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

Nguyen, Ngoc Cuong

doi:10.1090/tran/9241

Regularity of the Siciak-Zaharjuta extremal function on compact Kähler manifolds
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by Ngoc Cuong Nguyen;

Trans. Amer. Math. Soc. 377 (2024), 8091-8123

DOI: https://doi.org/10.1090/tran/9241

Published electronically: August 30, 2024

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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 $L$-regularity and the locally Hölder continuous property in pluripotential theory. As a consequence we give an effective characterization of the $(\mathscr {C}^\alpha , \mathscr {C}^{\alpha ’})$-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 $\mathbb {R}^n$ are shown to be regular in this sense.

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