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

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.