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 $L$-regularity and the locally Hölder continuous property in pluripolential theory. As a consequence we give an effective characterization of the $({\text{Cc}}^{\text{al}},{\text{Cc}}^{{\text{al}}^{\mathrm{\prime }}})$-regularity of compact sets, the notion introduced by Dinh, Ma and Nguyen. Using this criterion all compact fat subanalytic subsets in ${\text{bR}}^n$ are shown to be regular in this sense.