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

Albert Boggess, Andrew Raich

Research output: Contribution to journalArticlepeer-review

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.

Original languageEnglish (US)
Pages (from-to)186-203
Number of pages18
JournalProceedings of the American Mathematical Society, Series B
Volume9
DOIs
StatePublished - 2022
Externally publishedYes

Keywords

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

ASJC Scopus subject areas

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

Fingerprint

Dive into the research topics of 'The fundamental solution to □b on quadric manifolds – part 1. general formulas'. Together they form a unique fingerprint.

Cite this