Stability and strong convergence for complex Hessian equations with $L^1$ data

Based on the provided manuscript, here is a detailed technical summary of the paper "Stability and Strong Convergence for Complex Hessian Equations with $L^1$ Data" by Truong Dinh Dat.

1. Problem Statement and Context

The paper investigates the complex $m$-Hessian equations on bounded $m$-hyperconvex domains $\Omega \subset \mathbb{C}^n$. The specific equation under study is:
$$(dd^c u)^m \wedge \beta^{n-m} = f \beta^n$$
where $\beta = dd^c |z|^2$ is the standard Kähler form, $1 \le m \le n$, and the right-hand side data $f$ belongs to $L^1(\Omega)$.

The study sits between classical potential theory ($m=1$) and pluripotential theory ($m=n$, the complex Monge–Ampère equation). While the existence and stability of solutions for data in $L^p(\Omega)$ with $p > 1$ are well-established (notably via Kołodziej's $L^p$ estimates and subsequent works by Dinew, Kołodziej, and others), the case of merely integrable data ($f \in L^1$) remains less understood for complex Hessian equations ($1 \le m < n$).

A primary challenge is that standard convergence in $m$-Hessian capacity, while known to hold for $L^1$ data, is a relatively weak topology. Convergence in capacity does not guarantee the convergence of the associated nonlinear measures $H_m(u_j)$, nor does it control the interaction between the potentials and their Hessian measures.

2. Methodology

The author employs a combination of approximation techniques, uniform energy estimates, and refined capacity arguments adapted from Cegrell's theory and Kołodziej's methods. The core methodological steps include:

• Approximation of Data: The $L^1$ data $f$ is approximated by an increasing sequence of bounded functions $f_j = \min(f, j)$. For each $f_j$, a unique solution $u_j$ exists in the energy class $\mathcal{E}_m^0(\Omega)$.
• Uniform Mass Control: The author establishes that the total Hessian mass $\int_\Omega H_m(u_j)$ is uniformly bounded by $\int_\Omega f \beta^n$, independent of $j$.
• Capacity Estimates: Using the capacity estimate $\text{Cap}_m(\{u < -t\}) \le t^{-m} \int_\Omega H_m(u)$, the author proves that the sequence of solutions $\{u_j\}$ is uniformly tight with respect to the $m$-Hessian capacity. This ensures compactness and convergence in capacity to a limit $u$.
• Truncation and Absolute Continuity: A key technical ingredient is Lemma 2.1, which provides a quantitative form of the absolute continuity of the Hessian measure with respect to the $m$-Hessian capacity. Specifically, for any Borel set $E$ and $t > 0$:
  $$H_m(u)(E) \le \int_{\{u < -t\}} H_m(u) + t^m \text{Cap}_m(E, \Omega)$$
  This inequality allows the control of the measure on sets where the potential difference is large by splitting the domain into a "deep sublevel set" (controlled by mass) and a "capacity set" (controlled by convergence in capacity).

3. Key Contributions and Results

A. Existence and Uniqueness (Section 4)

The paper recalls and formalizes the existence of a unique weak solution $u \in \mathcal{F}_m(\Omega)$ for the Dirichlet problem with $L^1$ data. This is achieved by taking the limit of the approximating sequence $u_j$ and utilizing the monotone convergence theorem for the complex Hessian operator.

B. Stability in Capacity (Section 4)

The author confirms that if $f_j \to f$ in $L^1(\Omega)$, the corresponding solutions $u_j$ converge to $u$ in $m$-Hessian capacity. This result relies on the comparison principle and the $L^1$ convergence of the densities, avoiding the need for non-admissible test functions.

C. Strong Convergence in Energy (Section 5 - Main Result)

The central contribution of the paper is Theorem 5.1, which establishes a strong stability result in the natural energy topology. The author proves that if $f_j \to f$ in $L^1(\Omega)$, then:
$$\int_\Omega |u_j - u| H_m(u_j) \longrightarrow 0$$
This convergence is significantly stronger than convergence in capacity. It implies that the nonlinear measures $H_m(u_j)$ interact well with the convergence of the potentials.

The proof strategy involves splitting the integral $\int |u_j - u| H_m(u_j)$ into two parts based on a threshold $\delta$:

1. Small difference region: Controlled by the uniform bound on the total mass.
2. Large difference region: Controlled by combining the capacity convergence (which makes the capacity of the set $\{|u_j - u| > \delta\}$ small) with the quantitative absolute continuity lemma (Lemma 2.1) and the uniform control of sublevel sets (Lemma 3.2).

D. Extension to Measure Data (Section 6)

The strong convergence result is extended to sequences of nonnegative Radon measures $\mu_j$ with uniformly bounded mass that converge weakly to $\mu$. The same strong convergence $\int_\Omega |u_j - u| H_m(u_j) \to 0$ holds in this setting.

4. Significance and Claims

The paper positions its main result as a borderline case extension of known stability theories:

• Comparison with Monge–Ampère: In the complex Monge–Ampère case ($m=n$), strong stability results for $L^1$ data are known (e.g., by Benelkourchi–Guedj–Zeriahi, Liu–Zhang). This paper establishes the Hessian analogue of these results for $1 \le m < n$.
• Overcoming Structural Limitations: The author notes that the Hessian operator lacks the full monotonicity and strong comparison framework available in the Monge–Ampère case ($m=n$). Consequently, arguments used in the $m=n$ setting do not directly transfer. The paper demonstrates that strong convergence can still be achieved by carefully combining capacity convergence, uniform sublevel set control, and the quantitative absolute continuity of Hessian measures.
• Novelty: To the best of the author's knowledge, such a strong energy convergence result for complex Hessian equations with merely integrable right-hand sides has not been explicitly established in the literature prior to this work.

The paper concludes that this result provides a "substantial strengthening" of the known convergence in capacity, offering a more robust framework for analyzing the behavior of solutions when the source data is only in $L^1$.