  In this paper, we prove that the slowly-rotating Kerr-de Sitter family of
black holes are linearly stable as a family of solutions to the Einstein vacuum
equations with $\Lambda>0$ in harmonic (wave) gauge. This article is part of a
series that provides a novel proof of the full nonlinear stability of the
slowly-rotating Kerr-de Sitter family. This paper and its follow-up offer a
self-contained alternative approach to nonlinear stability of the Kerr-de
Sitter family from the original work of Hintz and Vasy by interpreting
quasinormal modes as $H^k$ eigenvalues of an operator on a Hilbert space, and
using integrated local energy decay estimates to prove the existence of a
spectral gap. In particular, we avoid the construction of a meromorphic
continuation of the resolvent. We also do not compactify the spacetime, thus
avoiding the use of $b$-calculus and instead only use standard
pseudo-differential arguments in a neighborhood of the trapped set; and avoid
constraint damping altogether. The methods in the current paper offer an
explicit example of how to use the vectorfield method to achieve resolvent
estimates on a trapping background.
