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

Abstract

<p>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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L"> <mml:semantics> <mml:mi>L</mml:mi> <mml:annotation encoding="application/x-tex">L</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-regularity and the locally Hölder continuous property in pluripotential theory. As a consequence we give an effective characterization of the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis script upper C Superscript alpha Baseline comma script upper C Superscript alpha prime Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="script">C</mml:mi> </mml:mrow> <mml:mi>α</mml:mi> </mml:msup> <mml:mo>,</mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="script">C</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>α</mml:mi> <mml:mo>′</mml:mo> </mml:msup> </mml:mrow> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(\mathscr {C}^\alpha , \mathscr {C}^{\alpha ’})</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R Superscript n"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb {R}^n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are shown to be regular in this sense.</p>