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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 localprojective approximations to any smooth, strongly $\mathbb{C}$-convexhypersurface $\mathcal{S}_{\beta}$ to two orders of tangency. This work is thenexamined in the context of projective dual $CR$-structures and thecorresponding pair of canonical dual Hardy spaces associated to$\mathcal{S}_{\beta}$, leading to a universal description of the Leraytransform and a factorization of the transform through orthogonal projectiononto the conjugate dual Hardy space.

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