The equation of complex Monge-Ampère type

and stability of solutions

by

U. Cegrell and S. Kołodziej

IMUJ PREPRINT 2003/23

Abstract. We prove that in a family of plurisubharmonic functions with Monge-Ampère measures bounded from above by such a measure of one function weak convergence is equivalent to convergence in capacity. We also show a very general statement on the existence of solutions of the complex Monge-Ampère type equation.

0. Introduction

It is well known (see e.g. [H]) that a sequence of plurisubharmonic (psh) functions which converges in the sense of distributions is also convergent in L_{loc}^{p} for any finite p ≥ 1. This rather strong type of convergence of functions does not, however, imply the weak^{∗} convergence of Monge-Ampère measures associated to those functions (see [L][C1]).

An example from [CK] shows that there exists a sequence of measures in a ball dμ_{j} converging in weak^{∗} topology to dμ and such that the solutions to the Dirichlet problem for the Monge-Ampère equation

(dd^{c}u_{j})^{n} = dμ_{j},

Typeset by AMS-TEX

1991 Mathematics Subject Classification. primary 32U05, secondary 32U40.

Key words and phrases. plurisubharmonic function, complex Monge-Ampère operator.

with zero boundary data, are uniformly bounded, but u_{j} fail to converge to u which solves

(dd^{c}u)^{n} = dμ.

In this paper we show that on subsets of P SH(Ω) where the Monge-Ampère masses are under control the above phenomena: discontinuity of the Monge-Ampère operator and the lack of stability of the solutions of the Monge-Ampère equation no longer occur. Furthermore, the weak^{∗} convergence of functions within those subsets is equivalent to the convergence in capacity (see Preliminaries below for definitions). By a theorem of Xing [X] for a sequence of psh functions converging in capacity the Monge-Ampère measures corresponding to those functions tend to the Monge-Ampère measure of the limit function.

We shall consider families of psh functions on which the Monge-Ampère operator and its inverse restricted to the image are continuous. For a strictly pseudoconvex domain Ω, a continuous function φ on ∂Ω, and a psh function u from F(φ) (defined in [A], see Section 1) the family with the properties mentioned above consists of those psh functions v in Ω which satisfy

v ∈ F(φ), (dd^{c}v)^{n} ≤ dμ := (dd^{c}u)^{n},

provided dμ is a finite measure which vanishes on pluripolar sets. The condition v ∈ F(φ) implies that v has boundary data φ in the sense that

lim sup_{ζ→z} v(ζ) = φ(z), z ∈ ∂Ω.

The same holds for a hyperconvex Ω if F(φ) contains the maximal plurisubharmonic function which is continuous up to the boundary and equal φ on ∂Ω.

In the families described above the convergence u_{j} → u in the sense of distributions is the same as the convergence in capacity. Since they contain non continuous functions one cannot expect anything more. In [K4] it was shown that with suitable bounds on Monge-Ampère measures (for instance in L^{p} norm , p > 1) we may get equicontinuity of families of psh functions.

In the second part of the paper we prove a very general existence theorem on the Dirichlet problem for the complex equation of the Monge-Ampère type

(dd^{c}u)^{n} = F (u(·), ·) dμ

with dμ finite and vanishing on pluripolar sets and F ≥ 0 which is only continuous in the first variable and upper bounded by a function from L^{1}(dμ) . Our theorem strengthens the results from [BT2][C2] and [K3]. The case of smooth functions is done in [CKNS].

1. Preliminaries.

We call a domain in ℂ^{n} hyperconvex if there exists nonzero u ∈ P SH(Ω) ∩ C(\overline{Ω}) such that u = 0 on ∂Ω. Throughout the paper we shall assume that Ω is hyperconvex.

Definition [C3][C4][A]. The set of bounded psh functions u satisfying lim_{z→ζ} u(z) = 0 for any ζ ∈ ∂Ω, and ∫_{Ω}(dd^{c}u)^{n} < ∞ we denote by E_{0}. A psh function u belongs to F if there exists a sequence u_{j} ∈ E_{0} with u_{j} ↓ u in Ω and

sup_{j} ∫_{Ω}(dd^{c}u_{j})^{n} < ∞.

A negative psh function u belongs to E if for every z_{0} ∈ Ω there is a neighborhood Ω′ ⊂ Ω of z_{0} and a sequence u_{j} ∈ E_{0} with u_{j} ↓ u in Ω′ and

sup_{j} ∫_{Ω}(dd^{c}u_{j})^{n} < ∞.

Furthermore, for φ ∈ C(∂Ω) we denote by E_{0}(φ) (resp. F(φ) ) the set of those u ∈ P SH(Ω) for which

h ≥ u ≥ h + v

for some v ∈ E_{0} (resp. v ∈ F) and h := sup{w ∈ P SH∩L^{∞}(Ω) : lim sup_{z→ζ} w(z) ≤ φ(ζ), ζ ∈ ∂Ω} provided that h → φ on ∂Ω (by the Walsh theorem h is then continuous on the closure of Ω). The subclass F^{a}(φ) consists of those functions from F(φ) for which (dd^{c}v)^{n} vanishes on all pluripolar sets.

Finally, by F_{1} we denote the set of those u ∈ F for which (with u_{j} as above)

sup_{j} ∫_{Ω} −u_{j}(dd^{c}u_{j})^{n} < ∞.

By [A] F(φ) ⊂ E if φ is non positive.

Let us observe that functions from F are not necessarily locally bounded and, according to [C3], we have

lim sup_{z→ζ} u(z) = 0

for any ζ ∈ ∂Ω.

Given a bounded plurisubharmonic function u one can define the Monge-Ampère measure

(dd^{c}u)^{n} = dd^{c}u ∧ dd^{c}u ∧ ... ∧ dd^{c}u (n terms)

as in [BT1]. This is a nonnegative Borel measure.

According to [C4] the Monge-Ampère operator is also well defined for u ∈ F(φ) and then (dd^{c}u)^{n} = lim(dd^{c}u_{j})^{n}, where u_{j} ∈ E_{0}(φ), u_{j} ↓ u. The limit measure is independent of the choice of this sequence. Furthermore, by Theorem 3.8 from [C3] we have for u ∈ F_{1}

(1.1)  ∫_{Ω} −u_{j}(dd^{c}u_{j})^{n} ↑ ∫_{Ω} −u(dd^{c}u)^{n}, j → ∞.

The following capacity was introduced in [BT3] for Borel sets E ⊂ Ω:

cap(E, Ω) = sup{∫_{E}(dd^{c}u)^{n} : u ∈ P SH(Ω), −1 ≤ u < 0}.

We call it the Bedford-Taylor capacity. It is closely related to the relative extremal function of the given compact set K:

u_{K}(z) = sup{u(z) : u ∈ P SH ∩ L^{∞}(Ω), u < 0 in Ω, u ≤ −1 on K}.

Its upper semicontinuous regularization u_{K}^{∗}(z) := \overline{lim}_{ζ→z}u_{K}(ζ) is a plurisubharmonic function and, by [BT3]

(1.2)  cap(K, Ω) = ∫_{K}(dd^{c}u_{K}^{∗})^{n} = ∫_{Ω}(dd^{c}u_{K}^{∗})^{n}.

It is a subadditive Choquet capacity and in bounded Ω it vanishes exactly on pluripolar sets (see [BT3]). Recall that a set E in ℂ^{n} is pluripolar if for any z ∈ E there exists a neighbourhood V of z and v ∈ P SH(V) such that E∩V ⊂ {v = −∞}.

Definition. A sequence u_{j} of functions defined in Ω is said to converge in capacity to u if for any t > 0 and K ⊂⊂ Ω

lim_{j→∞} cap(K ∩ {|u − u_{j}| > t}, Ω) = 0.

With respect to this type of convergence the Monge-Ampère operator is continuous.

Theorem 1.1 (Convergence theorem)[X]. Let {u_{k}^{j}}_{j=1}^{∞} be a locally uniformly bounded sequence of psh functions in Ω for k = 1, 2, ..., n; and let u_{k}^{j} → u_{k} ∈ P SH ∩ L_{loc}^{∞}(Ω) in capacity as j → ∞ for k = 1, 2, ..., n. Then

dd^{c}u_{1}^{j} ∧ ... ∧ dd^{c}u_{n}^{j} → dd^{c}u_{1} ∧ ... ∧ dd^{c}u_{n}

in the weak topology of currents.

This result has been generalized in [C5] in the following sense.

Theorem 1.2 [C5]. Suppose that u_{j} ∈ F, j = 0, 1, .... and u_{0} ≤ u_{j}. If u_{j} converges to u in capacity then (dd^{c}u_{j})^{n} tends to (dd^{c}u)^{n} as j → ∞.

One of the basic properties of plurisubharmonic functions is their quasicontinuity.

Theorem 1.3 [BT3]. For any psh function u defined in Ω and any positive number ϵ one can find an open set U ∈ Ω with cap(U, Ω) < ϵ and such that u restricted to Ω \ U is continuous.

We shall use the well known comparison principle from [BT3] as well as its generalization to the class F_{1}.

Comparison principle [BT3][C3]. If u, v ∈ P SH ∩ L^{∞}(Ω) and lim inf_{z→∂Ω}(u(z) − v(z)) ≥ 0, then

∫_{{u<v}}(dd^{c}v)^{n} ≤ ∫_{{u<v}}(dd^{c}u)^{n}.

The inequality remains valid if u and v belong to F_{1}.

The following lemma has been shown in [C5] for functions from F. The proof carries over to E since a function from E is locally in F.

Lemma 1.4. Assume that a finite positive measure μ in Ω vanishes on all pluripolar sets. If u_{0}, u_{j} ∈ E, u_{0} ≤ u_{j}, ∫ −u_{0}dμ < ∞, and u_{j} → u as distributions then

lim_{j→∞} ∫_{Ω} u_{j}dμ = ∫_{Ω} u dμ.

Corollary. Any Borel measure μ in Ω which vanishes on pluripolar sets and such that μ(Ω) is finite has the following property. Given a sequence u_{j} ∈ E converging weakly to u one can extract a subsequence u_{j_{k}} converging pointwise dμ almost everywhere.

Proof. Indeed, for any positive integer N one can find, using Lemma 1.4, a subsequence of max(u_{j}, −N) which tends to max(u, −N) dμ almost everywhere. Then using diagonal procedure we get the desired subsequence.

2. Stability.

First we shall prove stability results in a strictly pseudoconvex domain for the class of bounded functions with arbitrary continuous boundary data. Next we extend this result first to the case of hyperconvex domains and then to the bigger class of functions F(φ). Finally we show that the Monge-Ampère operator is continuous on F_{1}.

Lemma 2.1. Let Ω be a bounded strictly pseudoconvex domain in ℂ^{n}, φ ∈ C(∂Ω), and let dV denote the Lebesgue measure. Suppose u_{j}, v_{j} are uniformly bounded sequences of plurisubharmonic functions in Ω with

lim_{ζ→z} u_{j}(ζ) = φ(z) for z ∈ ∂Ω, φ ∈ C^{∞}(∂Ω),

and such that v_{j} → v in capacity, v, v_{j} are bounded and psh in a neighbourhood of the closure of Ω, v_{j} = v = 0 on ∂Ω, and v_{j} = v outside Ω. Furthermore, u_{j} → u in L^{1}(Ω) for some u ∈ P SH ∩ L^{∞}(Ω), with lim_{ζ→z} u(ζ) = φ(z) for z ∈ ∂Ω. If

(dd^{c}u_{j})^{n} ≤ (dd^{c}v_{j})^{n} = g_{j}dV, g_{j} ∈ L^{∞}(Ω)

then a) ∫ |u − u_{j}|(dd^{c}u_{j})^{n} → 0 as j → ∞.

b) (dd^{c}u_{j})^{n} → (dd^{c}u)^{n}.

c) cap({|u − u_{j}| ≥ a}) → 0 as j → ∞ for any a > 0.

Proof. Part a). It is proved in [K1, Lemma 2.7 ] or [K2, Lemma 3.4.5] that under our assumptions

(2.1)  ∫_{{u−u_{j}>a}}(dd^{c}u_{j})^{n} → 0 for any a > 0.

Those proofs are conducted for some specific sequences, so for the sake of completeness the proof of (2.1) is given in Appendix.

We use notation f_{+} = max(f, 0), f_{−} = − min(f, 0). By a version of Hartogs’ lemma [H] (u − u_{j})_{−} tends uniformly to zero on any set where u is continuous.

Since u is continuous off a set of arbitrarily small capacity (see Theorem 1.3) and the sequence u_{j} is uniformly bounded we can conclude that

∫ (u − u_{j})_{−}(dd^{c}u_{j})^{n} → 0.

This combined with (2.1) and the fact that by the assumptions and Chern-Levine -Nirenberg inequalities [CLN] the integrals ∫ (dd^{c}u_{j})^{n} are uniformly bounded, give a).

Part b) follows from a) and [K1] (see also Proposition 3.2.1 in [K2]).

For the proof of part c) let us fix a > 0 and denote by U_{j} the set {u_{j} < u − a}. Since cap is an inner capacity (see [BT3]) one can choose for every j a compact set K_{j} ⊂ U_{j} with

cap(K_{j}, Ω_{j}) ≥ cap(U_{j}, Ω) − 1/j.

We denote by u_{K_{j}} the relative extremal function of K_{j} . By (1.2)

cap(K_{j}, Ω) = ∫_{K_{j}}(dd^{c}u_{K_{j}}^{∗})^{n}.

Observe that

K_{j} ⊂ U_{j} ⊂ \{u_{j} < u + \frac{a}{3}u_{K_{j}}^{∗} − \frac{a}{3}\} =: V_{j} ⊂ \{u_{j} < u − \frac{a}{3}\} ⊂⊂ Ω.

Using this and the comparison principle we get

\begin{aligned}(\frac{a}{3})^{n}(cap(U_{j}, Ω) − 1/j) & ≤ (\frac{a}{3})^{n}cap(K_{j}, Ω) ≤ (\frac{a}{3})^{n} ∫_{V_{j}}(dd^{c}u_{K_{j}}^{∗})^{n} \\ & ≤ ∫_{V_{j}}(dd^{c}u_{j})^{n} ≤ ∫_{\{u_{j}<u− \frac{a}{3} \}}(dd^{c}u_{j})^{n}. \end{aligned}

Since, by part a), the right hand side tends to 0 we conclude that

cap(U_{j}, Ω) → 0,

which completes the proof.

Theorem 2.2 (Stability theorem). Let Ω be a strictly pseudoconvex domain. Suppose dμ = (dd^{c}v)^{n} for v ∈ P SH ∩ L^{∞}(Ω) with

lim_{ζ→z} v(ζ) = 0 for z ∈ ∂Ω, ∫_{Ω} dμ < ∞.

Let f_{j} be a sequence of dμ- measurable functions satisfying 0 ≤ f_{j} ≤ 1 and such that f_{j}dμ → f dμ as measures. Suppose that u_{j} solves the Dirichlet problem

(∗) u_{j} ∈ P SH ∩ L^{∞}(Ω), (dd^{c}u_{j})^{n} = f_{j}dμ,

lim_{ζ→z} u_{j}(ζ) = φ(z) for z ∈ ∂Ω,

for some continuous φ. Then for u = lim u_{j} (in L^{1}(Ω)) we have (dd^{c}u)^{n} = f dμ. Furthermore u_{j} tends to u in capacity.

Proof. We shall reduce the proof to the case of smooth φ, continuous f_{j} and compactly supported μ. Suppose first that we have shown the statement for smooth φ. Then we can approximate the given φ ∈ C(∂Ω) by a decreasing sequence φ_{s} ∈ C^{∞}(∂Ω) with ||φ_{s}−φ|| < 1/s (where ||·|| is the L^{∞} norm) and denote by u_{j}^{s} the solution of (∗) with φ_{s} in place of φ. By the comparison principle ||u_{j}^{s} − u_{j}|| < 1/s and for u^{s} = lim_{j→∞} u_{j}^{s} we have ||u^{s} − u|| ≤ 1/s. Thus

\begin{aligned}||u_{j} − u|| & ≤ ||u_{j}^{s} − u_{j}|| + ||u_{j}^{s} − u^{s}|| + ||u^{s} − u|| \\ & ≤ ||u_{j}^{s} − u^{s}|| + 2/s. \end{aligned}

Given δ > 0 and ϵ > 0 choose s > 6/δ and, using the convergence in capacity u_{j}^{s} → u^{s} , j → ∞, find j_{0} such that for j > j_{0}

cap({|u_{j}^{s} − u^{s}| > δ/3}, Ω) < ϵ.

Then, by the above inequalities

cap({|u_{j} − u| > δ}, Ω) < ϵ, j > j_{0}.

Therefore in what follows we can assume that φ is smooth.

The following proposition allows to reduce the proof to the case of continuous f_{j}’s.

Proposition. Given dμ measurable f with 0 ≤ f ≤ 1 and δ > 0 one can find continuous g such that for u, v the solutions of the Dirichlet problem (∗) corresponding to measures f dμ and g dμ respectively, we have

cap({|u − v| ≥ δ}, Ω) < δ.

Proof. Fix ϵ = δ^{n+1}/2 and use Egorov’s theorem to find continuous g with 0 ≤ g ≤ 1 and μ(E) < ϵ, where E = {f ≠ g}. Denote by ρ the solution of (∗) with measure 2dμ_{|E} and zero boundary data. Then by the comparison principle

|u − v| ≤ −ρ.

Consider a regular compact K ⊂ {ρ < −δ} (regular means that the extremal function is continuous). Then K ⊂ {ρ < δu_{K}} and by the comparison principle

\begin{aligned}δ^{n}cap(K, Ω) & = δ^{n} ∫_{K}(dd^{c}u_{K})^{n} \\ & ≤ ∫_{Ω}(dd^{c}ρ)^{n} < 2ϵ. \end{aligned}

Thus from the choice of ϵ and K the statement follows.

Using this proposition we can find for any f_{j} continuous f̃_{j} such that the solutions u_{j}, ũ_{j} corresponding to those measures obey

cap({|u_{j} − ũ_{j}| ≥ 1/j}, Ω) < 1/j.

Since cap is subadditive the last inequality says that the convergence in capacity of u_{j} is equivalent to the same type of convergence of ũ_{j}. So we may assume that f_{j} are continuous.

In the next step we prove (dd^{c}u)^{n} = f dμ under extra assumption that μ is compactly supported. Then we can replace v by max(v − 1, ρ) with ρ being the defining function of Ω such that ρ < v − 2 on K ⊂⊂ Ω, where K contains supp μ in its interior. Then this new function is defined and smooth in a neighbourhood of ∂Ω. Note that the Monge-Ampère measure of new v is equal to dμ plus a measure supported on Ω \ int K and thus f_{j}dμ are not changed if we take f_{j} = 0 off K. For such v one can define a sequence of smooth psh functions ρ_{j} ↓ v in the closure of Ω. Set α_{j} = sup_{Ω} ρ_{j}. Consider a regularizing sequence m_{j} for the function m(x, y) = max(x, y) of two real variables such that m_{j} are smooth, non decreasing in both variables and convex, with m_{j} ↓ m, 0 ≤ m_{j} − m ≤ α_{j}, and m_{j} = m for |x − y| < α_{j}/2. Then the functions v_{j} = m_{j}(ρ_{j} − 2α_{j}, ρ) are smooth and plurisubharmonic in a neighbourhood of the closure of Ω. Since

ρ_{j} − 2α_{j} ≤ v_{j} ≤ ρ_{j} + α_{j}

and α_{j} → 0 we conclude that v_{j} → v uniformly on each compact set where the restriction of v is continuous. Thus, in view of Theorem 1.3, it is convergent in capacity.

Define g_{j} by (dd^{c}v_{j})^{n} = g_{j}dV. For fixed j we apply [BT1] to find w̃_{s} such that

(dd^{c}w̃_{s})^{n} = f_{j}g_{s}dV,

lim_{ζ→z} w̃_{s}(ζ) = φ(z) for z ∈ ∂Ω. Then by Lemma 2.1 and the uniqueness of the solution of the Dirichlet problem we have the convergence in capacity w̃_{s} → u_{j} as s → ∞. Thus for s(j) large enough the function w_{j} = w̃_{s(j)} satisfies

(2.2)  \begin{aligned}||u_{j} − w_{j}||_{L^{1}(Ω)} & ≤ 1/j, \\ cap(\{|w_{j} − u_{j}| ≥ 1/j\}) & ≤ 1/j. \end{aligned}

By (2.2) and the convergence u_{j} → u in L^{1}(Ω) we get that w_{j} → u in L^{1}(Ω). Applying Lemma 2.1 once more with w_{j} in place of u_{j} we conclude that w_{j} → u in capacity. Therefore, passing to a subsequence, we can assume that

cap({|w_{j} − u| ≥ 1/j}) ≤ 1/j.

This estimate combined with (2.2) and the subadditivity of cap give

cap({|u_{j} − u| ≥ 2/j}) ≤ 2/j.

Thus u_{j} → u in capacity. (Precisely speaking, we have shown that one can extract from u_{j} a subsequence converging in capacity but this is enough to get the statement.) Furthermore, applying Xing’s result (Theorem 1.1) we conclude that (dd^{c}u_{j})^{n} → (dd^{c}u)^{n} which gives the statement for compactly supported μ. To get the general case take an exhaustion sequence K_{s} ↑ Ω of compact sets and define μ_{s} as the restriction of μ to K_{s}. Denote by u_{j}^{s} the solution of (∗) corresponding to the measure f_{j}dμ_{s}. By the first part of the proof u_{s} = lim_{j→∞} u_{j}^{s} satisfies (dd^{c}u_{s})^{n} = f dμ_{s}. Since u_{s} ↓ u we get by the convergence theorem [BT3] (dd^{c}u)^{n} = f dμ. The proof is complete.

We can show a version of this theorem in hyperconvex domains for suitable boundary data.

Theorem 2.2’. Let Ω be hyperconvex, φ ∈ ∂Ω, and dμ = (dd^{c}v)^{n} for v ∈ P SH ∩ L^{∞}(Ω) with

lim_{ζ→z} v(ζ) = φ(z) for z ∈ ∂Ω, ∫_{Ω} dμ < ∞.

Then the statement of Theorem 2.2 remains true.

Proof. Let h ∈ P SH ∩ C(Ω̄) be such that (dd^{c}h)^{n} = 0 and h = φ on ∂Ω (it exists by the assumptions). Take an exhaustion sequence of strictly pseudoconvex sets Ω_{k} ↑ Ω. Fix E ⊂⊂ Ω and δ, ϵ > 0. Choose k_{0} so large that E ⊂ Ω_{k_{0}} and

(2.3)  h − v < δ/2 on ∂Ω_{k}, k ≥ k_{0}.

Applying [K1] solve u_{jk} ∈ P SH ∩ L^{∞}(Ω_{k}), (dd^{c}u_{jk})^{n} = f_{j}dμ_{|Ω_{k}},

lim_{ζ→z} u_{jk}(ζ) = h(z) for z ∈ ∂Ω_{k},

u ∈ P SH ∩ L^{∞}(Ω), (dd^{c}u)^{n} = fdμ,

lim_{ζ→z} u(ζ) = h(z) for z ∈ ∂Ω.

and v_{k} ∈ P SH ∩ L^{∞}(Ω_{k}), (dd^{c}v_{k})^{n} = fdμ_{|Ω_{k}},

lim_{ζ→z} v_{k}(ζ) = h(z) for z ∈ ∂Ω_{k}.

By the comparison principle and (2.3) we have

u ≤ v_{k} < u + δ/2, u_{j} ≤ u_{jk} < u_{j} + δ/2 in Ω_{k}.

Therefore

(2.4)  \begin{aligned}u_{j} − u & ≤ u_{jk} − u = (u_{jk} − v_{k}) + (v_{k} − u) < (u_{jk} − v_{k}) + δ/2, \\ u − u_{j} & ≤ (v_{k} − u_{jk}) + (u_{jk} − u_{j}) < v_{k} − u_{jk} + δ/2. \end{aligned}

Fix k > k_{0} and use Theorem 2.2 to find j_{0} with

(2.5)  cap({|u_{jk} − v_{k}| > δ/2} ∩ E) < ϵ/2

for j > j_{0}. Using subadditivity of cap we infer from (2.4) and (2.5) that

cap({|u_{j} − u| > δ} ∩ E) ≤ 2cap({|u_{jk} − v_{k}| > δ/2} ∩ E) < ϵ.

This proves the statement.

In the most general version of the stability theorem we consider the family of functions from F(φ) with a bound on their Monge-Ampère masses.

Theorem 2.2” (stability in F^{a}(φ)). Let Ω be hyperconvex and let dμ be a nonnegative Borel measure which vanishes on pluripolar sets, such that dμ = (dd^{c}v)^{n} for v ∈ F(φ), φ ∈ C(∂Ω), and satisfying ∫_{Ω} dμ < ∞. Then for u_{j}, u ∈ F(φ) which solve

(dd^{c}u_{j})^{n} = f_{j}dμ, (dd^{c}u)^{n} = fdμ

for f, f_{j} measurable with values in [0, 1], and satisfying f_{j}dμ → f dμ, we have u_{j} → u in capacity.

Proof. By [C3] there exists ψ ∈ E_{0} and g ∈ L^{1}((dd^{c}ψ)^{n}) with dμ = g(dd^{c}ψ)^{n}. Set

dμ_{k} = min(g, k)(dd^{c}ψ)^{n}.

Then, by [K1][C3][C4] there exist v_{k}, u_{jk} ∈ F(φ) ∩ L^{∞}(Ω) and η_{k} ∈ F solving

\begin{aligned}(dd^{c}v_{k})^{n} & = fdμ_{k}, (dd^{c}u_{jk})^{n} = f_{j}dμ_{k} \\ (dd^{c}η_{k})^{n} & = [g − min(g, k)](dd^{c}ψ)^{n}. \end{aligned}

By the comparison principle u_{jk} ↓ u_{j}, v_{k} ↓ u. Hence

(2.6)  \begin{aligned}u_{j} − u & ≤ u_{jk} − u = (u_{jk} − v_{k}) + (v_{k} − u) \\ u − u_{j} & ≤ (v_{k} − u_{jk}) + (u_{jk} − u_{j}) ≤ v_{k} − u_{jk} − η_{k}, \end{aligned}

where the last inequality follows from the comparison principle.

Fix E ⊂⊂ Ω and δ, ϵ > 0. Using the fact that η_{k} ↑ 0 (in capacity) choose k so large that

cap({η_{k} < −δ/4} ∩ E) < ϵ/4, cap({|v_{k} − u| > δ/4} ∩ E) < ϵ/4.

By Theorem 2.2’ u_{jk} → v_{k} in capacity as j → ∞. Find j_{0} with

cap({|u_{jk} − v_{k}| > δ/4} ∩ E) < ϵ/4

for j > j_{0}. Those inequalities and (2.6) yield

\begin{matrix}cap(\{|u_{j} − u| > δ\} ∩ E) & ≤ 2cap(\{|u_{jk} − v_{k}| > δ/4\} ∩ E) \\ +cap(\{|v_{k} − u| > δ/4\} ∩ E) & + cap(\{η_{k} < −δ/4\} ∩ E) < ϵ. \end{matrix}

The proof is completed.

Corollary. Let μ be a measure and φ ∈ C(∂Ω) a function satisfying the hypothesis of Theorem 2.2”. Denote by A(φ, μ) the set of all solutions of the Dirichlet problem

\begin{matrix}u ∈ F^{a}(φ), \\ (dd^{c}u)^{n} = fdμ, \end{matrix}

where f varies over all dμ measurable functions with 0 ≤ f ≤ 1. Then in A(φ, μ) the weak^{∗} convergence is equivalent to the convergence in capacity.

Proof. Let u_{j} → u in the sense of distributions and let (dd^{c}u_{j})^{n} = f_{j}dμ. Choose a subsequence f_{j_{k}} converging in weak^{∗} topology to some f. By Theorem 2.2” (dd^{c}u)^{n} = f dμ and u_{j_{k}} → u in capacity. Since this argument works for any subsequence extracted from the original one we get u_{j} → u in capacity. Thus the corollary follows.

We end this section proving the continuity of the Monge-Ampère operator in the class F_{1}.

Theorem 2.3. Let u, u_{j} ∈ F_{1} with u_{j} → u as distributions. If

lim_{j→∞} ∫_{Ω} u_{j}(dd^{c}u_{j})^{n} = ∫_{Ω} u(dd^{c}u)^{n}

then a) ∫ |u − u_{j}|(dd^{c}u_{j})^{n} → 0 as j → ∞.

b) (dd^{c}u_{j})^{n} → (dd^{c}u)^{n}.

c) u_{j} → u in capacity.

Proof. Denote by v_{j} the function which is the upper semicontinuous regularization of sup_{k≥j} u_{k}. Then v_{j} ≥ u_{j} and v_{j} ↓ u. So

(2.7)  ∫_{Ω} |u − u_{j}|(dd^{c}u_{j})^{n} ≤ ∫_{Ω}[(v_{j} − u_{j}) + (v_{j} − u)](dd^{c}u_{j})^{n}.

The inequality v_{j} ≥ u_{j} and integration by parts, applied repeatedly, give

\begin{aligned}∫_{Ω} −v_{j}(dd^{c}v_{j})^{n} & ≤ ∫_{Ω} −u_{j}(dd^{c}v_{j})^{n} \\ & = ∫_{Ω} −v_{j}dd^{c}u_{j} ∧ (dd^{c}v_{j})^{n−1} ≤ ... ≤ ∫_{Ω} −v_{j}(dd^{c}u_{j})^{n}. \end{aligned}

Hence by (1.1) we have

∫_{Ω} u_{j}(dd^{c}u_{j})^{n} ≤ ∫_{Ω} v_{j}(dd^{c}u_{j})^{n} ≤ ∫_{Ω} v_{j}(dd^{c}v_{j})^{n} → ∫_{Ω} u(dd^{c}u)^{n}.

From [C3, Theorem 3.2] and the assumptions one obtains

∫_{Ω} −u(dd^{c}u_{j})^{n} ≤ (∫_{Ω} −u(dd^{c}u)^{n})^{1/n+1}(∫_{Ω} −u_{j}(dd^{c}u_{j})^{n})^{n/n+1} → ∫_{Ω} −u(dd^{c}u)^{n}.

Therefore, in view of the last two formulae, the right hand side of (2.7) goes to zero.

Part b) of the theorem is a consequence of the first part, [C3, Lemma 5.3] and (1.1). The convergence in capacity we prove in the same way as in Lemma 2.1 using the fact that the comparison principle works in F_{1}.

3. The equation of complex Monge-Ampère type.

In this section we show a very general existence theorem for the complex equation of Monge-Ampère type. We shall need the following two results from [C6].

Theorem 3.1. Suppose Ω is a bounded hyperconvex domain and φ is a continuous function on ∂Ω such that φ = u for some function u which is psh in Ω and continuous along ∂Ω. Let μ be a finite positive measure in Ω which vanishes on all pluripolar sets. Then there is a unique v ∈ F^{a}(φ) with

(dd^{c}v)^{n} = dμ.

Theorem 3.2. The comparison principle is valid in F^{a}(φ).

Now we are in a position to prove the following existence result.

Theorem 3.3. Let Ω be a bounded hyperconvex domain and let μ and φ satisfy the assumptions of Theorem 3.1. Suppose further that F (t, z) is a dV × dμ- measurable function on (−∞, 0] × Ω which is continuous in t and which satisfies

0 ≤ F (t, z) ≤ g(z)

for some g ∈ L^{1}(dμ). Then there exists u ∈ F^{a}(φ) such that

(dd^{c}u)^{n} = F (u(·), ·) dμ.

Proof. It is no loss of generality to assume that φ is negative. Denote by A the set of u ∈ F^{a}(φ) such that u ≥ u_{0} where u_{0} ∈ F^{a}(φ) satisfies

(dd^{c}u_{0})^{n} = g dμ.

The set is convex and compact in L^{1}(dV) topology. The solution u_{0} exists and is uniquely determined by Theorem 3.1. We define a map G : A → A so that G(u) = v if

(dd^{c}v)^{n} = F (u(·), ·) dμ.

Then (dd^{c}v)^{n} ≤ g dμ and so, by Theorem 3.2, v ≥ u_{0} and v ∈ A. By Schauder’s fixed point theorem [S] we are done as soon as we show that G is continuous. So suppose u_{j} ∈ A with u_{j} → u. Applying the corollary to Lemma 1.4 one can extract a subsequence (still denoted by u_{j}) converging to u dμ almost everywhere. Let w_{k} ∈ F^{a}(φ) be the solution of

(dd^{c}w_{k})^{n} = sup_{j≥k} F (u_{j}(·), ·)dμ ≥ F (u_{k}(·), ·)dμ.

Then, by the comparison principle (Theorem 3.2), w_{k} ≤ v_{k}, where v_{k} = G(u_{k}). Furthermore, let w̃_{k} solve

(dd^{c}w̃_{k})^{n} = inf_{j≥k} F (u_{j}(·), ·)dμ ≤ F (u_{k}(·), ·)dμ.

Then w̃_{k} ≥ v_{k}. Since lim_{k→∞} w_{k} and lim_{k→∞} w̃_{k} have the same Monge-Ampère mass both limits define the same function v = G(u) (see Theorem 3.1). From the above inequalities we get

v = lim_{k→∞} w_{k} ≤ lim_{k→∞} v_{k} ≤ lim_{k→∞} w̃_{k} = v.

Thus G(u_{j}) → G(u) and the theorem follows.

Appendix

Proof of formula (2.1).

Lemma A. Suppose that, after passing to a subsequence (which does not change u since u_{j} converge in L^{1}) we have

∫_{E_{j}(a_{0})}(dd^{c}u_{j})^{n} > A_{0}, A_{0} > 0, a_{0} > 0,

with the notation E_{j}(a_{0}) = {u_{j} < u − a_{0}}. Then there exist a_{m} > 0, A_{m} > 0, k_{1} > 0 such that

(A.1)  ∫_{E_{j}(a_{m})}(dd^{c}v_{j})^{n−m} ∧ (dd^{c}v_{k})^{m} > A_{m}, j > k > k_{1}.

Proof. We shall proceed by induction over m. For m = 0 the statement holds by the hypothesis. We assume that (A.1) is true for some fixed m < n and now we shall prove it for m + 1.

Let us observe that by the Chern-Levine-Nirenberg inequalities [CLN] there exists C > 0 such that

(A.2)  ∫_{Ω} T ≤ C,

for currents T which are wedge products of dd^{c}v_{j} or dd^{c}u_{j}. Indeed, v_{j} are bounded and plurisubharmonic in a fixed neighbourhood of Ω̄ and each u_{j} can be extended to the same set as a plurisubharmonic function by h + v_{j}, where h is a smooth psh function in a neighbourhood of Ω̄ which is equal φ on ∂Ω. On this larger domain we apply the Chern-Levine-Nirenberg inequalities. We also assume that all the functions involved have L^{∞} norm bounded by one.

Denote by T = T (j, k, m) the current (dd^{c}v_{j})^{n−m−1} ∧ (dd^{c}v_{k})^{m}. For fixed ϵ ∈ (0, \frac{a_{m}A_{m}}{4C+4} ) we choose an open set U such that

cap(U, Ω) < ϵ/2^{n+1},

and both u and v are continuous on Ω \ U. Note that, assuming for simplicity, that −1 < v_{j}, u_{j} < 0, we have

(A.3)  ∫_{U}(dd^{c}(v_{j} + v_{k}))^{n} ≤ 2^{n}cap(U, Ω) < ϵ/2.

Moreover one can replace v_{j} by u_{j} in this inequality. Then for k > k_{0} and we have

(A.4)  \begin{aligned}v_{k} & ≤ v + ϵ, \\ u_{k} & ≤ u + ϵ, \end{aligned}

on Ω \ U. Indeed, the inequalities are valid in a neighbourhood of ∂Ω because all u_{j} (resp. v_{j}) are bounded from above by the maximal function in Ω with boundary data φ (resp. 0). On the remaining part of Ω \ U one obtains (A.4) by the Hartogs lemma. Set

\begin{matrix}J′(j, k) := ∫_{Ω}(u − u_{j})dd^{c}v_{j} ∧ T, \\ J(j, k) := ∫_{Ω}(u − u_{j})dd^{c}v_{k} ∧ T, j > k > k_{0}. \end{matrix}

Integrating by parts we get

J′(j, k) − J(j, k) = ∫_{Ω}(v_{j} − v_{k})dd^{c}(u − u_{j}) ∧ T = ∫_{Ω\U} ... + ∫_{U} ... .

Since v_{j} → v uniformly away from U one can find k_{1} > k_{0} such that ||v_{j} − v_{k}|| < ϵ/2C on Ω \ U for j > k > k_{1}. Thus, using (A.2) and (A.3), we conclude that each integral on the right hand side does not exceed ϵ/2 for such j, k. So

(A.5)  J′(j, k) − J(j, k) ≤ ϵ ≤ \frac{a_{m}A_{m}}{4} , j > k > k_{1}.

Using the induction hypothesis, (A.2), (A.3), and (A.4) we have

\begin{aligned}J′(j, k) & ≥ a_{m} ∫_{E_{j}(a_{m})} dd^{c}v_{j} ∧ T − ϵ ∫_{Ω\U} dd^{c}v_{j} ∧ T − ∫_{U} dd^{c}v_{j} ∧ T \\ & ≥ a_{m} ∫_{E_{j}(a_{m})} dd^{c}v_{j} ∧ T − ϵ(C + 1) ≥ a_{m}A_{m} − ϵ(C + 1) ≥ \frac{3a_{m}A_{m}}{4} , \end{aligned}

for j > k > k_{2} > k_{1}. Combined with (A.5) this gives

(A.6)  J(j, k) ≥ \frac{a_{m}A_{m}}{2} , j > k > k_{2}.

Fixing d > 0 one can estimate J(j, k) from above as follows

\begin{aligned}J(j, k) & ≤ ∫_{\{u_{j}<u−d\}} dd^{c}v_{k} ∧ T + d ∫_{Ω} dd^{c}v_{k} ∧ T \\ & ≤ ∫_{\{u_{j}<u−d\}} dd^{c}v_{k} ∧ T + dC. \end{aligned}

Setting a_{m+1} := d = \frac{a_{m}A_{m}}{4C} in the last formula and combining it with (A.6) we finally arrive at

∫_{E_{j}(a_{m+1})} dd^{c}v_{k} ∧ T ≥ \frac{a_{m}A_{m}}{4} := A_{m+1}, j > k > k_{2},

which concludes the proof of the inductive step. Thus Lemma A follows.

Now we shall prove formula (2.1) reasoning by contradiction. So, suppose the hypothesis of Lemma A is valid. Then using its statement for m = n we can fix k > k_{1} such that

∫_{E_{j}(a_{n})}(dd^{c}v_{k})^{n} > A_{n} if j > k.

Since, by the construction, (dd^{c}v_{k})^{n} ≤ M_{k}dV for some M_{k} > 0 one infers from the last inequality that

V (E_{j}(a_{n})) ≥ M_{k}^{−1} ∫_{E_{j}(a_{n})}(dd^{c}v_{k})^{n} > \frac{A_{n}}{M_{k}} , j > k,

which contradicts the fact that u_{j} → u in L_{loc}^{1}. Thus (2.1) follows.

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Department of Mathematics, Umeå University S-901 87 Umeå, SWEDEN and FMI, Mid Sweden University, S-851 70 Sundsvall, SWEDEN

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