  We compute the exact norms of the Leray transforms for a family
$\mathcal{S}_{\beta}$ of unbounded hypersurfaces in two complex dimensions. The
$\mathcal{S}_{\beta}$ generalize the Heisenberg group, and provide local
projective approximations to any smooth, strongly $\mathbb{C}$-convex
hypersurface $\mathcal{S}_{\beta}$ to two orders of tangency. This work is then
examined in the context of projective dual $CR$-structures and the
corresponding pair of canonical dual Hardy spaces associated to
$\mathcal{S}_{\beta}$, leading to a universal description of the Leray
transform and a factorization of the transform through orthogonal projection
onto the conjugate dual Hardy space.
