We consider the Cauchy problem for the massless scalar wave equation in the extreme Kerr geometry with the smooth initial data compactly supported outside the event horizon. Firstly, we derive an integral representation of the solution as an infinite sum over the angular momentum modes, each of which is an integral of the energy variable omega on the real axis. This integral representation involves solutions of the radial and angular ordinary differential equations which arise in the separation of variables. Furthermore, based on this integral representation, we prove that the solution of every azimuthal mode decays pointwise in time in L-loc(infinity).
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