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.