Hidden Boundary Trace Regularity and an Observability Estimate with Interior Remainder for Boundary-Degenerate Hyperbolic Equations

We study hidden boundary trace regularity for two-dimensional hyperbolic equations with boundary degeneracy governed by $\text{mcA}\text{vp}=-\text{Div}(A\mathrm{\nabla }\text{vp})$, where $A=\text{diag}(1,r^{\text{al}})$ and $\text{al}\in (0,1)$. We establish well-posedness in weighted Sobolev spaces and prove an $L^2$ trace estimate for the normal derivative on the nondegenerate side $r=1$. Using truncated geometries and Carleman weights adapted to the anisotropic degeneracy, we derive a large-time observability estimate with a lower-order interior remainder. We also identify a framework-level obstruction at the critical threshold $\text{al}=1$: the weighted Dirichlet coercivity underlying the subcritical analysis loses uniformity and exhibits a logarithmic loss on truncated domains.