Equicontinuity of families of plurisubharmonic

functions with bounds on their Monge-Ampère masses.

by

Sławomir Kołodziej

IMUJ PREPRINT 2001/06

Abstract. It is proved that families of solutions of the complex Monge-Ampère equation with fixed boundary values and certain bounds on the right hand side are equicontinuous.

0. Introduction. It is a well known property of plurisubharmonic functions that if a sequence u_{j} ∈ P SH(Ω) tends to u ∈ P SH(Ω) in the sense of distribution theory then it is convergent in a much stronger sense; for instance, u_{j} → u in L_{loc}^{p}(Ω) for any p ∈ [p, ∞) (see e.g. [H]). However, the Monge-Ampère measures (dd^{c}u_{j})^{n} may still have no limit (see [L], [C], [CK]). To be sure that (dd^{c}u_{j})^{n} converge as measures we need to know that u_{j} → u with respect to capacity [X]. In the present paper we study the convergence of plurisubharmonic functions with fixed boundary values and their Monge-Ampère measures controlled in terms of the Bedford-Taylor capacity. Then rather weak assumptions force the uniform convergence.

We shall work throughout in a strictly pseudoconvex domain Ω ⊂ ℂ^{n}. In this paper we call an increasing function h : ℝ_{+} → (1, ∞) admissible if it satisfies

∫_{1}^{∞}(xh^{1/n}(x))^{−1} dx < ∞,

and if for some a > 1 b > 1 and x_{0} > 0 we have

h(ax) ≤ bh(x) for x > x_{0}.

Typeset by AMS-TEX

2000 Mathematics Subject Classification. primary 32U05, secondary 32U40.

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

Partially supported by KBN Grant No. PO3A 010 19

Let us define the family of nonnegative Borel measures in Ω associated to an admissible function h and a positive constant A:

F(A, h) = \{μ : μ(K) ≤ F (cap(K, Ω)) for F (x) = \frac{Ax}{h(x^{−1/n})}

and any compact K ⊂ Ω}.

Here cap stands for the Bedford-Taylor capacity (see the next section). In [K1], [K2] and [K3] it was proved that the Dirichlet problem for the complex Monge-Ampère equation in Ω

(0.1) u ∈ P SH ∩ C(\overline{Ω}), (dd^{c}u)^{n} = dμ,

lim_{z→x} u(z) = φ(x) for x ∈ ∂Ω.

has a solution for any measure in F(A, h) and any given continuous boundary data φ. Furthermore, given φ and F(A, h) we have a uniform bound for L^{∞} norm of those solutions.

For a function ψ : ℝ_{+} → ℝ_{+} such that \frac{ψ(x)}{x} increases to ∞ as x → ∞ we define

L^{ψ}(c_{0}) = {f ∈ L^{1}(Ω) : f ≥ 0, ∫_{Ω} ψ(f) dV ≤ c_{0}}.

We shall prove that any weakly convergent sequence in the family of functions

\begin{matrix}P(A, h, ψ, c_{0}; φ) \\ = \{u ∈ P SH(Ω) ∩ C(\overline{Ω}) : (dd^{c}u)^{n} ∈ F(A, h) ∩ L^{ψ}(c_{0}), u = φ on ∂Ω\} \end{matrix}

(φ continuous) is in fact uniformly convergent. As a consequence the family P(A, h, ψ, c_{0}; φ) is an equicontinuous family. In particular, families of solutions of (0.1) corresponding to dμ = f dV (dV denoting the Lebesgue measure) with f ∈ L^{ψ}(c_{0}) for ψ(t) = |t|(log(1 + |t|))^{m}, m > n as well as those corresponding to {f : ||f||_{p} ≤ c_{0}}, p > 1 (where || · ||_{p} denotes the norm in L^{p}(Ω) for p ∈ [1, ∞]) are equicontinuous.

In the last theorem we obtain a stability estimate for plurisubharmonic functions with fixed boundary values and densities of their Monge-Ampère measures uniformly bounded in L^{p}(Ω) norm. From [K1] we know that for p > 1 such a family satisfies

(0.2)  ||u − v||_{∞} ≤ const.||f − g||_{p}

if (dd^{c}u)^{n} = f dV and (dd^{c}v)^{n} = g dV. The constant in (0.2) depends on p and L^{∞} norm of the boundary data. Here we obtain an estimate which involves also the L^{q} norm of |u − v| for q and p conjugate. For ||f||_{p} and ||g||_{p} bounded by c_{p} and α ∈ (0, 1) we have

||u − v||_{∞} ≤ const.cp^{\frac{1+α}{n+αn+1}} ||u − v||q^{\frac{1}{n+αn+1}} ,

where const. depends on p, n, α, Ω.

Similar stability estimates are also valid for the solutions of the complex Monge-Ampère equation on compact Kähler manifolds. The topic is treated in a separate paper [K5]. For more information on the methods of pluripotential theory which are used in the proofs below we refer to [KL][K2][K4].

1. Preliminaries.

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 Borel nonnegative measure.

The following capacity was introduced in [BT2] for Borel sets E ⊂ Ω:

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

It is now called 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 [BT2]

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

The following result of Bedford and Taylor will be often used in the sequel.

(Comparison principle). [BT2]. 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}.

2. Results.

First lemma is a quantitative version of [K2, Lemma 2.3.1]

Lemma 1. Let Ω be a strictly pseudoconvex domain in ℂ^{n}. Assume u ∈ P SH ∩ C(Ω), and v ∈ P SH ∩ L^{∞}(Ω). Suppose that for some positive number A and an admissible function h the following inequality holds

(2.1)  ∫_{K}(dd^{c}v)^{n} ≤ Acap(K, Ω)h^{−1}((cap(K, Ω))^{−1/n})

for any compact set K. If the sets U(s) := {v + s < u} are nonempty and relatively compact in Ω for s ∈ [t, t + δ] Then

δ ≤ κ(cap(U(t), Ω)),

where

κ(s) = c(n)A^{1/n}[∫_{s^{−1/n}}^{∞} y^{−1}h^{−1/n}(y) dy + h^{−1/n}(s^{−1/n})]

and the constant c(n) depends only on n.

Proof. The estimate follows from (2.3.4) and (2.3.5) in the proof of Lemma 2.3.1 in [K2].

Theorem 1. Let Ω be a strictly pseudoconvex domain in ℂ^{n}. Assume that u_{j} ∈ P SH(Ω) is a sequence converging weakly to u ∈ P SH(Ω) and for any j

lim inf_{z→∂Ω}(u_{j} − u)(z) ≥ 0.

Suppose further that

(dd^{c}u_{j})^{n} = f_{j} dV

with f_{j} ∈ F(A, h) ∩ L^{ψ}(c_{0}), where L^{ψ}(c_{0}) is defined as in Introduction with \frac{ψ(x)}{x} increasing to ∞ as x goes to ∞. Then u_{j} → u uniformly in Ω.

Proof. Denote by a_{j}(δ) = cap(E_{j}(2δ), Ω) the Bedford-Taylor capacity of the set E_{j}(2δ) = {u_{j} + 2δ ≤ u}. Since, by [K2] u and u_{j} are continuous the set E_{j}(2δ) is compact in view of our assumptions. Let us denote by v_{j} the relative extremal function of the set E_{j}(2δ). By [BT2] (see Section 1)

∫_{E_{j}(2δ)}(dd^{c}v_{j})^{n} = a_{j}(δ).

Observe that for V = {u_{j} ≤ δv_{j} + u − δ} the following inclusions hold

E_{j}(2δ) ⊂ V ⊂ E_{j}(δ).

Applying the comparison principle we thus get

(2.2)  \begin{aligned}a_{j}(δ)δ^{n} & ≤ ∫_{E_{j}(2δ)}[dd^{c}(δv_{j} + u)]^{n} ≤ ∫_{V} (dd^{c}u_{j})^{n} \\ & ≤ ∫_{E_{j}(δ)} f_{j} dV. \end{aligned}

Hence for any M > 0 and u_{+} := max(u, 0) we have

(2.3)

a_{j}(δ)δ^{n+1} ≤ ∫_{Ω}(u − u_{j})_{+}f_{j} dV
= ∫_{{f_{j}>M}}(u − u_{j})_{+}f_{j} dV + ∫_{{f_{j}≤M}}(u − u_{j})_{+}f_{j} dV

≤ max_{Ω}(u − u_{j})_{+} ∫_{{f_{j}>M}} f_{j} dV + M ∫_{Ω}(u − u_{j})_{+} dV

≤ max_{Ω}(u − u_{j})_{+} \frac{M}{ψ(M)} ∫_{Ω} ψ(f_{j}) dV + M ∫_{Ω}(u − u_{j})_{+} dV

For the last inequality we use the assumption that \frac{ψ(x)}{x} is increasing. Fix ϵ > 0. By [K1] there exists c_{1} > 0 such that v ≥ −c_{1} for any v ∈ F(A, h) with v ≥ u on ∂Ω. So, in view of our assumptions the quantities

max_{Ω}(u − u_{j})_{+} ∫_{Ω} ψ(f_{j}) dV

are uniformly bounded. Using the assumptions on ψ we can make \frac{M}{ψ(M)} arbitrarily small by taking M big enough. We choose M so that the first term on the right hand side (2.3) is less than ϵ/2 for any j. Since u_{j} → u in L^{1}(Ω) (see e.g. [H]) the other term is less than ϵ/2 for j > j_{0}. Therefore

a_{j}(δ) ≤ ϵδ^{−n−1} for j > j_{0}.

Suppose for a while that E_{j}(3δ) were nonempty. Then, applying Lemma 1 we would get

δ ≤ κ(a_{j}(δ)) ≤ κ(ϵδ^{−n−1}), j > j_{0}.

Since, by the assumption on h we have lim_{s→0} κ(s) = 0 the last inequality yields a contradiction if we take ϵ small enough. Thus E_{j}(3δ) is empty for j > j_{0} which together with Hartogs’ lemma implies the uniform convergence of the sequence u_{j}.

Theorem 2. The set P(A, h, ψ, c_{0}; φ) is equicontinuous.

Proof. Arguing by contradiction suppose that for some ϵ > 0, u_{j} ∈ F(A, h)∩L^{ψ}(c_{0}) and two sequences z_{j}, w_{j} of points in Ω we had ||z_{j} − w_{j}|| < j^{−1} and u_{j}(z_{j}) − u_{j}(w_{j}) > ϵ. One can pick subsequences z_{j_{k}}, w_{j_{k}} converging to z ∈ \overline{Ω} and u_{j_{k}} converging in L^{1} norm to u ∈ P SH(Ω) ∩ C(\overline{Ω}). By Theorem 1 u_{j_{k}} converges uniformly. Thus for k large enough we have the following inequalities

\begin{aligned}|u(z_{j_{k}}) − u(z)| & < ϵ/4, \\ |u(w_{j_{k}}) − u(z)| & < ϵ/4, \\ |u_{j_{k}}(z_{j_{k}}) − u(z_{j_{k}})| & < ϵ/4, \\ |u_{j_{k}}(w_{j_{k}}) − u(w_{j_{k}})| & < ϵ/4. \end{aligned}

Combined they yield

|u_{j_{k}}(z_{j_{k}}) − u_{j_{k}}(w_{j_{k}})| < ϵ

which contradicts the choice of the sequences.

Let ψ : [0, ∞) → [0, ∞) be an increasing function fulfilling the inequalities

2t ≤ ψ(2t) ≤ aψ(t),

for some a > 0 and any t > t_{0}. Then the Orlicz space corresponding to ψ is defined by

L^{ψ}(Ω) = {f ∈ L^{1}(Ω) : ∫_{Ω} ψ(|f|) dV < ∞}.

Take

ψ(t) = |t|(log(1 + |t|))^{n}h(log(1 + |t|)),

for some admissible h. Then, by [K1] L^{ψ}(c_{0}) ⊂ F(A, h) for some positive A. For a continuous function φ defined on ∂Ω and ψ as above we denote by P(ψ, c_{0}; φ) the set of plurisubharmonic functions u in Ω which are equal to φ on ∂Ω and such that

(dd^{c}u)^{n} = f dV

for some f ∈ L^{ψ}(c_{0}).

Corollary. The functions from P(ψ, c_{0}; φ) are equicontinuous.

In particular the family P(p, c_{0}; φ) := P(ψ, c_{0}; φ) with ψ(t) = t^{p}, p > 1 is equicontinuous. Below we find some more precise estimates for those families.

Lemma 2. Let u_{j} be a sequence converging to u in the sense of distributions theory, such that u_{j} ∈ P(p, c_{0}; φ) and

lim inf_{z→∂Ω}(u_{j} − u)(z) ≥ 0.

Then for δ > 0 and a_{j}(δ), E_{j}(2δ) defined as in Theorem 1, we have

a_{j}(δ) ≤ c_{0}δ^{−n−1}||(u − u_{j})_{+}||_{q}

with p^{−1} + q^{−1} = 1.

Proof. We use (2.2) and Hölder’s inequality to obtain

\begin{aligned}a_{j}(δ)δ^{n+1} & ≤ ∫_{Ω}(u − u_{j})_{+}f_{j} dV \\ & ≤ (∫ f_{j}^{p} dV)^{1/p}(∫ (u − u_{j})_{+}^{q}dV)^{1/q} ≤ c_{0}||(u − u_{j})_{+}||_{q}. \end{aligned}

The proof is completed.

To estimate avereges of L^{p} functions over compact sets in terms of Bedford-Taylor capacity we need the following lemma (Lemma 4.3 from [K4]) which is a generalized version of Zeriahi’s result [Z]. The estimate in [Z] holds on compact subsets of Ω. Zeriahi pointed out to me that the proof of Lemma 2.5.1 in [K2] (which is an ingredient of the proof of the estimate we seek to establish) has a gap. The proof is corrected in [K4, Lemma 2.8].

Lemma 3. Suppose u ∈ P SH(Ω) ∩ C(\overline{Ω}), u = 0 on ∂Ω, ∫ (dd^{c}u)^{n} ≤ 1. Then for any α < 2 the Lebesgue measure V (Ω_{s}) of the set Ω_{s} := {u < s} is bounded from above by c exp(−2πα|s|), where c does not depend on u.

Now we can prove the aforementioned estimate.

Lemma 4. Let p > 1, t < min(4π−1, p−1) and f ∈ L^{p}(Ω). Then for any compact K in Ω

∫_{K} f dV ≤ c(p, n, Ω)||f||_{p}a(K) exp(−ta^{−1/n}(K))

where we use the notation a(K) = cap(K, Ω) and the constant c(p, n, Ω) depends on p, n and the set Ω.

Proof. It is enough to show the inequality for f such that ||f||_{p} = 1. Take α such that \frac{1+t}{2π} < α < 2. Observe that one can pick α depending only on p since t depends only on p. Set h(s) = e^{ts}, G(s) = s^{n}e^{ts} and ψ(s) = s(s + 1)^{t} log^{n}(s + 1). First we shall prove that

(2.4)  ∫_{Ω} |v|^{n}h(|v|)f dV ≤ A,

with A = ∫_{Ω} ψ(f) dV + c ∑_{1}^{∞} s^{n}e^{(t+1−2πα)s}, for v ∈ P SH(Ω) of the form v = cap^{−1/n}(K, Ω)u_{K}, where u_{K} is the relative extremal function of K with respect to Ω and c is the constant from Lemma 3. Applying Young’s inequality (see e.g. [H]) one obtains

G(|v(x)|)f(x) ≤ ∫_{0}^{f(x)} G(log(1 + r)) dr + ∫_{0}^{G(|v(x)|)}[exp(G^{−1}(t)) − 1] dt
≤ f(x)G(log(1 + f(x))) + ∫_{0}^{|v(x)|} e^{s}G′(s) ds
≤ ψ(f(x)) + G(|v(x)|)e^{|v(x)|}.

Integrating this inequality over Ω we get

∫ G(|v|)f dV ≤ ∫ ψ(f) dV + ∫ G(|v|)e^{|v|}dV.

Observe that, by Minkowski’s inequality

∫ ψ(f) dV ≤ ||f + 1||_{p} ≤ ||f||_{p} + (∫_{Ω} dV)^{1/n}.

Thus to get (2.4) we need to find a suitable estimate on the second integral on the right hand side. To this end we apply Lemma 3. Then

∫_{Ω} G(|v(x)|)e^{|v(x)|} dV = ∑_{s=0}^{∞} ∫_{{−s−1<v≤−s}} G(|v(x)|)e^{|v(x)|} dV

≤ ∑_{s=0}^{∞}(s + 1)^{n}e^{(t+1)(s+1)}V ({v < −s}) ≤ c ∑_{s=0}^{∞}(s + 1)^{n}e^{(t+1)(s+1)}e^{−2παs}

≤ce^{t+1} ∑_{s=1}^{∞}(s + 1)^{n}e^{(t+1−2πα)s}

and the series is convergent due to the choice of α. Finally, one infers the statement from (2.3) as follows. Since

|v(z)|^{n} = a^{−1}(K)|u_{K}(z)| = a^{−1}(K)

for z ∈ K we get

\begin{aligned}A & ≥ ∫_{Ω} |v|^{n}h(|v|)f dV ≥ ∫_{K} |v|^{n}h(|v|)f dV \\ & ≥ \frac{1}{a(K)} exp(ta^{−1/n}(K)) ∫_{K} f dV, \end{aligned}

which is the desired estimate as we recall that ||f||_{p} = 1.

We shall use the following simple proposition in the proof of Theorem 3 below. Proposition. Given C_{0} > 0 and α ∈ (0, 1) we have

x exp(Cx^{−α}) ≥ C_{0}

for any C ≥ (αe)^{−1}C_{0}^{α} and any x > 0. Proof. The derivative of the function g(x) = x exp(Cx^{−α}) has its zero at x_{0} = (αC)^{1/α} and g(x_{0}) = (αeC)^{1/α}. Hence for C ≥ (αe)^{−1}C_{0}^{α} one gets g(x) ≥ [αe(αe)^{−1}C_{0}^{α}]^{1/α} = C_{0}.

We are now ready to show the main estimate. Theorem 3. Let Ω be a strictly pseudoconvex domain in ℂ^{n}. Assume that u and v belong to P SH(Ω) and

lim inf_{z→∂Ω}(v − u)(z) ≥ 0.

Suppose further that

(dd^{c}v)^{n} = g dV, (dd^{c}u)^{n} = f dV

with ||f||_{p}, ||g||_{p} ≤ c_{p}, p > 1. Then for given α ∈ (0, 1) there exists a constant c′ = c′(p, n, α, Ω) such that

||(u − v)_{+}||_{∞} ≤ c′cp^{\frac{α+1}{n+αn+1}} ||(u − v)_{+}||q^{\frac{1}{n+αn+1}} .

where p^{−1} + q^{−1} = 1. Proof. We shall apply Lemma 1 with h(s) = e^{ts}, where t is chosen so that t < min(4π − 1, p − 1). For this choice of h one has

(2.5)  ∫_{s}^{∞} y^{−1}h^{−1/n}(y) dy ≤ \frac{n}{t} exp(\frac{−ts}{n}).

Now, Lemma 4 tells us that (2.1) is satisfied with A = c(p, n, Ω)c_{p}. Set

δ = \frac{1}{2}||(u − v)_{+}||_{∞}.

Then U(s) = {v + s < u} is nonempty for s ∈ (δ, 2δ). By Lemma 1 and (2.5)

δ ≤ c(n)A^{\frac{1}{n}} (\frac{n}{t} + 1) exp(−\frac{t}{n}a^{−1/n}(δ)), with a(δ) = cap(U(δ), Ω).

Coupling this with the estimate from Lemma 2 one obtains

(2.6)  δ ≤ c_{1}(p, n, Ω)cp^{\frac{1}{n}} exp(−\frac{t}{n}(δ/2)^{(n+1)/n}||(u − v)_{+}||_{q}^{−1/n}c_{p}^{−1/n}).

Making use of the proposition above we infer that for α ∈ (0, 1) we have

δ exp(Cδ^{−α}) ≥ c_{1}(p, n, Ω)c_{p}^{1/n}

for C = (αe)^{−1}c_{1}^{α}(p, n, Ω)c_{p}^{α/n}. Inserting this into (2.6) and taking logarithms of both sides one obtains

δ^{α+ \frac{n+1}{n}} ≤ 4(αe)^{−1} \frac{n}{t}c_{1}^{α}(p, n, Ω)cp^{\frac{1+α}{n}} ||(u − v)_{+}||q^{\frac{1}{n}} .

So

||(u − v)_{+}||_{∞} ≤ c′(p, n, α, Ω)cp^{\frac{1+α}{n+αn+1}} ||(u − v)_{+}||q^{\frac{1}{n+αn+1}} .

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postal address: Jagiellonian University, Institute of Mathematics Reymonta 4, 30-059 Kraków, Poland e-mail: kolodzie@ im.uj.edu.pl