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Transactions of the American Mathematical Society

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The Kohn-Laplace equation on abstract CR manifolds: Global regularity


Authors: Tran Vu Khanh and Andrew Raich
Journal: Trans. Amer. Math. Soc. 373 (2020), 7575-7606
MSC (2010): Primary 32V20, 32V35, 32W10; Secondary 35N15, 32U05
DOI: https://doi.org/10.1090/tran/8206
Published electronically: August 28, 2020
MathSciNet review: 4169668
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Abstract: Let $ M$ be a compact, pseudoconvex-oriented, $ (2n+1)$-dimensional, abstract CR manifold of hypersurface type, $ n\geq 2$. We prove the following:

(i) If $ M$ admits a strictly CR-plurisubharmonic function on $ (0,q_0)$-forms, then the complex Green operator $ G_q$ exists and is continuous on $ L^2_{0,q}(M)$ for degrees $ q_0\le q\le n-q_0$. In the case that $ q_0=1$, we also establish continuity for $ G_0$ and $ G_n$. Additionally, the $ \bar {\partial }_{b}$-equation on $ M$ can be solved in $ C^\infty (M)$.

(ii) If $ M$ satisfies ``a weak compactness property'' on $ (0,q_0)$-forms, then $ G_q$ is a continuous operator on $ H^s_{0,q}(M)$ and is therefore globally regular on $ M$ for degrees $ q_0\le q\le n-q_0$; and also for the top degrees $ q=0$ and $ q=n$ in the case $ q_0=1$.

We also introduce the notion of a ``plurisubharmonic CR manifold'' and show that it generalizes the notion of ``plurisubharmonic defining function'' for a domain in $ \mathbb{C}^N$ and implies that $ M$ satisfies the weak compactness property.


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Additional Information

Tran Vu Khanh
Affiliation: School of Mathematics and Applied Statistics, University of Wollongong, New South Wales 2522, Australia
Address at time of publication: School of Engineering, Tan Tao University, Long An, Vietnam
MR Author ID: 815734
Email: tkhanh@uow.edu.au, khanh.tran@ttu.edu.vn

Andrew Raich
Affiliation: Department of Mathematical Sciences, SCEN 327, 1 University of Arkansas, Fayetteville, Arkansas 72701
MR Author ID: 634382
ORCID: 0000-0002-3331-9697
Email: araich@uark.edu

DOI: https://doi.org/10.1090/tran/8206
Received by editor(s): March 29, 2017
Received by editor(s) in revised form: October 27, 2017, August 3, 2018, December 4, 2018, and March 25, 2019
Published electronically: August 28, 2020
Additional Notes: The first author was supported by ARC grant DE160100173 and NAFOSTED grant 101.02-2019.319
The second author was partially supported by NSF grant DMS-1405100.
Article copyright: © Copyright 2020 American Mathematical Society