Relatively non-degenerate integrated decay estimates on subextremal Kerr de Sitter

We study the Klein--Gordon equation $\Box\psi-\mu^2_{\textit{KG}}\psi=0$ on subextremal Kerr de Sitter black hole backgrounds with parameters $(a,M,l)$, where $l^2=\frac{3}{\Lambda}$. We prove a "relatively non degenerate integrated" decay estimate assuming an appropriate mode stability statement for real frequency solutions of Carter's radial ode. Our results, in particular, apply unconditionally in the very slowly rotating case $|a|\ll M,l$, and in the case where $\psi$ is axisymmetric. Exponential decay for $\psi$ to a constant is a consequence of this estimate. To prove our result, we introduce a novel pseudodifferential commutation operator $\mathcal{G}$ that generalizes our previous purely physical space commutation \cite{mavrogiannis} and we use it in conjunction with the Morawetz estimate of our companion \cite{mavrogiannis4}. This pseudodifferential operator is defined using Fourier decomposition with respect to time frequencies $\omega$ and azimuthal frequencies $m$, but does not require Carter's full separation.