Improved decay rates with small regularity loss for the wave equation about a Schwarzschild black hole

We continue our study of the decoupled wave equation in the exterior of a spherically symmetric, Schwarzschild, black hole. Because null geodesics on the photon sphere orbit the black hole, extra effort must be made to show that the high angular momentum components of a solution decay sufficiently fast, particularly for low regularity initial data. Previous results are rapid decay for regular ($H^3$) initial data \cite{BSterbenz} and slower decay for rough ($H^{1+\epsilon}$) initial data \cite{BlueSoffer3}. Here, we combine those methods to show boundedness of the conformal charge. From this, we conclude that there are bounds for global in time, space-time norms, in particular \int_I |\tilde\phi|^4 d^4vol < C for $H^{1+\epsilon}$ initial data with additional decay towards infinite and the bifurcation sphere. Here $\tilde\phi$ refers to a solution of the wave equation. $I$ denotes the exterior region of the Schwarzschild solution, which can be expressed in coordinates as $r>2M$, $t\in\Reals, \omega\in S^2$, and $d^4\text{vol}$ is the natural 4-dimensional volume induced by the Schwarzschild pseudo-metric. We also demonstrate that the photon sphere has the same influence on the wave equation as a closed geodesic has on the wave equation on a Riemannian manifold. We demonstrate this similarity by extending our techniques to the wave equation on a class of Riemannian manifolds. Under further assumptions, the space-time estimates are sufficient to prove global bounds for small data, nonlinear wave equations on a class of Riemannian manifolds with closed geodesics. We must use global, space-time integral estimates since $L^\infty$ estimates cannot hold at this level of regularity.