Hölder estimates for the \(\overline\partial\)-equation on singular quotient varieties
type: Item
instance of: scholarly article
title: Hölder estimates for the \(\overline\partial\)-equation on singular quotient varieties
zbMATH Open document ID: 1216.32024
author: Martín Solis
author: E. S. Zeron
published in: Boletín de la Sociedad Matemática Mexicana. Third Series
publication date: 2010-10-01T00:00:00Z
review text: This paper is a continuation of the authors' previous work [\textit{F. Acosta} and \textit{E. S. Zeron}, Bol. Soc. Mat. Mex., III. Ser. 13, No. 1, 73--86 (2007; Zbl 1195.32017)], they consider the Hölder estimates for the \(\overline \partial\)-equation on singular quotient varieties. Let \(\mathcal G\) be a finite subgroup of unitary \(n\times n\) matrices acting on \(\mathbb C^n\) by matrix multiplication. Assume that \(\mathbb C^n/\mathcal G\) is well embedded in \(\mathbb C^p\) for some \(p>n\) and let \(\mathbb C^n/\mathcal G\) be identified with its embedded image in \(\mathbb C^p\). Let \(\pi: \mathbb C^n \to \mathbb C^n/\mathcal G\) be the quotient map and let \(D_{\mathcal G}(x,y)\) be the natural distance on \(\mathbb C^n/\mathcal G\) induced by \(\pi\) (the distance between \(\pi^{-1}(x)\) and \(\pi^{-1}(y)\) in \(\mathbb C^n\)). Fix \(R>0\) and let \(B_R\) be the open ball in \(\mathbb C^n\) of radius \(R\) and center at the origin. Let \(\lambda\) be a continuous (0,1)-form on \(\pi (B_R)\) which is \(\overline \partial\) closed on the regular part of \(\pi (B_R)\). In the first part of this paper, the authors -- using the quotient map combined with Henkin's construction -- build a continuous solution of \(\overline \partial h=\lambda\) on \(\pi(B_R)\) which satisfies the following estimates in terms of the natural distance on \(\pi(B_R)\). \[ \begin{aligned}\|h\|_{\pi(B_R)} +\sup_{x,y \in \pi(B_R)} \frac {|h(x)-h(y)|}{D_{\mathcal G}(x,y)^{1/2}} &\leq C_1(R)\|\lambda\|_{\pi(B_R)},\\ \sup_{x,y \in \pi(B_{R/2})} \frac {|h(x)-h(y)|}{D_{\mathcal G}(x,y)^{\delta}} &\leq C_2(R,\delta)\|\lambda\|_{\pi(B_R)}, \end{aligned} \] where \(0<\delta < 1\). In the second part of this paper, assuming that the embedding functions are homogeneous polynomials, the authors find a unified way to establish the following relation between the embedded distance (in \(\mathbb C^p\)) and the natural distance on \(\pi (B_R)\). \[ \|x-y\| \geq \frac {C_3\cdot (D_{\mathcal G}(x,y))^N…
Mathematics Subject Classification ID: 32W05
Mathematics Subject Classification ID: 35N15
zbMATH DE Number: 5794423
zbMATH Keywords: \(\overline\partial\)-equation
zbMATH Keywords: Hölder estimates
zbMATH Keywords: singular quotient varieties
reviewed by: Lee Taichung
MaRDI profile type: Publication
Recommended article: Hölder estimates for the \(\overline \partial\)-equation on surfaces with simple singularities
Recommended article: Hölder estimates for the \(\overline\partial\)-equation in some domains of finite type
Recommended article: Singular quasilinear elliptic equations and Hölder regularity
Recommended article: Hölder estimates for the \(\overline\partial\)-equation on surfaces with singularities of the type \(E_6\) and \(E_7\)
Recommended article: Hölder estimates for the ∂̄ problem for (p,q) forms on product domains
Recommended article: Singular quasilinear elliptic systems and Hölder regularity
Recommended article: The \({\overline{\partial}}\)-equation on homogeneous varieties with an isolated singularity
Recommended article: Q3397543
Recommended article: \(L^{2}\) estimates for \(\bar{\partial}\) across a divisor with Poincaré-like singularities
Recommended article: Quantitative Hölder estimates for even singular integral operators on patches
description: scientific article; zbMATH DE number 5794423