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 $\mathcal{C}$, we constructed in Part I a family of metrics $g_{\epsilon }$ on the complement $M_{\epsilon }\subset M$ of an $\epsilon$-neighborhood of $\mathcal{C}$ with the following behavior: away from $\mathcal{C}$ one has $g_{\epsilon }\to g$ as $\epsilon \to 0$, while the ${\epsilon }^{-1}$-rescaling of $g_{\epsilon }$ around every point of $\mathcal{C}$ tends to a fixed subextremal Kerr metric; and $g_{\epsilon }$ solves the Einstein vacuum equation modulo $\mathcal{O}({\epsilon }^{\mathrm{\infty }})$ errors. The ultimate goal, achieved in Part III, is to correct $g_{\epsilon }$ to a true solution on any fixed precompact subset of $M$ by addition of a size $\mathcal{O}({\epsilon }^{\mathrm{\infty }})$ 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 $(M_{\epsilon },g_{\epsilon })$ which, on a suitable scale of Sobolev spaces, are uniform on $\epsilon$-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 $(M_{\epsilon },g_{\epsilon })$ for uniform timescales and with full control in all asymptotic regimes as $\epsilon \to 0$.