LAUSR

We prove boundedness and polynomial decay statements for solutions to the spin $\pm1$ Teukolsky-type equation projected to the $\ell=1$ spherical harmonic on Reissner-Nordstr{\"o}m spacetime. The equation is verified by a gauge-invariant quantity which we identify and which involves the electromagnetic and curvature tensor. This gives a first description in physical space of gauge-invariant quantities transporting the \textit{electromagnetic radiation} in perturbations of a charged black hole. a#13; a#13; The proof is based on the use of derived quantities, introduced in previous works on linear stability of Schwarzschild (\cite{DHR}). The derived quantity verifies a Fackerell-Ipser-type equation, with right hand side vanishing at the $\ell=1$ spherical harmonics. The boundedness and decay for the projection to the $\ell\geq 2$ spherical harmonics are implied by the boundedness and decay for the Teukolsky system of spin $\pm2$ obtained in \cite{Giorgi4}.a#13; a#13; a#13; The spin $\pm1$ Teukolsky-type equation is verified by the curvature and electromagnetic components of a gravitational and electromagnetic perturbation of the Reissner-Nordstr{\"o}m spacetime. Consequently, together with the estimates obtained in \cite{Giorgi4}, these bounds allow to prove the full linear stability of Reissner-Nordstr{\"o}m metric for small charge to coupled gravitational and electromagnetic perturbations.