Item

Abstract

This first part of the series treats the Maxwell equations in the exterior of
a slowly rotating Kerr black hole. By performing a first-order differential
operator on each extreme Newman-Penrose (N-P) scalar in a Kinnersley tetrad,
the resulting equation and the Teukolsky master equation for the extreme N-P
component are both in the form of an inhomogeneous
\textquotedblleft{spin-weighted Fackerell-Ipser equation\textquotedblright}
(SWFIE) and constitute a weakly coupled system. We first prove energy estimate
and integrated local energy decay (Morawetz) estimate for this type of
inhomogeneous SWFIE following the method in (Dafermos and Rodnianski in Decay
for solutions of the wave equation on Kerr exterior spacetimes I-II: the cases
$|a|\ll M$ or axisymmetry, 2010, arXiv:1010.5132), and then utilize these
estimates to achieve both a uniform bound of a positive definite energy and a
Morawetz estimate for the coupled system of each extreme N-P component. The
same type of estimates for the regular extreme N-P components defined in the
regular Hawking-Hartle tetrad is also proved. The hierarchy here is generalized
in our second part (Ma in Uniform energy bound and Morawetz estimates for
extreme components of spin fields in the exterior of a slowly rotating Kerr
black hole II: linearized gravity, 2017, arXiv:1708.07385) of this series to
treat the extreme components of linearized gravity.