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We establish a new family of Carleman inequalities for wave operators oncylindrical spacetime domains containing a potential that is criticallysingular, diverging as an inverse square on all the boundary of the domain.These estimates are sharp in the sense that they capture both the naturalboundary conditions and the natural $H^1$-energy. The proof is based aroundthree key ingredients: the choice of a novel Carleman weight with rathersingular derivatives on the boundary, a generalization of the classicalMorawetz inequality that allows for inverse-square singularities, and thesystematic use of derivative operations adapted to the potential. As anapplication of these estimates, we prove a boundary observability property forthe associated wave equations.

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