  In this paper we consider solutions to the linear wave equation on higher
dimensional Schwarzschild black hole spacetimes and prove robust nondegenerate
energy decay estimates that are in principle required in a nonlinear stability
problem. More precisely, it is shown that for solutions to the wave equation
\Box_g\phi=0 on the domain of outer communications of the Schwarzschild
spacetime manifold (M^n_m, g) (where n >= 3 is the spatial dimension, and m > 0
is the mass of the black hole) the associated energy flux E[\phi](\Sigma_\tau)
through a foliation of hypersurfaces (\Sigma_\tau) (terminating at future null
infinity and to the future of the bifurcation sphere) decays,
E[\phi](\Sigma_\tau) <= CD/\tau^2, where C is a constant only depending on n
and m, and D < \infty is a suitable higher order initial energy on \Sigma_0;
moreover we improve the decay rate for the first order energy to
E[\partial_t\phi](\Sigma_\tau^R) <= CD/\tau^(4-2\delta) for any \delta > 0
where \Sigma_\tau^R denotes the hypersurface (\Sigma_\tau) truncated at an
arbitrarily large fixed radius R < \infty provided the higher order energy
D_\delta on \Sigma_0 is finite. We conclude our paper by interpolating between
these two results to obtain the pointwise estimate |\phi|_{\Sigma_\tau^R} <= (C
D'_\delta) / \tau^(3/2-\delta). In this work we follow the new physical-space
approach to decay for the wave equation of Dafermos and Rodnianski.
