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This paper is the first of a series regarding the Teukolsky equation of spin$\pm 1$ and spin $\pm 2$ on Kerr backgrounds in the full subextremal range ofparameters $|a|<M$. In the present paper, we study fixed frequency solutions ofthe transformed system of equations introduced by Dafermos, Holzegel andRodnianski, obtaining estimates which are uniform in the separation parameters.A corollary of our result, to be laid out in the second paper of the series, isthat solutions of the Teukolsky equation on subextremal Kerr arising fromregular initial data remain bounded and decay in time. This is a key step inestablishing the full linear stability of Kerr under electromagnetic andgravitational perturbations. Our estimates can also be applied to understandingmore delicate features of the Teukolsky equation, such as their scatteringproperties.

Boundedness and decay for the Teukolsky equation on Kerr in the full subextremal range $|a|