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

2021/11/27 by David E. Barrett, Barrett, David E., Luke D. Edholm +1
Mathematics · Medicine · #32A25 #32A26 (Primary) #32F17 #32V05 #42A38 (Secondary) #Advanced Harmonic Analysis Research #Complex Variables (math.CV) #Dupuytren's Contracture and Treatments #FOS: Mathematics

paper · doi:10.48550/arxiv.2111.13954

openalex publication_date 2021/11/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The Leray transform \bfL 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 L²-boundedness of \bfL, along with an exact spectral description of \bfL^*\bfL. This yields both the norm and high-frequency norm of \bfL, the latter giving an affirmative answer to an unbounded analogue of an open conjecture relating the essential norm of \bfL to a projective invariant on a bounded hypersurface. \bfL 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.

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