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

TL;DR

This paper analyzes the Leray transform on unbounded hypersurfaces in complex two-dimensional space, providing conditions for boundedness, spectral descriptions, and linking function theory with projective geometry.

Contribution

It offers necessary and sufficient conditions for the $L^2$-boundedness of the Leray transform and establishes a connection between its spectral properties and projective invariants.

Findings

Derived exact spectral description of $f{L}^*f{L}$

Established conditions for $L^2$-boundedness of $f{L}$

Constructed projectively invariant Hardy spaces

Abstract

The Leray transform $\bf{L}$ 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 $\bf{L}$, along with an exact spectral description of $\bf{L}^*\bf{L}$. This yields both the norm and high-frequency norm of $\bf{L}$, the latter giving an affirmative answer to an unbounded analogue of an open conjecture relating the essential norm of $\bf{L}$ to a projective invariant on a bounded hypersurface. $\bf{L}$ 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.