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In this paper, we prove that the slowly-rotating Kerr-de Sitter family ofblack holes are linearly stable as a family of solutions to the Einstein vacuumequations with $\Lambda>0$ in harmonic (wave) gauge. This article is part of aseries that provides a novel proof of the full nonlinear stability of theslowly-rotating Kerr-de Sitter family. This paper and its follow-up offer aself-contained alternative approach to nonlinear stability of the Kerr-deSitter family from the original work of Hintz and Vasy by interpretingquasinormal modes as $H^k$ eigenvalues of an operator on a Hilbert space, andusing integrated local energy decay estimates to prove the existence of aspectral gap. In particular, we avoid the construction of a meromorphiccontinuation of the resolvent. We also do not compactify the spacetime, thusavoiding the use of $b$-calculus and instead only use standardpseudo-differential arguments in a neighborhood of the trapped set; and avoidconstraint damping altogether. The methods in the current paper offer anexplicit example of how to use the vectorfield method to achieve resolventestimates on a trapping background.

Linear stability of the slowly-rotating Kerr-de Sitter family