Carleman estimates with sharp weights and boundary observability for wave operators with critically singular potentials

We establish a new family of Carleman inequalities for wave operators on cylindrical spacetime domains involving a potential that is critically singular, diverging as an inverse square on all the boundary of the domain. These estimates are sharp in the sense that they capture both the natural boundary conditions and the natural H1-energy. The proof is based around three key ingredients: the choice of a novel Carleman weight with rather singular derivatives on the boundary, a generalization of the classical Morawetz inequality that allows for inverse-square singularities, and the systematic use of derivative operations adapted to the potential. As an application of these estimates, we prove a boundary observability property for the associated wave equations.A.E. is supported by the ERC Consolidator Grant 862342 and by the ICMAT– Severo Ochoa grant CEX2019-000904-S. A.S. is supported by the EPSRC grant EP/R011982/1. B.V. is supported by the ERC Starting Grant 801867