The Leray transform: factorization, dual $CR$ structures and model hypersurfaces in $\mathbb{C}{\mathbb{P}}^2$

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.