A Vector Field Approach to Almost-Sharp Decay for the Wave Equation on Spherically Symmetric, Stationary Spacetimes

摘要

We present a new vector field approach to almost-sharp decay for the waveequation on spherically symmetric, stationary and asymptotically flatspacetimes. Specifically, we derive a new hierarchy of higher-order weightedenergy estimates by employing appropriate commutator vector fields. In caseswhere an integrated local energy decay estimate holds, like in the case ofsub-extremal Reissner-Nordstrom black holes, this hierarchy leads toalmost-sharp global energy and pointwise time-decay estimates with decay ratesthat go beyond those obtained by the traditional vector field method. Ourestimates play a fundamental role in our companion paper where preciselate-time asymptotics are obtained for linear scalar fields on suchbackgrounds.