[Paper Review] Convergence in capacity

Urban Cegrell|ArXiv.org|May 11, 2005

Geometry and complex manifolds1 references4 citations

TL;DR

This paper establishes the convergence of Monge-Ampère measures $(dd^c u_j)^n$ to $(dd^c u)^n$ in the weak* topology under the assumption that a sequence of plurisubharmonic functions $u_j$ in the class $\mathcal{F}$ converges to $u$ in $\mathbb{C}^n$-capacity. The key result generalizes earlier convergence theorems by linking capacity convergence to weak* convergence of associated Monge-Ampère measures via approximation and capacity estimates in hyperconvex domains.

ABSTRACT

The purpose of this paper is to study convergence of Monge-Ampere measures associated to sequences of plurisubharmonic functions defined on a hyperconvex subset of ${\mathbb C^n}$.

Motivation & Objective

• To generalize the convergence of Monge-Ampère measures for sequences of plurisubharmonic functions in the class $\mathcal{F}$.
• To establish that convergence in $\mathbb{C}^n$-capacity implies weak* convergence of the associated Monge-Ampère measures.
• To extend previous results from uniformly bounded sequences to the broader class $\mathcal{F}$, which includes functions with controlled growth and finite Monge-Ampère mass.
• To provide a functional analytic framework using capacity and approximation to control the convergence of nonlinear measures.

Proposed method

• Utilizes the concept of convergence in $\mathbb{C}^n$-capacity, defined as $\mathrm{cap}(\{z \in K : |u - u_j| > \delta\}) \to 0$ for all compact $K \subset \subset \Omega$ and $\delta > 0$.
• Applies approximation by functions in $\mathcal{E}_0 \cap C(\bar{\Omega})$ to reduce the problem to continuous functions with compact support in the boundary-regularized class.
• Employs a truncation technique via measures $\mu_p = \min(f, p)(dd^c \psi)^n$ to approximate general positive measures vanishing on pluripolar sets.
• Uses the weak* convergence of integrals $\int u_j \, d\mu_p \to \int u \, d\mu_p$ and applies Fatou's lemma and monotone convergence to pass to the limit.
• Employs a recursive argument on the degree of the Monge-Ampère form, proving convergence for $p+1$-forms by assuming it for $p$-forms and estimating error terms via capacity and sublevel sets.
• Constructs a sequence of capacity-based majorants $h_N^*$ to control the difference $u_j - u$ on sets where $|u_j - u| > \varepsilon$, ensuring integrability and decay of error terms.

Experimental results

Research questions

• RQ1 Does convergence in $\mathbb{C}^n$-capacity of a sequence $u_j \in \mathcal{F}$ imply weak* convergence of the associated Monge-Ampère measures $(dd^c u_j)^n$?
• RQ2 Can the convergence result for uniformly bounded sequences be extended to the class $\mathcal{F}$, which includes unbounded plurisubharmonic functions with finite Monge-Ampère mass?
• RQ3 How can capacity estimates be used to control the convergence of nonlinear measures in pluripotential theory?
• RQ4 What role do approximation by $\mathcal{E}_0$-functions and truncation of measures play in proving weak* convergence of Monge-Ampère measures?

Key findings

• If $u_j \in \mathcal{F}$, $u_0 \leq u_j$, and $u_j$ converges to $u$ in $\mathbb{C}^n$-capacity, then $(dd^c u_j)^n$ converges weak* to $(dd^c u)^n$ as $j \to \infty$.
• The convergence holds for all test functions $h \in \mathcal{E}_0$, which is sufficient to imply weak* convergence of the measures by standard duality in pluripotential theory.
• The proof relies on a recursive argument on the degree of the Monge-Ampère form, establishing convergence for $p+1$-forms under the inductive assumption for $p$-forms.
• Error terms in the difference $\int (u_j - u)(dd^c u_j)^p \land \cdots$ are controlled using capacity-based majorants $h_N^*$ and truncation, ensuring they vanish in the limit.
• The key estimate involves bounding $|I_{j_k}|$, $|II_{j_k}|$, and $|III_{j_k}|$ in terms of $\int (-h_N)(dd^c u)^p \land \cdots$ and $\varepsilon \int (dd^c u_0)^p \land \cdots$, both of which vanish as $N \to \infty$ and $\varepsilon \to 0$.
• The result is stable under subsequences: every subsequence of $u_j$ contains a further subsequence along which $(dd^c u_j)^n$ converges weak* to $(dd^c u)^n$, proving the full convergence.