We prove that the regularity of the extremal function of a compact subset of a compact K\"ahler manifold is a local property, and that the continuity and H\"older continuity are equivalent to classical notions of the local $L$-regularity and the locally H\"older continuous property in pluripolential theory. As a consequence we give an effective characterization of the $(\Cc^\al, \Cc^{ \al'})$-regularity of compact sets, the notion introduced by Dinh, Ma and Nguyen. Using this criterion all compact fat subanalytic subsets in $\bR^n$ are shown to be regular in this sense.
Categories: math.CV math.DG