On the blowup of quantitative unique continuation estimates for waves and applications to stability estimates

In this paper we are interested in the blowup of a geometric constant $\mathfrak{C}(\delta)$ appearing in the optimal quantitative unique continuation property for wave operators. In a particular geometric context we prove an upper bound for $\mathfrak{C}(\delta)$ as $\delta$ goes to $0$. Here $\delta>0$ denotes the distance to the maximal unique continuation domain. As applications we obtain stability estimates for the unique continuation property up to the maximal domain. Using our abstract framework~\cite{FO25abstract} we also derive a stability estimate for a hyperbolic inverse problem. The proof is based on a global explicit Carleman estimate combined with the propagation techniques of Laurent-L\'eautaud.