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

## TL;DR

This paper develops sharp Carleman estimates for wave operators with critically singular inverse-square potentials on cylindrical domains, enabling boundary observability results despite the singularities.

## Contribution

It introduces a new family of Carleman inequalities with sharp weights tailored for wave operators with inverse-square boundary singularities, advancing control theory for such PDEs.

## Key findings

- Established sharp Carleman inequalities for singular potentials
- Proved boundary observability for wave equations with inverse-square potentials
- Generalized Morawetz inequality to handle critical singularities

## Abstract

We establish a new family of Carleman inequalities for wave operators on cylindrical spacetime domains containing 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 $H^1$-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.