Generalized Cegrell Classes in Nonlinear Potential Theory

• Generalized Cegrell classes are advanced energy-type function spaces that broaden classical plurisubharmonic domains to analyze nonlinear PDEs via Monge–Ampère and Hessian operators.
• They are defined through rigorous convex cone structures and integration-by-parts formulas, supporting approximation techniques and capacity theory for variational tools.
• The framework enables precise operator definitions, boundary regularity estimates, and comparison principles essential for solving complex Hessian and Monge–Ampère equations.

Generalized Cegrell classes are advanced energy-type function spaces designed for the paper of nonlinear potential theory in complex, real, and quaternionic analysis. Originating in pluripotential theory for plurisubharmonic (PSH) functions, these classes provide a canonical domain for nonlinear PDEs such as the complex Monge–Ampère and Hessian equations, and their generalizations to m-positive closed currents, weighted energies, superformalism, and quaternionic settings. The development and extension of Cegrell-type classes enable new analytic, geometric, and variational tools, including comparison and uniqueness principles, envelope constructions, and capacity theory.

1. Foundational Definitions and Structures

Generalized Cegrell classes extend classical PSH energy classes to broader function spaces and operator contexts. In the complex setting, for an m-positive closed current $T$, the class $E_{m,T}(\Omega)$ consists of negative m-subharmonic functions $u$ on $\Omega$ vanishing outside $\text{Supp}(T)$ and with finite p-th energy: $$E_{m,T}(\Omega) = \{ u \in P_m(\Omega) : u(z) \to 0 \text{ on } \Omega \setminus \text{Supp}(T),\ J(u) := \int_\Omega (dd^c u)^p \wedge T < +\infty \}$$ with $1 \leq p \leq m$.

The associated class $F_{m,T}(\Omega)$ comprises $u$ approximable by a decreasing sequence $(u_j) \subset E_{m,T}(\Omega)$ for which $\sup_j \int_\Omega (dd^c u_j)^p \wedge T$ remains bounded.

In the quaternionic setting, analogs include $\mathcal{E}_p(\Omega)$: $$\mathcal{E}_p(\Omega) = \{ \varphi \in \text{QPSH}^-(\Omega) : \lim_{q \to \partial \Omega} \varphi(q) = 0,\ \int_\Omega (-\varphi)^p (\Delta \varphi)^n < +\infty \}$$ Other subsequent generalizations introduce weighted versions, e.g., $E_{\chi, m}(\Omega)$, using an increasing weight $\chi$ to define the energy integrals in the complex Hessian setting.

These classes are convex cones and support integration-by-parts formulas, Blocki-type inequalities, and approximation results via decreasing sequences that preserve the energy bounds (Dhouib et al., 2015, Wan, 2018, Do et al., 2022).

2. Operator Theory: Complex Hessian, SuperHessian, and Quaternionic Monge–Ampère

The extension of classical Monge–Ampère theory to m-subharmonic and more general energy classes relies on defining the appropriate nonlinear elliptic differential operators. In the complex-m-positive context, the operator $(dd^c u)^p \wedge T$ is shown to be well-defined, lower semicontinuous under decreasing sequences, and extendable by induction: $$\text{If } (u_j) \searrow u \text{ near } \text{Supp}(T),\ \text{then } (dd^c u_j)^p \wedge T \to (dd^c u)^p \wedge T \text{ weakly}.$$

The real superformalism setting introduces the $m$-superHessian operator for m-convex functions: $$T \wedge B_{n-m} \wedge dd^\# u_1 \wedge \cdots \wedge dd^\# u_k$$ preserving positivity and converging under smoothing or decreasing sequences (Elkhadhra et al., 2020).

In quaternionic theory, the Monge–Ampère operator is modeled through the Moore determinant and acts on QPSH functions: $(\Delta u)^n = n! \cdot \det(u) \cdot \Omega_{2n}$ with extension to weak limits along the energy class approximating sequences, enabling analysis on unbounded domains and functions (Wan, 2018, Do et al., 2023).

3. Capacity, Potential Currents, and Weighted Classes

Advanced pluripotential and Hessian theories demand robust notions of capacity and potential. The m-relative capacity $\text{cap}_{m,T}(K,\Omega)$ is defined by supremizing the mass of $(dd^c u)^p \wedge T$ over test functions $u \in P_m(\Omega),\ 0 \leq u \leq 1$: $$\text{cap}_{m,T}(K,\Omega) = \sup \left\{ \int_K (dd^c u)^p \wedge T : u \in P_m(\Omega),\ 0 \leq u \leq 1 \right\}$$ It admits the expected subadditivity, monotonicity, and quasicontinuity required for regularization results.

The m-potential current $U_m$ generalizes the Lelong–Skoda construction and is defined by convolution with a Riesz kernel of the form $h_m(x)$, yielding negative currents whose traces reflect the underlying m-subharmonicity.

Weighted classes $E_{\chi,m}(\Omega)$ allow for refined control of singularities and boundary behavior via the function $\chi$: the limit function of a uniformly energy-bounded sequence lies in the class; integration and averaging preserve membership.

4. Comparison Principles, Uniqueness, Rooftop Envelopes, and Geodesic Structures

Generalized comparison principles govern uniqueness and stability. The extended Bedford–Taylor comparison principle asserts: for $u,v \in N(H)$ with $u \preceq v$ and $(dd^n u) \leq (dd^n v)$, it follows that $u \geq v$; equality of measures implies equality of functions.

Rooftop envelopes $P(u,v)$ and Green–Poisson residual operators $g_u$ encode singularity profiles, with the idempotency conjecture $g_{g_u} = g_u$ verified under broad conditions (Åhag et al., 7 May 2024).

Geodesic connectivity describes optimal interpolations—segments $u_t$ below $u_0$ and $u_1$—in Cegrell classes and is characterized by conditions on residual operators. Propagation of rooftop equality and generalizations to envelopes under boundary data strengthen control over singularity structures in the class.

5. Hölder-Type Energy Estimates and Obstructions to the Variational Method

The critical Hölder-type energy estimates for mutual and individual energies play a foundational role in variational approaches to nonlinear PDEs. In the quaternionic theory: $$e_p(u_0,\dots,u_n) \leq D \prod_{k=0}^n e_p(u_k)^{1/(n+p)}$$ where $D > 1$ for $p \neq 1$ (explicitly, $D = p^{1/(1-p)}$ if $0 < p < 1$ and $p^{\alpha(p,n)/(p-1)}$ if $p > 1$).

This fact—proved via Beta and Gamma functions for explicit families $u_a(q) = |q|^{2a} - 1$—shows the optimal constant exceeds unity, preventing application of the classical variational method to the Monge–Ampère equation in these classes (Do et al., 2023). The obstruction is structural and parallels similar phenomena in extended (non-complex) Cegrell settings.

6. Boundary Behavior, Regularity, and Sufficient Criteria

Detailed near-boundary estimates for functions in $\mathcal{F}(\Omega)$ sharpen understanding of regularity and mass control. For strictly pseudoconvex $\Omega$ and $u \in \mathcal{F}(\Omega)$,

$$\operatorname{Vol}_{2n}(\{z \in W_a: u(z) < -\varepsilon\}) \leq C \frac{d^{n+1-n a}}{\varepsilon^n},$$

illustrating strong decay as $d \to 0$ near the boundary (Do et al., 2019). In the unit ball $B_{2n}$, radial averages and densities yield sufficient criteria: if

$$\lim_{d \to 0^+} \frac{1}{d} \operatorname{Vol}_{2n}(\{|z| > 1-d,\ u(z) < -A d\}) = 0,$$

then $u \in \mathcal{F}(B_{2n})$, enabling practical membership tests and regularity analysis.

7. Applications, Impact, and Open Problems

Generalized Cegrell classes inform the solvability of complex Hessian and Monge–Ampère equations with right-hand sides given by singular measures or currents. Quasicontinuity and appropriate capacities allow for regularization, stability, and existence theorems under degenerate scenarios.

These frameworks have been extended to geometric analysis, hyperkähler metrics, tropical and superformalism potential theory, and quaternionic settings relevant to theoretical physics.

Key open problems include:

• Full existence results for the Dirichlet problem when data may charge pluripolar sets.
• Precise understanding of boundary values and whether all $u \in E$ belong to $N(\tilde{u})$ for their maximal majorants.
• Extension of operator definitions to broader singular classes, possibly leveraging modern distributional theory (Åhag et al., 7 May 2024).

Generalized Cegrell classes, through envelope constructions, operator theory, and analytic estimates, constitute an active area of research with ongoing developments in comparison principles, variational obstructions, singularity classification, and complex geometric analysis.