Hölder regularity of Siciak-Zaharjuta extremal functions on compact Hermitian manifolds

Published 2 Apr 2026 in math.CV | (2604.01887v2)

Abstract: For a compact subset in a compact Hermitian manifold, we prove that the Hölder continuity of the extremal function at a given point in the set is a local property and that the Hölder continuity of a weighted extremal function follows from the Hölder continuities of the extremal function and the weight function with a uniform density in capacity. The second result can be seen as a continuation of a result of Lu, Phung and Tô \cite{LPT21}. Moreover, for a compact subset in a compact Hermitian manifold, we prove that the Hölder continuity of the extremal function with the uniform density in capacity is equivalent to the local Hölder continuity property, which is also equivalent to the weak local Hölder continuity property. These results are generalizations of the results of Nguyen \cite{Ng24} on compact Kähler manifolds. We also show that the (μ)-Hölder continuity property of a convex compact subset in (\mathbb{C}ⁿ⁾ implies the local (μ)-Hölder continuity property of order (1).

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Authors (1)

Hyunsoo Ahn

Summary

The paper establishes that the Hölder continuity of Siciak-Zaharjuta extremal functions depends solely on the local geometry of compact Hermitian manifolds.
It unifies various regularity notions by proving the equivalence between local, weak local, and density-based Hölder continuity properties.
The work quantifies the degradation in regularity for weighted extremal functions using explicit exponents derived from capacity estimates and advanced regularization methods.

Hölder Regularity of Siciak-Zaharjuta Extremal Functions on Compact Hermitian Manifolds

Overview and Motivation

This work addresses the fine regularity properties of Siciak-Zaharjuta extremal functions associated to compact subsets within compact Hermitian manifolds. The paper establishes sharp Hölder continuity results and their local and global characterizations for extremal functions in this context, generalizing previously known results from the Kähler setting to the more general Hermitian framework. Additionally, the interplay between Hölder regularity, weighted extremal functions, and uniform capacities is rigorously analyzed, with precise quantitative estimates and equivalence statements.

Main Results

The central contributions are twofold:

Locality of Hölder Regularity: Hölder continuity at a fixed point is shown to be a local property—the extremal function $V_K$ is $\mu$ -Hölder at $a\in K$ if and only if the localized extremal function $V_{K \cap \overline{B}(a,r)}$ is $\mu$ -Hölder at $a$ for some holomorphic coordinate ball $\overline{B}(a,r)$ . Furthermore, continuity of weighted extremal functions is shown to depend on properties (Hölder regularity and uniform capacity density) of both the set and the weight.
Equivalence of Continuity Notions and Uniform Capacity Density: Several continuity and regularity properties—including various forms of local and weak local Hölder continuity and uniform capacity density—are shown to be equivalent. In particular, Hölder continuity with uniform capacity density is equivalent to possessing a (weak) local Hölder continuity property, precisely quantified in terms of capacity and Hölder exponent orders. These equivalences, with explicit controls on exponents and quantifiers, extend the recent compact Kähler case theorems to the Hermitian category.

An ancillary but important result establishes that for a convex compact subset $F$ in $\mathbb{C}^n$ , the global $\mu$ -Hölder continuity property implies local $\mu$ 0-Hölder continuity property (with respect to balls centered in $\mu$ 1) with order $\mu$ 2.

Technical Foundations

The analysis is grounded in pluripotential theory on complex manifolds beyond the Kähler class. The central object is the Siciak-Zaharjuta extremal function, which on a compact Hermitian manifold $\mu$ 3 and a subset $\mu$ 4 is defined as

$\mu$ 5

where $\mu$ 6 are $\mu$ 7-plurisubharmonic functions. The regularity properties of $\mu$ 8 reflect geometric and potential-theoretic features of $\mu$ 9 (e.g., pluripolarity). For weighted extremal functions, a real-valued weight function $a\in K$ 0 is incorporated, and the dependence on the regularity of $a\in K$ 1 is tracked.

Key technical components and estimates include:

Hölder Moduli and Local Properties: The paper introduces rigorous notions of various moduli of continuity (pointwise, local, global) and establishes their interrelation and the implications for the regularity of extremal functions.
Uniform Density in Capacity: The concept of uniform density in (Bedford-Taylor or global) capacity plays a central role: it quantifies "thickness" of the set in potential-theoretic terms, which, together with continuity, is necessary and sufficient for preservation of Hölder regularity under natural operations.
Demailly-type Regularization: The proof leverages Demailly's smoothing techniques for $a\in K$ 2-psh functions on compact Hermitian manifolds, which allow for precise control of how regularization and approximation affect Hölder exponents in the absence of standard Kähler structure.

Explicit comparison inequalities between "local" (relative) and "global" capacities are established both in local charts and in global terms on $a\in K$ 3, facilitating the transfer of potential-theoretic estimates from $a\in K$ 4 to general Hermitian manifolds.

Explicit Quantitative Statements

There are sharp, explicitly stated estimates connecting the exponents and “orders” of Hölder continuity and uniform capacity density. For example, if $a\in K$ 5 and a weight $a\in K$ 6 are each $a\in K$ 7-Hölder and $a\in K$ 8 has uniform capacity density of order $a\in K$ 9, then $V_{K \cap \overline{B}(a,r)}$ 0 is Hölder continuous with exponent $V_{K \cap \overline{B}(a,r)}$ 1. The paper clarifies the precise degradation in exponent in passing from unweighted to weighted extremal functions and relates these to notions of $V_{K \cap \overline{B}(a,r)}$ 2-regularity as in [DMN17].

Additionally, corresponding necessary and sufficient conditions are formulated for the equivalence between several notions of local Hölder continuity, capacity density, and continuity of extremal functions. The proofs yield explicit dependencies of Hölder coefficients and exponents on geometric and capacity data.

Implications and Theoretical Consequences

These results have substantive implications for pluripotential theory, several complex variables, and complex dynamics:

Generalization From Kähler to Hermitian: By extending regularity equivalences and the capacity-based characterization of regularity to the Hermitian context, the paper opens these techniques to a much broader class of manifolds (since many natural complex manifolds lack Kähler metrics).
Applications to Complex Monge-Ampère Equations: The established regularity properties facilitate the development of well-posedness and regularity theories for degenerate complex Monge-Ampère equations on compact Hermitian manifolds, as regular $V_{K \cap \overline{B}(a,r)}$ 3-psh “subsolutions” play a key role.
Quantitative Pluripotential Estimates: The explicit modulus estimates can be expected to influence further quantitative studies in equidistribution, complex dynamics, and approximation theory on Hermitian and complex-analytic varieties.
Capacity and Geometry: The deep interplay between potential-theoretic uniform “fatness” (capacity density) and fine function regularity clarifies the necessary geometric constraints for function regularity beyond the classical Euclidean or projective case.

Speculatively, these precise regularity conditions may also find roles in future advances: geometric flows on non-Kähler manifolds, fine equidistribution results for Fekete or interpolation arrays, and the classification theory of degenerate complex Monge-Ampère equations.

Conclusion

This work develops a comprehensive and quantitatively sharp theory of Hölder regularity for Siciak-Zaharjuta extremal functions on compact Hermitian manifolds, subsuming and strictly extending Kähler-manifold results. The equivalence of regularity, continuity, and uniform capacity conditions is established in full detail, including weighted settings and for general compact subsets. The analysis is notable both for its technical depth (relying on pluripotential estimates, capacity theory, and regularization) and its explicit control over moduli and exponents.

The results provide foundational tools for further pluripotential-theoretic and geometric-analytic research on complex manifolds without the Kähler condition, offering new perspectives on regularity, approximation, and potential theory in several complex variables.

References

H. Ahn, "Hölder regularity of Siciak-Zaharjuta extremal functions on compact Hermitian manifolds" (2604.01887)
T.-C. Dinh, X. Ma, V.-A. Nguyen, "Equidistribution speed for Fekete points associated with an ample line bundle" [DMN17]
N.C. Nguyen, "Regularity of the Siciak-Zaharjuta extremal function on compact Kähler manifolds" [Ng24]
C.H. Lu, T.T. Phung, T.D. Tô, "Stability and Hölder regularity of solutions to complex Monge–Ampère equations on compact Hermitian manifolds" [LPT21]
S. Kołodziej, N.C. Nguyen, "Hölder continuous solutions of the Monge–Ampère equation on compact Hermitian manifolds" [KN18]