Teukolsky on slowly-rotating Kerr-de Sitter in the vanishing $\mathrm{\Lambda }$ limit

We prove energy, Morawetz and $r^p$-weighted estimates for solutions to the Teukolsky equation set on a slowly-rotating Kerr-de Sitter background. The main feature of our estimates is their uniformity with respect to the cosmological constant $\mathrm{\Lambda }>0$ (thus allowed to tend to $0$), while they hold on the whole domain of outer communications, extending up to $r\sim {\mathrm{\Lambda }}^{-\frac{1}{2}}$. 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 $\mathrm{\Lambda }\to 0$.