The basic setup consists of a complex flag manifold $Z=G/Q$ where $G$ is a
complex semisimple Lie group and $Q$ is a parabolic subgroup, an open orbit $D
= G_0(z) \subset Z$ where $G_0$ is a real form of $G$, and a $G_0$--homogeneous
holomorphic vector bundle $\mathbb E \to D$. The topic here is the double
fibration transform ${\cal P}: H^q(D;{\cal O}(\mathbb E)) \to H^0({\cal
M}_D;{\cal O}(\mathbb E'))$ where $q$ is given by the geometry of $D$, ${\cal
M}_D$ is the cycle space of $D$, and $\mathbb E' \to {\cal M}_D$ is a certain
naturally derived holomorphic vector bundle. Schubert intersection theory is
used to show that ${\cal P}$ is injective whenever $\mathbb E$ is sufficiently
negative.