High frequency behavior of the Leray transform: model hypersurfaces and projective duality
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| Format: | Preprint |
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2021
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| _version_ | 1866909624159961088 |
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| author | Barrett, David E. Edholm, Luke D. |
| author_facet | Barrett, David E. Edholm, Luke D. |
| contents | The Leray transform $\bf{L}$ is studied on a family $M_γ$ 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 $M_γ$, along with the realization of their duals as invariant Hardy spaces on the dual hypersurfaces. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2111_13954 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | High frequency behavior of the Leray transform: model hypersurfaces and projective duality Barrett, David E. Edholm, Luke D. Complex Variables 32A26 (Primary), 32F17, 32A25, 32V05, 42A38 (Secondary) The Leray transform $\bf{L}$ is studied on a family $M_γ$ 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 $M_γ$, along with the realization of their duals as invariant Hardy spaces on the dual hypersurfaces. |
| title | High frequency behavior of the Leray transform: model hypersurfaces and projective duality |
| topic | Complex Variables 32A26 (Primary), 32F17, 32A25, 32V05, 42A38 (Secondary) |
| url | https://arxiv.org/abs/2111.13954 |