Fourier supports of K-finite Bessel integrals on classical tube domains

Let (Formula presented.) be the symmetric tube domain associated with the Jordan algebra (Formula presented.), (Formula presented.), (Formula presented.), or (Formula presented.), and (Formula presented.) be its Shilov boundary. Also, let (Formula presented.) be a degenerate principal series representation of (Formula presented.). Then we investigate the Bessel integrals assigned to functions in general (Formula presented.)-types of (Formula presented.). We give individual upper bounds of their supports, when (Formula presented.) is reducible. We also use the upper bounds to give a partition for the set of all (Formula presented.)-types in (Formula presented.), that turns out to explain the (Formula presented.)-module structure of (Formula presented.). Thus, our results concretely realize a relationship observed by Kashiwara and Vergne [(Formula presented.)-types and singular spectrum, in Noncommutative Harmonic analysis, Lecture Notes in Mathematics, Vol. 728 (Springer, 1979), pp. 177–200] between the Fourier supports and the asymptotic (Formula presented.)-supports assigned to (Formula presented.)-submodules in (Formula presented.).