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

We construct solution operators to the $\bar{\mathrm{\partial }}$-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.