Boundary Rigidity of Line-Bundle Shilov--Poisson Transforms on Bounded Symmetric Domains

Let $\mathrm{\Omega }=G\mathrm{/}K$ be an irreducible bounded symmetric domain other than the unit disk, and let $S$ be its Shilov boundary. We study the two-parameter family of Shilov–Poisson transforms between homogeneous line bundles over $S$ and $\mathrm{\Omega }$. For a transform $F$ of $L^2$ boundary data, we determine what finite Euclidean regularity up to the full boundary $\mathrm{\partial }\mathrm{\Omega }$ forces in the interior. Outside five explicit exceptional parameter families, sufficiently high finite regularity forces $F=0$. In each exceptional family, the conclusion is an interior covariant differential equation. In the scalar Poisson–Szegő case, it yields a finite-order Graham type theorem: the stated boundary regularity forces a Bergman-harmonic function to be pluriharmonic.