Gluing small black holes along timelike geodesics II: uniform analysis on glued spacetimes

Given a smooth globally hyperbolic (3+1) -dimensional spacetime (M, g) satisfying the Einstein vacuum equations (possibly with cosmological constant) and an inextendible timelike geodesic C, we constructed in Part I a family of metrics g_ on the complement M_ M of an -neighborhood of C with the following behavior: away from C one has g_ g as 0, while the ^-1-rescaling of g_ around every point of C tends to a fixed subextremal Kerr metric; and g_ solves the Einstein vacuum equation modulo O (^) errors. The ultimate goal, achieved in Part III, is to correct g_ to a true solution on any fixed precompact subset of M by addition of a size O (^) metric perturbation which needs to satisfy a quasilinear wave equation (the Einstein vacuum equations in a suitable gauge). The present paper lays the necessary analytical foundations. We develop a framework for proving estimates for solutions of (tensorial) wave equations on (M_, g_) which, on a suitable scale of Sobolev spaces, are uniform on -independent precompact subsets of the original spacetime M. These estimates are proved by combining two ingredients: the spectral theory for the corresponding wave equation on Kerr; and uniform microlocal estimates governing the propagation of regularity through the small black hole, including radial point estimates reminiscent of diffraction by conic singularities and long-time estimates near perturbations of normally hyperbolic trapped sets. As an illustration of this framework, we construct solutions of a toy nonlinear scalar wave equation on (M_, g_) for uniform timescales and with full control in all asymptotic regimes as 0.