H\"older regularity of Siciak-Zaharjuta extremal functions on compact Hermitian manifolds

TL;DR

This paper investigates the H"older regularity of extremal functions on compact Hermitian manifolds, establishing local-global equivalences and extending previous results from K"ahler to Hermitian settings.

Contribution

It generalizes known H"older regularity results from K"ahler to Hermitian manifolds and explores local and global properties of extremal functions with capacity density.

Findings

H"older continuity at a point is a local property.

H"older continuity of weighted extremal functions follows from extremal and weight functions.

H"older regularity is equivalent to weak local H"older continuity.

Abstract

For a compact subset in a compact Hermitian manifold, we prove that the H\"older continuity of the extremal function at a given point in the set is a local property and that the H\"older continuity of a weighted extremal function follows from the H\"older 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 T\^o \cite{LPT21}. Moreover, for a compact subset in a compact Hermitian manifold, we prove that, both at the point level and at the global level, the H\"older continuity of the extremal function with the uniform density in capacity is equivalent to the local H\"older continuity property, which is also equivalent to the weak local H\"older continuity property. These results are generalizations of the results of Nguyen \cite{Ng24} on compact K\"ahler manifolds. We also show that the \(\mu\)-H\"older continuity property of a convex compact subset in \(\mathbb{C}^n\) and of a compact subset in \(\mathbb{C}^n\) at a star center imply the local \(\mu\)-H\"older continuity property of order \(\mu\) of the convex compact subset and of the compact subset at the point, respectively.