0 60 80 73 v 2 1 8 Se p 20 06 Errata for “ Global existence and scattering for the nonlinear Schrödinger equation on Schwarzschild manifolds ” , “ Semilinear wave

In [7], [1], [2], local decay estimates were proven for the (decoupled) Schrödinger, wave, and Regge-Wheeler equations on the Schwarzschild manifold, using commutator methods. Here, we correct a step in the commutator argument. The corrected argument works either for radial semilinear equations or general linear equations. This recovers the results in [7] and [2], but does not recover the non radial, large data, semilinear result asserted in [1]. In [7], [1], [2], various equations are considered on the Schwarzschild manifold, with the aim of making progress towards understanding the stability of the Schwarzschild solution. These papers use vector field methods and similar commutator arguments to get local decay results. We report here that these papers all contain an error in the calculation of a commutator of the form i[−∂ 2 r * , (1/2)(g(−i∂ r *) + (−i∂ r *)g)] involving the multiplier (1/2)(g(−i∂ r *) + (−i∂ r *)g). These papers have been followed by further work by ourselves and others [3], [4], [5], using additional vector field arguments to prove stronger decay results for wave equations. Local decay arguments are a necessary part of these results, and both [4] and [5] provide corrected arguments at the analogous stage. Interesting decay results have also been proven using very different methods in [6]. Here, we have modified the multiplier, (1/2)(g(−i∂ r *) + (−i∂ r *)g). By local decay estimates, we mean space time integral estimates of the form, for σ > 1, ∞ 1 R×S 2