Hölder estimates for the \(\overline\partial\)-equation on surfaces with singularities of the type \(E_6\) and \(E_7\) (Q1012411)

Let \(\Sigma\subset \mathbb{C}^3\) be a two dimensional subvariety with simple isolated singularity at the origin. The main result of the paper under review is ``the Hölder estimates for the \(\overline \partial \)-equation on a neighborhood of the isolated singularity''. There are only two series types and three exceptional types of simple isolated singularities, and all of them are of the form of a quotient space \(\mathbb{C}^2/{\mathcal G}\), where \(\mathcal G\) is a finite subgroup of \(SL_2(\mathbb{C}).\) The quotient map induces a covering map which is given by a polynomial map from \(\mathbb{C}^2\) to \(\mathbb{C}^3\). The authors' approach to this problem is: first lift the \(\overline \partial\)-closed form to the covering space (which is a ball with center at the origin); then, apply the result on the ball to get a solution with the Hölder estimate there. The final step is to translate the solution on the covering space to a solution of the original equation. In order to do that, they need a good control of the relation between the distance of two points on the covering space and the distance of their images on \(\Sigma\). That is the following: Theorem. Let \(\pi\) be the polynomial quotient mapping from \(\mathbb{C}^2\) over the singular surface \(\Sigma \cong \mathbb{C}^2/{\mathcal G}\) embedded in \(\mathbb{C}^3\), where \(\mathcal G <SL_2(|\mathbb{C})\) is the subgroup \( {E_6,E_7,D_{n+2}}\) or \(\mathbb Z_n\), with \(n\geq 2\). Define \(\beta=1/{\mathcal G|}\). Given an open ball \(B_R\subset \mathbb{C}^2\) of radius \(R>0\) and centre at the origin, there exists a finite positive constant \(C(R)\) such that: For each pair of points \(z\) and \(\zeta\) in \(B_R\) with \(\|z-\zeta\|\) less than or equal to \(\|z-H\zeta\|\) for every \(H\in \mathcal G\), the following inequality holds, \[ \|\pi(z)-\pi(\zeta)\|^{2\beta}\geq C(R)\|z-\zeta\|(\|z\|+\|\zeta\|). \] Once the above inequality is established, the main theorem of this paper follows easily. That is: Theorem. Let \(\pi\) be the quotient mapping from \(\mathbb{C}^2\) over the singular surface \(\Sigma \cong \mathbb{C}^2/{\mathcal G}\) emdedded in \(\mathbb{C}^3\) where \(\mathcal G <SL_2(\mathbb{C})\) is the subgroup \(E_6, E_7, D_{n+2}\) or \(\mathbb Z_n\), with \(n\geq 2\). Fix \(\delta < 1/{|\mathcal G|}\). Given an open ball \(B_R\subset \mathbb{C}^2\) of radius \(R>0\) and centre at the origin, there exists a finite positive constant \(C(R, \delta)\) such that: For every continuous (0,1) differential form \(\lambda\) defined on the compact set \(\pi (\overline {B_R})\subset \Sigma\) , and \(\overline \partial\)-closed on the regular part of \(\pi (B_R)\), there exists a continuous function \(h\) on \(\pi (B_R)\) which satisfies both the equation \(\overline \partial h= \lambda\) on the regular part of \(\pi(B_R)\) and the Hölder estimate \[ \|h\|_{\pi(B_R)} +\sup_{x,w \in \pi(B_R)} \frac {|h(x)-h(w)|}{\|x-w\|^\delta} \leq C(R,\delta)\|\lambda\|_{\pi(B_R)}. \]