The fundamental solution to □b on quadric manifolds – part 1. general formulas

Albert Boggess; Andrew Raich

doi:10.1090/bproc/77

The fundamental solution to □_b on quadric manifolds – part 1. general formulas

Albert Boggess
, Andrew Raich

Mathematical and Statistical Sciences, School of (SoMSS)

Research output: Contribution to journal › Article › peer-review

5 Scopus citations

Abstract

This paper is the first of a three part series in which we explore geometric and analytic properties of the Kohn Laplacian and its inverse on general quadric submanifolds of Cⁿ × C^m. In this paper, we present a streamlined calculation for a general integral formula for the complex Green operator N and the projection onto the nullspace of □_b . The main application of our formulas is the critical case of codimension two quadrics in C⁴ wherewediscuss the known solvability and hypoellipticity criteria of Peloso and Ricci [J. Funct. Anal. 203 (2003), pp. 321–355] We also provide examples to show that our formulas yield explicit calculations in some well-known cases: the Heisenberg group and a Cartesian product of Heisenberg groups.

Original language	English (US)
Pages (from-to)	186-203
Number of pages	18
Journal	Proceedings of the American Mathematical Society, Series B
Volume	9
DOIs	https://doi.org/10.1090/bproc/77
State	Published - 2022

Keywords

Complex Green operator
Heisenberg group
Quadric submanifolds
Szegö kernel
Szegö projection
Tangential Cauchy-Riemann operator
fundamental solution
∂

ASJC Scopus subject areas

Algebra and Number Theory
Analysis
Discrete Mathematics and Combinatorics
Geometry and Topology

Access to Document

10.1090/bproc/77

Cite this

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title = "The fundamental solution to □b on quadric manifolds – part 1. general formulas",

abstract = "This paper is the first of a three part series in which we explore geometric and analytic properties of the Kohn Laplacian and its inverse on general quadric submanifolds of Cn × Cm. In this paper, we present a streamlined calculation for a general integral formula for the complex Green operator N and the projection onto the nullspace of □b . The main application of our formulas is the critical case of codimension two quadrics in C4 wherewediscuss the known solvability and hypoellipticity criteria of Peloso and Ricci [J. Funct. Anal. 203 (2003), pp. 321–355] We also provide examples to show that our formulas yield explicit calculations in some well-known cases: the Heisenberg group and a Cartesian product of Heisenberg groups.",

keywords = "Complex Green operator, Heisenberg group, Quadric submanifolds, Szeg{\"o} kernel, Szeg{\"o} projection, Tangential Cauchy-Riemann operator, fundamental solution, ∂",

author = "Albert Boggess and Andrew Raich",

note = "Funding Information: Received by the editors May 15, 2020, and, in revised form, October 27, 2020, December 3, 2020, and December 24, 2020. 2020 Mathematics Subject Classification. Primary 32W10, 35R03, 32V20, 42B37, 43A80. Key words and phrases. Quadric submanifolds, tangential Cauchy-Riemann operator, ∂b¯, complex Green operator, Szeg{\"o} kernel, Szeg{\"o} projection, fundamental solution, Heisenberg group. This work was supported by a grant from the Simons Foundation (707123, ASR). Publisher Copyright: {\textcopyright} 2022 by the author(s) under Creative Commons Attribution-NonCommercial 3.0 License (CC BY NC 3.0)",

year = "2022",

doi = "10.1090/bproc/77",

language = "English (US)",

volume = "9",

pages = "186--203",

journal = "Proceedings of the American Mathematical Society, Series B",

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publisher = "American Mathematical Society",

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AU - Raich, Andrew

N1 - Funding Information: Received by the editors May 15, 2020, and, in revised form, October 27, 2020, December 3, 2020, and December 24, 2020. 2020 Mathematics Subject Classification. Primary 32W10, 35R03, 32V20, 42B37, 43A80. Key words and phrases. Quadric submanifolds, tangential Cauchy-Riemann operator, ∂b¯, complex Green operator, Szegö kernel, Szegö projection, fundamental solution, Heisenberg group. This work was supported by a grant from the Simons Foundation (707123, ASR). Publisher Copyright: © 2022 by the author(s) under Creative Commons Attribution-NonCommercial 3.0 License (CC BY NC 3.0)

PY - 2022

Y1 - 2022

N2 - This paper is the first of a three part series in which we explore geometric and analytic properties of the Kohn Laplacian and its inverse on general quadric submanifolds of Cn × Cm. In this paper, we present a streamlined calculation for a general integral formula for the complex Green operator N and the projection onto the nullspace of □b . The main application of our formulas is the critical case of codimension two quadrics in C4 wherewediscuss the known solvability and hypoellipticity criteria of Peloso and Ricci [J. Funct. Anal. 203 (2003), pp. 321–355] We also provide examples to show that our formulas yield explicit calculations in some well-known cases: the Heisenberg group and a Cartesian product of Heisenberg groups.

AB - This paper is the first of a three part series in which we explore geometric and analytic properties of the Kohn Laplacian and its inverse on general quadric submanifolds of Cn × Cm. In this paper, we present a streamlined calculation for a general integral formula for the complex Green operator N and the projection onto the nullspace of □b . The main application of our formulas is the critical case of codimension two quadrics in C4 wherewediscuss the known solvability and hypoellipticity criteria of Peloso and Ricci [J. Funct. Anal. 203 (2003), pp. 321–355] We also provide examples to show that our formulas yield explicit calculations in some well-known cases: the Heisenberg group and a Cartesian product of Heisenberg groups.

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KW - Heisenberg group

KW - Quadric submanifolds

KW - Szegö kernel

KW - Szegö projection

KW - Tangential Cauchy-Riemann operator

KW - fundamental solution

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