Cauchy transforms and Szegő projections in dual Hardy spaces: Inequalities and Möbius invariance

Journal of Functional Analysis · 2025-04-02

Cauchy transforms and Szegő projections in dual Hardy spaces: inequalities and Möbius invariance

arXiv (Cornell University) · 2024-07-17

Dual pairs of interior and exterior Hardy spaces associated to a simple closed Lipschitz planar curve are considered, leading to a Möbius invariant function bounding the norm of the Cauchy transform $\bf{C}$ from below. This function is shown to satisfy strong rigidity properties and is closely connected via the Berezin transform to the square of the Kerzman-Stein operator. Explicit example calculations are presented. For ellipses, a new asymptotically sharp lower bound on the norm of $\bf{C}$ is produced.

Invariant rectification of non-smooth planar curves

Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry · 2023-08-12

Sums of CR and projective dual CR functions

Pure and Applied Mathematics Quarterly · 2022-01-01

A smooth, strongly $\mathbb{C}$-convex, real hypersurface $S$ in $\mathbb{CP}^n$ admits a projective dual CR structure in addition to the standard CR structure. Given a smooth function $u$ on $S$, we provide characterizations for when $u$ can be decomposed as a sum of a CR function and a dual CR function. Following work of Lee on pluriharmonic boundary values, we provide a characterization using differential forms. We further provide a characterization using tangential vector fields in the style of Audibert and Bedford.

Sums of CR and projective dual CR functions

arXiv (Cornell University) · 2021-09-02

A smooth, strongly $\mathbb{C}$-convex, real hypersurface $S$ in $\mathbb{CP}^n$ admits a projective dual CR structure in addition to the standard CR structure. Given a smooth function $u$ on $S$, we provide characterizations for when $u$ can be decomposed as a sum of a CR function and a dual CR function. Following work of Lee on pluriharmonic boundary values, we provide a characterization using differential forms. We further provide a characterization using tangential vector fields in the style of Audibert and Bedford.

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

arXiv (Cornell University) · 2021 · 1 citations

• Mathematics
• Pure mathematics
• Mathematical analysis

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.

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Advances in Mathematics · 2020 · 6 citations

• Mathematics
• Pure mathematics
• Combinatorics

Projective-umbilic points of circular real hypersurfaces in $\mathbb {C}^2$

Proceedings of the American Mathematical Society · 2020-05-20

We show that the boundary of any bounded strongly pseudoconvex complete circular domain in $\mathbb {C}^2$ must contain points that are exceptionally tangent to a projective image of the unit sphere.

Projective-umbilic points of circular real hypersurfaces in $\\mathbb\n C^2$

arXiv (Cornell University) · 2019-09-20

We show that the boundary of any bounded strongly pseudoconvex complete\ncircular domain in $\\mathbb C^2$ must contain points that are exceptionally\ntangent to a projective image of the unit sphere.\n

The Clemson Juvenile Delinquency Project: Major Findings from a Multi-Agency Study

Journal of Child and Family Studies · 2017-06-05 · 18 citations