Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Hyunsoo Ahn

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 \(1\).

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

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 -Hölder continuity property of a convex compact subset in implies the local -Hölder continuity property of order .

Paper Structure

This paper contains 15 sections, 21 theorems, 156 equations.

Table of Contents

  1. Introduction
  2. Basic properties of extremal functions
  3. Local Hölder continuity property and uniform density in capacity
  4. Moduli of continuity of extremal functions on complex coordinate spaces
  5. Local Hölder continuity property
  6. Uniform density in capacity
  7. Relation of extremal functions and uniform density in capacity in the complex coordinate space
  8. Extension of Hölder continuity of extremal functions on compact subsets
  9. Demailly estimates
  10. Hölder continuity of extremal functions
  11. Proofs of the main theorems
  12. Proof of the first theorem
  13. Proof of the second theorem
  14. Proof of the third Theorem
  15. Appendix

Key Result

Theorem 1.1

Let $K$ be a compact subset of a compact Hermitian manifold $X$ and $a\in K$. Let $\bar{B}(a,r)$ be a closed holomorphic coordinate ball in $X$ with the center $a\in K$ and the finite radius $r>0$. Then the following items hold: $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (47)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Definition 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Definition 1.8
  • Corollary 1.9
  • Proposition 2.1
  • ...and 37 more