Non-modal linear stability of the Schwarzschild black hole

A proof is given that the space $\L$ of solutions of the linearized vacuum Einstein's equation around a Schwarzschild black hole is parameterized by two scalar fields which are gauge invariant combinations of perturbed algebraic and differential invariants of the Weyl tensor, and encode the information on the odd (-) and even (+) sectors $\L_{\pm}$ These fields measure the distortion of the geometry caused by a generic perturbation, and are shown to be pointwise bounded on the outer region $r \geq 2M$.