Teukolsky on slowly-rotating Kerr-de Sitter in the vanishing $Λ$ limit

As a first step towards resolving a vanishing cosmological constant black hole stability conjecture, we prove energy, Morawetz and rp-weighted estimates for solutions to the Teukolsky equations on a slowly-rotating Kerr-de Sitter background, which we derive using an extension of the non-integrable formalism of [GKS24]. The main feature of our estimates is their uniformity with respect to the cosmological constant $Λ>0$ (thus allowed to tend to 0), while they hold on the whole domain of outer communications, extending up to $Λ^{-\frac{1}{2}}$. As an application of our result, we recover well-known corresponding estimates for solutions to Teukolsky on a slowly-rotating Kerr background in the limit $Λ\to 0$.