Decay and non-decay of the local energy for the wave equation on the De Sitter-Schwarzschild metric

We describe an expansion of the solution of the wave equation on the De Sitter-Schwarzschild metric in terms of resonances. The principal term in the expansion is due to a resonance at 0. The error term decays polynomially if we permit a logarithmic derivative loss in the angular directions and exponentially if we permit an e derivative loss in the angular directions.