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 $H^1$-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.

中文翻译：

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具有临界奇异势的波算子的具有尖锐权重和边界可观测性的卡尔曼估计

我们为圆柱时空域上的波算子建立了一个新的卡尔曼不等式族，涉及一个临界奇异的势，在域的所有边界上发散为反平方。这些估计是尖锐的，因为它们同时捕获了自然边界条件和自然 $H^1$-能量。该证明基于三个关键要素：选择在边界上具有相当奇异导数的新卡尔曼权重、允许反平方奇异性的经典莫拉维茨不等式的推广，以及系统地使用适用于潜在的导数运算. 作为这些估计的应用，我们证明了相关波动方程的边界可观测性。