Charged scalar fields on Reissner--Nordström spacetimes I: integrated energy estimates

This is the first part of a series of papers deriving the precise, late-time behaviour and (in)stability properties of charged scalar fields on near-extremal Reissner–Nordström spacetimes via energy estimates. In this paper, we establish global, weighted integrated energy decay and energy boundedness estimates for solutions to the charged scalar field equation on (near-)extremal Reissner–Nordström(–de Sitter) spacetimes. These estimates extend to Reissner–Nordström spacetimes away from extremality under the assumption of mode stability on the real axis. Together with the companion paper [Gaj26], this paper forms the first global quantitative analysis of the charged scalar field equation on asymptotically flat black hole spacetimes, without a smallness assumption on the scalar field charge. Due to a coupling of the degeneration of the red-shift effect with the presence of superradiance at the linearized level, charged scalar fields on Reissner–Nordström spacetimes also probe some of the main difficulties encountered when studying the (neutral) wave equation on extremal Kerr spacetimes.