The Hölder continuous subsolution theorem for complex Hessian equations

Abstract

Cet article est mis à disposition selon les termes de la licence LICENCE INTERNATIONALE D'ATTRIBUTION CREATIVE COMMONS BY 4.0.

Erratum to “The Hölder continuous subsolution theorem for complex Hessian equations”

“…Here we will give a new version of [BZ20,Lem. 4.2] and its complete proof following essentially the same scheme.…”

“…We approximate ϕ by smooth functions. As in [BZ20], we extend ϕ as a Hölder continuous function of order α on C n and denote by ϕ δ (0 < δ < δ 0 ) the usual smooth approximants of ϕ on C n . We know that ϕ δ ∈ SH m (Ω δ ) ∩ C ∞ (C n ).…”

“…and u δ u on Ω and u δ = u on Ω Ω δ for δ such that 0 < δ < δ 0 . We refer to [BZ20] for this construction.…”

“…B] and is essentially correct when the boundary datum g belongs to C 1,1 (∂Ω). However its proof in the general case relies on [BZ20,Lem. 4.2] whose proof is not complete (see Lemma 2.2 below).…”

Weighted Green functions for complex Hessian operators

“…Complex Hessian equations have received increasing attention in recent years as they appear in many geometric problems. They provide important examples of fully non-linear PDE's of second order on complex manifolds which interpolate between (linear) complex Laplace-Poisson equations and (non linear) complex Monge-Ampère equations (see [BZ20] and the references therin).…”