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Price's Law states that linear perturbations of a Schwarzschild black holefall off as $t^{-2\ell-3}$ for $t \to \infty$ provided the initial data decaysufficiently fast at spatial infinity. Moreover, if the perturbations areinitially static (i.e., their time derivative is zero), then the decay ispredicted to be $t^{-2\ell-4}$. We give a proof of $t^{-2\ell-2}$ decay forgeneral data in the form of weighted $L^1$ to $L^\infty$ bounds for solutionsof the Regge--Wheeler equation. For initially static perturbations we obtain$t^{-2\ell-3}$. The proof is based on an integral representation of thesolution which follows from self--adjoint spectral theory. We apply twodifferent perturbative arguments in order to construct the correspondingspectral measure and the decay bounds are obtained by appropriate oscillatoryintegral estimates.

A proof of Price's Law on Schwarzschild black hole manifolds for all angular momenta