A Parameter Version of Forstneri\v{c}'s Splitting Lemma

TL;DR

This paper develops a parameter-dependent solution operator for the ar-equation and applies it to extend Forstneri's splitting lemma, ensuring continuous dependence on parameters for varying maps and domains.

Contribution

It introduces a parameter version of Forstneri's splitting lemma by constructing continuous solution operators for the ar-equation that depend on domain parameters.

Findings

Constructed solution operators for the ar-equation depending continuously on the domain.

Extended Forstneri's splitting lemma to parameter-dependent settings.

Ensured continuous dependence of splitting maps on parameters.

Abstract

We construct solution operators to the $\overline{\partial}$-equation that depend continuously on the domain. This is applied to derive a parameter version of Forstneri\v{c}'s splitting lemma: If both the maps and the domains they are defined on vary continuously with a parameter, then the maps obtained from Forstneri\v{c}'s splitting will depend continuously on the parameter as well.