Annals of Mathematical Sciences and Applications

Volume 8 (2023)

Number 1

The Kohn–Laplacian and Cauchy–Szegö projection on model domains

Pages: 111 – 155

DOI: https://dx.doi.org/10.4310/AMSA.2023.v8.n1.a4

Authors

Der-Chen Chang (Department of Mathematics and Statistics, Georgetown University, Washington, D.C., U.S.A.; and Graduate Institute of Business Administration, College of Management, Fu Jen Catholic University, Taipei, Taiwan)

Ji Li (School of Mathematical and Physical Sciences, Macquarie University, Sydney, NSW, Australia)

Jingzhi Tie (Department of Mathematics, University of Georgia, Athens, Ga., U.S.A.)

Qingyan Wu (Department of Mathematics, Linyi University, Shandong, China)

Abstract

We study the Kohn–Laplacian and its fundamental solution on some model domains in $\mathbb{C}^{n+1}$, and further discuss the explicit kernel of the Cauchy–Szegö projections on these model domains using the real analysis method. We further show that these Cauchy–Szegö kernels are Calderón–Zygmund kernels under the suitable quasi-metric.

Keywords

CR manifolds, Kohn–Laplacian, Cauchy–Szegö projection, Heisenberg group

2010 Mathematics Subject Classification

Primary 32V05, 32V20. Secondary 53C56.

Full Text (PDF format)

All authors contributed equally to this work which included mathematical theory and analysis. The authors have read and agreed to the published version of the manuscript.

Der-Chen Chang is partially supported by an NSF grant DMS-1408839 and a McDevitt Endowment Fund at Georgetown University.

Ji Li is partially supported by the Australian Research Council (ARC) through the research grant DP220100285.

Qingyan Wu is supported by the National Science Foundation of China (grant nos. 12171221 and 12071197), the Natural Science Foundation of Shandong Province (grant nos. ZR2021MA031 and 2020KJI002).

Received 27 November 2022

Accepted 6 December 2022

Published 30 March 2023