High frequency behavior of the Leray transform: model hypersurfaces and projective duality

The Leray transform $\mathbf{L}$ is studied on a family $M_{\gamma }$ of unbounded hypersurfaces in two complex dimensions. For a large class of measures, we obtain necessary and sufficient conditions for the $L^2$-boundedness of $\mathbf{L}$, along with an exact spectral description of ${\mathbf{L}}^{\ast }\mathbf{L}$. This yields both the norm and high-frequency norm of $\mathbf{L}$, the latter giving an affirmative answer to an unbounded analogue of an open conjecture relating the essential norm of $\mathbf{L}$ to a projective invariant on a bounded hypersurface. $\mathbf{L}$ is also shown to play a central role in bridging the function theoretic and projective geometric notions of duality. Our work leads to the construction of projectively invariant Hardy spaces on the $M_{\gamma }$, along with the realization of their duals as invariant Hardy spaces on the dual hypersurfaces.