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The Kohn-Laplace equation on abstract CR manifolds: Global regularity
Transactions of the American Mathematical Society ( IF 1.2 ) Pub Date : 2020-08-28 , DOI: 10.1090/tran/8206
Tran Khanh , Andrew Raich

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 a domain in $\mathbb C^N$ and implies that $M$ satisfies the weak compactness property.

中文翻译:

抽象 CR 流形上的 Kohn-Laplace 方程:全局规律

令 $M$ 是一个紧凑的、面向伪凸的、$(2n+1)$ 维的、超曲面类型的抽象 CR 流形,$n\geq 2$。我们证明如下: (i) 如果 $M$ 在 $(0,q_0)$-形式上承认严格的 CR-plurisubharmonic 函数,那么复格林算子 $G_q$ 存在并且在 $L^2_{0 上是连续的, q}(M)$ 度数 $q_0\le q\le n-q_0$。在 $q_0=1$ 的情况下,我们也建立了 $G_0$ 和 $G_n$ 的连续性。此外,$M$ 上的 $\bar\partial_b$ 方程可以在 $C^\infty(M)$ 中求解。(ii) 如果 $M$ 满足 $(0,q_0)$-forms 上的“弱紧致性性质”,则 $G_q$ 是 $H^s_{0,q}(M)$ 上的连续算子,因此度数 $q_0\le q\le n-q_0$ 的全球常规 $M$;以及在 $q_0=1$ 的情况下,最高度数 $q=0$ 和 $q=n$。我们还介绍了“多次谐波 CR 流形”的概念
更新日期:2020-08-28
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