Long time energy averages and a lower resolvent estimate for damped waves

We consider the damped wave equation on a compact manifold. We propose different ways of measuring decay of the energy (time averages of lower energy levels, decay for frequency localized data...) and exhibit links with resolvent estimates on the imaginary axis. As an application we prove a universal logarithmic lower resolvent bound on the imaginary axis for the damped wave operator when the Geometric Control Condition (GCC) is not satisfied. This is to be compared to the uniform boundedness of the resolvent on that set when GCC holds. The proofs rely on (i) various (re-)formulations of the damped wave equation as a conservative hyperbolic part perturbed by a lower order damping term;(ii) a "Plancherel-in-time" argument as in classical proofs of the Gearhart-Huang-Pr{\"u}ss theorem; and (iii) an idea of Bony-Burq-Ramond of propagating a coherent state along an undamped trajectory up to Ehrenfest time.