Hölder estimates for the \(\overline\partial\)-equation on singular quotient varieties (Q1959050)

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 \(\pi (B_R)\). \[ \|x-y\| \geq \frac {C_3\cdot (D_{\mathcal G}(x,y))^N}{p\cdot\max [1,(2R)^{N-2}]}. \] So, the Hölder estimates above can also be formulated in terms of the embedded distance. In the final part of this paper, they apply the results above to all cases of surfaces with simple singularities, which even cover the missing part of the previous work mentioned above.