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This first part of the series treats the Maxwell equations in the exterior ofa slowly rotating Kerr black hole. By performing a first-order differentialoperator on each extreme Newman-Penrose (N-P) scalar in a Kinnersley tetrad,the resulting equation and the Teukolsky master equation for the extreme N-Pcomponent 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 estimateand integrated local energy decay (Morawetz) estimate for this type ofinhomogeneous SWFIE following the method in (Dafermos and Rodnianski in Decayfor 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 theseestimates to achieve both a uniform bound of a positive definite energy and aMorawetz estimate for the coupled system of each extreme N-P component. Thesame type of estimates for the regular extreme N-P components defined in theregular Hawking-Hartle tetrad is also proved. The hierarchy here is generalizedin our second part (Ma in Uniform energy bound and Morawetz estimates forextreme components of spin fields in the exterior of a slowly rotating Kerrblack hole II: linearized gravity, 2017, arXiv:1708.07385) of this series totreat the extreme components of linearized gravity.

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Uniform energy bound and Morawetz estimate for extreme components of  spin fields in the exterior of a slowly rotating Kerr black hole I: Maxwell  field