High frequency behavior of the Leray transform: model hypersurfaces and projective duality
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.
- Publication:
-
arXiv e-prints
- Pub Date:
- November 2021
- DOI:
- 10.48550/arXiv.2111.13954
- arXiv:
- arXiv:2111.13954
- Bibcode:
- 2021arXiv211113954B
- Keywords:
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- Mathematics - Complex Variables;
- 32A26 (Primary);
- 32F17;
- 32A25;
- 32V05;
- 42A38 (Secondary)
- E-Print:
- 54 pages