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

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 ′) (C^, C^ ’) -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 Rⁿ are shown to be regular in this sense.