Regularity of the Siciak-Zaharjuta extremal function on compact Kähler manifolds

Abstract

We prove that the regularity of the extremal function of a compact subset of a compact Kähler manifold is a local property, and that the continuity and Hölder continuity are equivalent to classical notions of the local L L -regularity and the locally Hölder continuous property in pluripotential theory. As a consequence we give an effective characterization of the ( C α , C α ′ ) (\mathscr {C}^\alpha , \mathscr {C}^{\alpha ’}) -regularity of compact sets, the notion introduced by Dinh, Ma and Nguyen [Ann. Sci. Éc. Norm. Supér. (4) 50 (2017), pp. 545–578]. Using this criterion all compact fat subanalytic sets in R n \mathbb {R}^n are shown to be regular in this sense.