Volume 55, Issue 1
A Kind of Integral Representation on Complex Manifold

J. Math. Study, 55 (2022), pp. 95-108.

Published online: 2022-01

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• Abstract

In this paper, by using the Hermitian metric and Chern connection, we study the case of a strictly pseudoconvex domain $G$ with non-smooth boundaries in a complex manifold. By constructing a new integral kernel, we obtain a new Koppelman-Leray-Norguet formula of type $(p,q)$ on $G$, and get the continuous solutions of $\bar{\partial}$-equations on $G$ under a suitable condition. The new formula doesn't involve integrals on the boundary, thus one can avoid complex estimations of the boundary integrals, and the density of integral may be not defined on the boundary but only in the domain. As some applications, we discuss the Koppelman-Leray-Norguet formula of type $(p,q)$ for general strictly pseudoconvex polyhedrons (unnecessarily non-degenerate) on Stein manifolds, also get the continuous solutions of $\bar{\partial}$-equations under a suitable condition.

• Keywords

Complex manifold, strictly pseudoconvex domain, non-smooth boundary, Koppelman-Leray-Norguet formula, $\bar{\partial}$-equation.

32A26, 32Q99, 32T15

94453595@qq.com (Teqing Chen)

wei2785801@qztc.edu.cn (Zhiwei Li)

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@Article{JMS-55-95, author = {Teqing and Chen and 94453595@qq.com and 22012 and School of Information Management, Minnan University of Science and Technology, Shishi 362700, China and Teqing Chen and Zhiwei and Li and wei2785801@qztc.edu.cn and 22013 and School of Mathematics and Computer Science, Quanzhou Normal University, Quanzhou 362000, China and Zhiwei Li}, title = {A Kind of Integral Representation on Complex Manifold}, journal = {Journal of Mathematical Study}, year = {2022}, volume = {55}, number = {1}, pages = {95--108}, abstract = {

In this paper, by using the Hermitian metric and Chern connection, we study the case of a strictly pseudoconvex domain $G$ with non-smooth boundaries in a complex manifold. By constructing a new integral kernel, we obtain a new Koppelman-Leray-Norguet formula of type $(p,q)$ on $G$, and get the continuous solutions of $\bar{\partial}$-equations on $G$ under a suitable condition. The new formula doesn't involve integrals on the boundary, thus one can avoid complex estimations of the boundary integrals, and the density of integral may be not defined on the boundary but only in the domain. As some applications, we discuss the Koppelman-Leray-Norguet formula of type $(p,q)$ for general strictly pseudoconvex polyhedrons (unnecessarily non-degenerate) on Stein manifolds, also get the continuous solutions of $\bar{\partial}$-equations under a suitable condition.

}, issn = {2617-8702}, doi = {https://doi.org/10.4208/jms.v55n1.22.08}, url = {http://global-sci.org/intro/article_detail/jms/20197.html} }
TY - JOUR T1 - A Kind of Integral Representation on Complex Manifold AU - Chen , Teqing AU - Li , Zhiwei JO - Journal of Mathematical Study VL - 1 SP - 95 EP - 108 PY - 2022 DA - 2022/01 SN - 55 DO - http://doi.org/10.4208/jms.v55n1.22.08 UR - https://global-sci.org/intro/article_detail/jms/20197.html KW - Complex manifold, strictly pseudoconvex domain, non-smooth boundary, Koppelman-Leray-Norguet formula, $\bar{\partial}$-equation. AB -

In this paper, by using the Hermitian metric and Chern connection, we study the case of a strictly pseudoconvex domain $G$ with non-smooth boundaries in a complex manifold. By constructing a new integral kernel, we obtain a new Koppelman-Leray-Norguet formula of type $(p,q)$ on $G$, and get the continuous solutions of $\bar{\partial}$-equations on $G$ under a suitable condition. The new formula doesn't involve integrals on the boundary, thus one can avoid complex estimations of the boundary integrals, and the density of integral may be not defined on the boundary but only in the domain. As some applications, we discuss the Koppelman-Leray-Norguet formula of type $(p,q)$ for general strictly pseudoconvex polyhedrons (unnecessarily non-degenerate) on Stein manifolds, also get the continuous solutions of $\bar{\partial}$-equations under a suitable condition.

Teqing Chen & Zhiwei Li. (2022). A Kind of Integral Representation on Complex Manifold. Journal of Mathematical Study. 55 (1). 95-108. doi:10.4208/jms.v55n1.22.08
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