ω) = |h|,

and, since by Proposition 2.1

μ_{i} log(ε/r) ≤ g^{ε,δ}(z) ≤ μ_{i} log \frac{|z − p^{i}|}{r} if ε ≤ |z − p^{i}| ≤ r,

we have

g^{ε,δ}(z + h) − g^{ε,δ}(z) ≤ μ_{i} log \frac{|z − p^{i} + h|}{ε} ≤ 2\frac{μ_{i}}{ε} |h|,

if z ∈ ∂B(p^{i}, ε + |h|), i = 1, …, k.

From the comparison principle we get

g^{ε,δ}(z + h) − g^{ε,δ}(z) ≤ 2 max\{B, \frac{M}{ε} \}|h| if ∥h| ≤ min\{ρ, dist(z, ∂Ω^{ε})\},

and thus

(3.1)  |∇g^{ε,δ}| ≤ \frac{C_{1}}{ε} in Ω^{ε}.

We will need a lemma:

Lemma 3.2. There exists a constant C̃ = C̃(k, n) such that for given p^{1}, …, p^{k} ∈ ℂ^{n}, a ∈ ℂ^{n} \ {p^{1}, …, p^{k}} we can orthonormally change variables in ℂ^{n} so that

|a − p^{i}| ≤ C̃|a_{1} − p_{1}^{i}|, i = 1, …, k.

Proof. By S denote the unit sphere in ℂ^{n}. We have to show that there exists b ∈ S such that

|a − p^{i}| ≤ C̃|⟨a − p^{i}, b⟩|, i = 1, …, k,

that is that

|⟨\frac{a − p^{i}}{|a − p^{i}|}, b⟩| ≥ \frac{1}{C̃} .

Define

C̃ := \frac{1}{min_{S^{k}} f} ,

where

f(ζ^{1}, …, ζ^{k}) := max_{b∈S} min_{i} |⟨ζ^{i}, b⟩|

is a continuous function on S^{k}. It remains to show that f > 0 on S^{k}. Fix ζ^{1}, …, ζ^{k} ∈ S and define K_{i} := {b ∈ S : ⟨b, ζ^{i}⟩ = 0}, i = 1, …, k. Then ⋃_{i} K_{i} ≠ S and thus for b ∈ S \ ⋃_{i} K_{i} we have

f(ζ^{1}, …, ζ^{k}) ≥ min_{i} |⟨ζ^{i}, b⟩| > 0. □

End of proof of Theorem 3.1. Fix a ∈ Ω^{ε} and choose variables as in Lemma 3.2. Set

P (λ) := (λ − p_{1}^{1}) … (λ − p_{1}^{k}),

so that

\frac{|P (z_{1})|}{|P (a_{1})|} ≤ C_{2} \frac{min_{i} |z − p^{i}|}{min_{i} |a − p^{i}|} ≤ \frac{C_{3}}{min_{i} |a − p^{i}|}, z ∈ Ω.

For h sufficiently small let

Ω′′ := \{z ∈ Ω : z + \frac{P (z_{1})}{P (a_{1})}h ∈ Ω\}

and

Ω′ := Ω′′ \ ⋃_{i} \overline{B}(p^{i}, ε + ε′),

where

ε′ = min{ε, r − ε, dist(a, ∂Ω^{ε}), ρ}.

Set

v(z) := g^{ε,δ} (z + \frac{P (z_{1})}{P (a_{1})}h) + \frac{C_{4}}{min_{i} |a − p^{i}|}(|z − p^{1}|^{2} − R^{2})|h|,

so that if f C_{4} is big enough then

Mv ≥ |1 + \frac{P′(z_{1})}{P (a_{1})}h_{1}|^{2} δ + \frac{C_{4}}{min_{i} |a − p^{i}|}|h| ≥ δ.

For z ∈ ∂Ω′′ we have

v(z) − g^{ε,δ}(z) ≤ 2Bdist(z, ∂Ω) ≤ 2B \frac{C_{3}}{min_{i} |a − p^{i}|}|h|,

whereas for z ∈ ∂B(p^{i}, ε + ε′)

v(z) − g^{ε}(z) ≤ \frac{C_{1}}{ε} \frac{|P (z_{1})|}{|P (a_{1})|}|h| ≤ C_{1}C_{2} \frac{ε + ε′}{ε min_{i} |a − p^{i}|} ≤ \frac{C_{5}}{min_{i} |a − p^{i}|}|h|.

Therefore, the comparison principle gives

g^{ε}(a + h) − g^{ε}(a) ≤ \frac{C_{6}}{min_{i} |a − p^{i}|}|h| if |h| ≤ ε′ \frac{min_{i} |a − p^{i}|}{C_{3}}

and the theorem follows. □

4. Estimates of the second derivative

Our goal will be to estimate |∇^{2}g^{ε,δ}| for small ε, δ. First, we need such an estimate on ∂Ω^{ε}. We will follow the method from [CKNS] (see also [Gu]). We shall prove two theorems.

Theorem 4.1. Let Ω be a bounded strictly pseudoconvex domain in ℂ^{n} and ψ a C^{∞} psh defining function for Ω. Assume that dd^{c}ψ ≥ dd^{c}|z|^{2} and that there are positive constants A, a such that

|ψ|, |∇ψ|, |∇^{2}ψ|, |∇^{3}ψ| ≤ A on \overline{Ω},
|∇ψ| ≥ a on ∂Ω.

For ρ > 0 denote U = \{z ∈ ℂ^{n} : dist(z, ∂Ω) < ρ\}. Let u ∈ P SH(Ω∩U)∩C^{∞}(\overline{Ω}∩U) be such that u = 0 on ∂Ω and u < 0, Mu = δ in Ω ∩ U, where 0 < δ ≤ δ_{0}. Assume also that there are positive constants b, B such that

|∇u| ≥ b on ∂Ω,
|∇u| ≤ B on \overline{Ω} ∩ U.

Then there is a constant C = C(n, ρ, a, A, b, B, δ_{0}) such that

|∇^{2}u| ≤ C on ∂Ω.

Theorem 4.2. Fix α > 1 and let Ω = {z ∈ ℂ^{n} : 1 < |z| < α}. Assume that u ∈ P SH(Ω) ∩ C^{∞}(\overline{Ω}) is such that u = 0 on ∂B_{1} (B_{α} = B(0, α)), u > 0, Mu = δ > 0 in Ω. Suppose moreover that there are positive constants β, b, B such that

u ≥ β on ∂B_{α},
|∇u| ≥ b on ∂B_{1},
|∇u| ≤ B on \overline{Ω}.

Then there exist positive constants δ_{0} = δ_{0}(n, α, β) and C = C(n, α, β, b, B) such that if 0 < δ ≤ δ_{0}, we have

|∇^{2}u| ≤ C on ∂B_{1}.

Proof of Theorem 4.1. If z_{0} ∈ ∂Ω then by N_{z_{0}} we will denote the outer normal of ∂Ω at z_{0}. For z ∈ \overline{Ω} we have

∂_{N_{z_{0}}} ψ(z) = ⟨∇ψ(z), ∇ψ(z_{0})/|∇ψ(z_{0})|⟩ ≥ |∇ψ(z_{0})| − A|z − z_{0}|.

Hence, there are positive constants ã, ρ̃ under control such that

∂_{N_{z_{0}}} ψ(z) ≥ ã, z_{0} ∈ ∂Ω, z ∈ \overline{Ω} ∩ \overline{B}(z_{0}, ρ̃).

Fix z_{0} ∈ ∂Ω. We may assume that N_{z_{0}} = (0, …, 0, 1), so that ∂_{N_{z_{0}}} = ∂/∂x_{n}. Since both ψ and u are C^{∞} defining functions for Ω, there exists a C^{∞} function v, defined in a neighborhood of ∂Ω such that u = vψ and v > 0 on \overline{Ω} ∩ U. Therefore, if t, s ∈ {x_{1}, y_{1}, …, x_{n−1}, y_{n−1}, y_{n}}, then

(4.1)  u_{ts}(z_{0}) = \frac{u_{x_{n}}(z_{0})ψ_{ts}(z_{0})}{ψ_{x_{n}}(z_{0})}

and thus

(4.2)  |u_{ts}(z_{0})| ≤ C_{1}.

Suppose now that we know that

(4.3)  |u_{tx_{n}}(z_{0})| ≤ C_{2}

and we want to estimate |u_{x_{n}x_{n}}(z_{0})|. We have

u_{x_{n}x_{n}} = 4u_{n\overline{n}} − u_{y_{n}y_{n}}

and, by (4.1), (4.2), (4.3) and since dd^{c}ψ ≥ dd^{c}|z|^{2},

δ_{0} ≥ δ = det(u_{i\overline{j}}(z_{0})) ≥ u_{n\overline{n}}(z_{0})(a/A)^{n−1} − C_{3}.

It thus remains to show (4.3). On \overline{Ω} ∩ \overline{B}(z_{0}, ρ̃) define

T := u_{t} − \frac{ψ_{t}}{ψ_{x_{n}}} u_{x_{n}},

so that

(4.4)  T = 0 on ∂Ω ∩ \overline{B}(z_{0}, ρ̃).

We have

T_{x_{n}}(z_{0}) = u_{tx_{n}}(z_{0}) − \frac{ψ_{tx_{n}}(z_{0})}{ψ_{x_{n}}(z_{0})}u_{x_{n}}(z_{0}),

and thus it is enough to prove that

|T_{x_{n}}(z_{0})| ≤ C_{4}.

Set f := ψ_{t}/ψ_{x_{n}}, then

(4.5)  |∇f|, |∇^{2}f| ≤ C_{5} in \overline{Ω} ∩ \overline{B}(z_{0}, ρ̃).

Since det(u_{i\overline{j}}) is constant, one can show that

u^{i\overline{j}}u_{i\overline{j}t} = u^{i\overline{j}}u_{i\overline{j}x_{n}} = 0.

(Here (u^{i\overline{j}}) denotes the inverse transposed matrix of (u_{i\overline{j}}).) Hence, we can compute

u^{i\overline{j}}T_{i\overline{j}} = −u_{x_{n}}u^{i\overline{j}}f_{i\overline{j}} − 2Re u^{i\overline{j}}u_{ix_{n}}f\overline{j} = −u_{x_{n}}u^{i\overline{j}}f_{i\overline{j}} − 2f_{x_{n}} − 2Im u^{i\overline{j}}u_{iy_{n}}f\overline{j}.

Since

u^{i\overline{j}}(u_{y_{n}}^{2})_{i\overline{j}} = 2u^{i\overline{j}}u_{iy_{n}}u_{\overline{j}y_{n}},

the Schwartz inequality and (4.5) give

u^{i\overline{j}}(±T + \frac{1}{2}u_{y_{n}}^{2})_{i\overline{j}} ≥ ∓u_{x_{n}}u^{i\overline{j}}f_{i\overline{j}} ∓ 2f_{x_{n}} − u^{i\overline{j}}f_{i}f\overline{j} ≥ −C_{6}(∑_{i} u^{i\overline{i}} + 1).

On ∂Ω we have u_{y_{n}} = u_{x_{n}}ψ_{y_{n}}/ψ_{x_{n}} and thus by (4.4)

| ± T + \frac{1}{2}u_{y_{n}}^{2}| ≤ C_{7}|z − z_{0}|^{2}, z ∈ ∂Ω ∩ \overline{B}(z_{0}, ρ̃).

Moreover,

| ± T + \frac{1}{2}u_{y_{n}}^{2}| ≤ C_{8} in \overline{Ω} ∩ \overline{B}(z_{0}, ρ̃),

and we obtain that if w = ±T + \frac{1}{2}u_{y_{n}}^{2} − C_{9}|z − z_{0}|^{2}, where C_{9} is big enough, then w ≤ 0 on ∂(Ω ∩ B(z_{0}, ρ̃)) and

u^{i\overline{j}}w_{i\overline{j}} ≥ −C_{10}(∑_{i} u^{i\overline{i}} + 1).

Therefore, if C_{11} and C_{12} are big enough then w+C_{11}ψ+C_{12}u ≤ 0 on ∂(Ω∩B(z_{0}, ρ̃)) and u^{i\overline{j}}(w + C_{11}ψ + C_{12}u)_{i\overline{j}} ≥ 0 in Ω ∩ B(z_{0}, ρ̃). By the maximum principle w + C_{11}ψ + C_{12}u ≤ 0 in Ω ∩ B(z_{0}, ρ̃) and thus

|T_{x_{n}}(z_{0})| ≤ C_{11}A + C_{12}B. □

Proof of Theorem 4.2. Set

ψ(z) = λ(|z|^{2} − 1),

where λ = β/(α^{2} − 1), so that ψ ≤ u in Ω for δ sufficiently small. We now follow the proof of Theorem 4.1. Fix z_{0} ∈ ∂B_{1}, we may assume that z_{0} = (0, …, 0, 1). We may reduce the problem to the estimate

|u_{tx_{n}}(z_{0})| ≤ C_{1}.

Similarly as before we get that if w = ±T + \frac{1}{2}u_{y_{n}}^{2} − C_{2}|z − z_{0}|^{2}, where C_{2} is big enough, then

u^{i\overline{j}}w_{i\overline{j}} ≥ −C_{3}(∑_{i} u^{i\overline{i}} + 1) in Ω ∩ B(z_{0}, 1)

and w ≤ 0 on ∂(Ω ∩ B(z_{0}, 1)).

Now by the inequality between arithmetic and geometric means we have

u^{i\overline{j}}(ψ − u)_{i\overline{j}} ≥ λ ∑_{i} u^{i\overline{i}} − n ≥ \frac{λ}{2} ∑_{i} u^{i\overline{i}} + n(\frac{λ}{2δ^{1/n}} − 1) ≥ \frac{λ}{2}(∑_{i} u^{i\overline{i}} + 1)

for δ small enough. Thus

u^{i\overline{j}}(w + C_{4}(ψ − u))_{i\overline{j}} ≥ 0 in Ω ∩ B(z_{0}, 1)

if C_{4} is sufficiently big and by the maximum principle we conclude that

|T_{x_{n}}(z_{0})| ≤ C_{4}B. □

Proof of Theorem 1.1. Let ψ be a C^{2,1} defining function for Ω with dd^{c}ψ ≥ dd^{c}|z|^{2} in Ω and

|ψ|, |∇ψ|, |∇^{2}ψ|, |∇^{3}ψ| ≤ A on \overline{Ω},
|∇ψ| > a on ∂Ω

for some positive a and A. We can find r̃ > 0 such that for every z_{0} ∈ ∂Ω there exists a ball B(z_{1}, 2r̃), contained in Ω and tangent to ∂Ω at z_{0}. Then

g(z) ≤ −\frac{γ}{log 2} log \frac{|z − z_{1}|}{2r̃} if r̃ ≤ |z − z_{1}| ≤ 2r̃,

where

γ = max_{dist(z,∂Ω)≥r̃}g(z).

Therefore we can find b with

lim inf_{z→∂Ω} \frac{|g(z)|}{dist(z, ∂Ω)} > b > 0.

Let ψ_{j} = ψ ∗ ρ_{1/j} be the standard regularization of ψ and let Ω_{j} = {ψ_{j} < 0}. If j is big enough then the constants A, a and b are good also for ψ_{j} and Ω_{j}. Thus, we may assume that ψ (and thus Ω) is C^{∞}, provided that we prove that the constant in Theorem 1.1 depends only on n, k, r, R, m, M, A, a and b.

By Proposition 2.2 g^{ε,δ} ∈ C^{∞}(\overline{Ω^{ε}}), if 0 < ε < r_{0}, 0 < δ ≤ 1. It is enough to show that for small positive ε and δ we have

|∇^{2}g^{ε,δ}(z)| ≤ \frac{C_{1}}{min_{i} |z − p^{i}|^{2}} , z ∈ Ω^{ε}.

Since |∇g^{ε,δ}| ≥ b on ∂Ω, by Theorems 3.1 and 4.1 we have

(4.6)  |∇^{2}g^{ε,δ}| ≤ C_{2} on ∂Ω.

For |w| ≥ 1 and fixed i = 1, …, k set

u(w) := g^{ε,δ}(p^{i} + εw) − μ_{i} log(ε/r).

By (2.2) and (2.3)

u(w) ≥ μ_{i} log |w| − C_{3}.

Thus, if α is so big that β := m log α − C_{3} > 0 then for sufficiently small ε u ≥ β on ∂B_{α}. Moreover, g^{ε,δ} ≥ −C_{4} on ∂B(p^{i}, r). Thus by the comparison principle, for sufficiently small ε we have

\frac{μ_{i}}{2} log \frac{|z − p^{i}|}{r} + \frac{μ_{i}}{2} log(ε/r) + |z − p^{i}|^{2} − ε^{2} ≤ g^{ε,δ}(z) if ε ≤ |z − p^{i}| ≤ r.

Therefore

|∇g^{ε,δ}| ≥ \frac{μ_{i}}{2ε} on ∂B(p^{i}, ε)

and |∇u| ≥ μ_{i}/2 on ∂B_{1}. From Theorem 4.2 it follows that for δ small enough

|∇^{2}u| ≤ C_{5} on ∂B_{1}

which means that

(4.7)  |∇^{2}g^{ε,δ}| ≤ \frac{C_{5}}{ε^{2}} on ∂B(p^{i}, 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 \ {p^{1}, …, p^{k}}. From the fact that g^{ε,δ} is psh it follows that

(4.8)  |∇^{2}g^{ε,δ}(a)| = lim sup_{h→0} \frac{g^{ε,δ}(a + h) + g^{ε,δ}(a − h) − 2g^{ε,δ}(a)}{|h|^{2}}.

Let P be as in the proof of Theorem 3.1 and let Ω′′ ⋐ Ω′ ⋐ Ω, ε′ > 0. For z ∈ \overline{Ω′} \ ⋃_{i} B(p^{i}, ε + ε′) and small h set

D(z, h) := g^{ε,δ} (z + \frac{P (z_{1})}{P (a_{1})}h)

and

v(z, h) = D(z, h) + D(z, −h) + \frac{C_{6}}{|P (a_{1})|^{2}} (|z − p^{1}|^{2} − R^{2})|h|^{2},

so that

D(z, 0) = g^{ε,δ}(z),
D(a, h) = g^{ε,δ}(a + h),

v is psh in z and

(4.9)  v(a, h) ≥ g^{ε,δ}(a + h) + g^{ε,δ}(a − h) − \frac{C_{6}R^{2}}{|P (a_{1})|^{2}} |h|^{2}.

If C_{6} is sufficiently big and h sufficiently small then

(Mv(·, h))^{1/n}

≥ (|1 + \frac{P′(z_{1})}{P (a_{1})}h_{1}|^{2/n} + |1 − \frac{P′(z_{1})}{P (a_{1})}h_{1}|^{2/n}) δ^{1/n} + \frac{C_{6}}{|P (a_{1})|^{2}} |h|^{2}

≥ (2δ)^{1/n}.

The Taylor expansion of D(z, ·) about the origin gives

v(z, h) ≤ D(z, h) + D(z, −h) ≤ 2g^{ε,δ}(z)||∇^{2}(D(z, ·))||_{\overline{B}(0,|h|)}|h|^{2}.

Since

|∇^{2}(D(z, ·))(h̃)| = \frac{|P (z_{1})|^{2}}{|P (a_{1})|^{2}} |∇^{2}g^{ε,δ} (z + \frac{P (z_{1})}{P (a_{1})}h̃)| ,

we get

v(z, h) ≤ 2g^{ε,δ}(z) + C′|h|^{2}, z ∈ ∂Ω′,
v(z, h) ≤ 2g^{ε,δ}(z) + C′_{i}|h|^{2}, z ∈ ∂B(p^{i}, ε + ε′),

where

C′ = C_{7} \frac{||∇^{2}g^{ε,δ}||_{Ω\\overline{Ω′′}}}{|P (a_{1})|^{2}} ,

C′_{i} = C_{8} \frac{(ε + ε′)^{2}||∇^{2}g^{ε,δ}||_{B(p^{i},ε+2ε′)∩Ω^{ε}}}{|P (a_{1})|^{2}}

for h small enough. Now we can apply the comparison principle to v and 2g^{ε,δ}. We obtain

v(a, h) ≤ 2g^{ε,δ}(a) + max{C′, C′_{1}, …, C′_{k}}|h|^{2}.

By (4.8) and (4.9)

|∇^{2}g^{ε,δ}(a)| ≤ max\{C′, C′_{1}, …, C′_{k}\} + \frac{C_{6}R^{2}}{|P (a_{1})|^{2}} .

If we let Ω′′ ↑ Ω, ε′ ↓ 0 and use (4.6), (4.7) then the desired estimate follows. □

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Jagiellonian University, Institute of Mathematics, Reymonta 4, 30-059 Kraków, Poland

E-mail address: blocki@im.uj.edu.pl