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

Abstract

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.

References in corpus (3)

• Cauchy-type integrals in several complex variables
• The Cauchy-Szeg\H o projection for domains in $\mathbb C^n% with minimal smoothness
• Sums of CR functions from competing CR structures

Cited by in corpus (4)

• High frequency behavior of the Leray transform: model hypersurfaces and projective duality
• Cauchy transforms and Szegő projections in dual Hardy spaces: inequalities and Möbius invariance
• The Laplace and Leray transforms on some (weakly) convex domains in ${\mathbb{C}}^2$
• The Leray transform: distinguished measures, symmetries and polygamma inequalities