  This is the second in a series of papers in which we take a systematic study
of gauge field theories such as the Maxwell equations and the Yang-Mills
equations, on curved space-times. In this paper, we study the Maxwell equations
in the domain of outer-communication of the Schwarzschild black hole. We show
that if we assume that the middle components of the non-stationary solutions of
the Maxwell equations verify a Morawetz type estimate supported around the
trapped surface, then we can prove uniform decay properties for components of
the Maxwell fields in the entire exterior of the Schwarzschild black hole,
including the event horizon, by making only use of Sobolev inequalities
combined with energy estimates using the Maxwell equations directly. This proof
is entirely gauge independent, and does not pass through the scalar wave
equation on the Schwarzschild black hole, and does not need to separate the
middle components for the Maxwell fields. However, proving a Morawetz estimate
directly using the Maxwell equations, without refering to the scalar wave
equation, seems to be out of reach of the mathematical community as of today;
which I was not able to solve yet in this work. If one is able to prove the
Morawetz estimate directly using the Maxwell equations, this combined with the
present work would give full conceptual proof of decay of the Maxwell fields on
the Schwarzschild black hole, and would then be in particular useful for the
non-abelian case of the Yang-Mills equations where the separation of the middle
components cannot occur. The whole manuscript is written in an expository way
where we detail all the calculations.
