ON THE SOBOLEV DOLBEAULT COHOMOLOGY OF A DOMAIN WITH PSEUDOCONCAVE BOUNDARIES

PDF (PC)

ON THE SOBOLEV DOLBEAULT COHOMOLOGY OF A DOMAIN WITH PSEUDOCONCAVE BOUNDARIES

ON THE SOBOLEV DOLBEAULT COHOMOLOGY OF A DOMAIN WITH PSEUDOCONCAVE BOUNDARIES

ON THE SOBOLEV DOLBEAULT COHOMOLOGY OF A DOMAIN WITH PSEUDOCONCAVE BOUNDARIES

ON THE SOBOLEV DOLBEAULT COHOMOLOGY OF A DOMAIN WITH PSEUDOCONCAVE BOUNDARIES

摘要/Abstract

参考文献

Metrics

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 6

Metrics

本文评价

推荐阅读 10

doi:10.1007/s10473-024-0203-2

数学物理学报(英文版) ›› 2024, Vol. 44 ›› Issue (2): 431-444.doi: 10.1007/s10473-024-0203-2

Jian CHEN

School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China

收稿日期:2022-12-26 修回日期:2023-05-22 出版日期:2024-04-25 发布日期:2023-12-06
作者简介:Jian CHEN, E-mail: jian-chen@whu.edu.cn

Jian CHEN

School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China

Received:2022-12-26 Revised:2023-05-22 Online:2024-04-25 Published:2023-12-06
About author:Jian CHEN, E-mail: jian-chen@whu.edu.cn

摘要： In this note, we mainly make use of a method devised by Shaw [15] for studying Sobolev Dolbeault cohomologies of a pseudoconcave domain of the type $\Omega=\widetilde{\Omega} \backslash \overline{\bigcup_{j=1}^{m}\Omega_j}$ , where $\widetilde{\Omega}$ and $\{\Omega_j\}_{j=1}^m\Subset\widetilde{\Omega}$ are bounded pseudoconvex domains in $\mathbb{C}^n$ with smooth boundaries, and $\overline{\Omega}_1,\cdots,\overline{\Omega}_m$ are mutually disjoint. The main results can also be quickly obtained by virtue of [5].

关键词: Cauchy-Riemann equations, pseudoconcave domains, $\bar{\partial}$ -Neumann operator, Bergman spaces

Abstract: In this note, we mainly make use of a method devised by Shaw [15] for studying Sobolev Dolbeault cohomologies of a pseudoconcave domain of the type $\Omega=\widetilde{\Omega} \backslash \overline{\bigcup_{j=1}^{m}\Omega_j}$ , where $\widetilde{\Omega}$ and $\{\Omega_j\}_{j=1}^m\Subset\widetilde{\Omega}$ are bounded pseudoconvex domains in $\mathbb{C}^n$ with smooth boundaries, and $\overline{\Omega}_1,\cdots,\overline{\Omega}_m$ are mutually disjoint. The main results can also be quickly obtained by virtue of [5].

Key words: Cauchy-Riemann equations, pseudoconcave domains, $\bar{\partial}$ -Neumann operator, Bergman spaces

中图分类号:

32W05

Jian CHEN. ON THE SOBOLEV DOLBEAULT COHOMOLOGY OF A DOMAIN WITH PSEUDOCONCAVE BOUNDARIES[J]. 数学物理学报(英文版), 2024, 44(2): 431-444.

Jian CHEN. ON THE SOBOLEV DOLBEAULT COHOMOLOGY OF A DOMAIN WITH PSEUDOCONCAVE BOUNDARIES[J]. Acta mathematica scientia,Series B, 2024, 44(2): 431-444.

[1] Bell S R, Boas H P. Regularity of the Bergman projection and duality of holomorphic function spaces. Math Ann, 1984, 267: 473-478
[2] Boas H P. The Szegö projection: Sobolev estimates in regular domains. Trans Amer Math Soc, 1987, 300: 109-132
[3] Chakrabarti D, Laurent-Thiébaut C, Shaw M C. On the

$L^2$ -Dolbeault cohomology of annuli. Indiana Univ Math J, 2018, 67: 831-857
[4] Chakrabarti D, Shaw M C.

$L^2$ Serre duality on domains in complex manifolds and applications. Trans Amer Math Soc, 2012, 364: 3529-3554
[5] Chakrabarti D, Harrington P. Exact sequences and estimates for the

$\bar{\partial}$ -problem. Math Z, 2021, 299: 1837-1873
[6] Chen S C, Shaw M C.Partial differential equations in several complex variables. Providence, RI: Amer Math Soc, 2001
[7] Fu S Q, Shaw M C. Sobolev estimates and duality for

$\bar{\partial}$ on domains in

$\mathbb{C}P^n$ . Pure Appl Math Q, 2022, 18: 503-529
[8] Hörmander L.

$L^2$ estimates and existence theorems for the

$\bar{\partial}$ operator. Acta Math, 1965, 113: 89-152
[9] Huang X J, Li X S.

$\bar{\partial}$ -equation on a lunar domain with mixed boundary conditions. Trans Amer Math Soc, 2016, 368: 6915-6937
[10] Laurent-Thiébaut C, Leiterer J. On Serre duality. Bull Sci Math, 2000, 124: 93-106
[11] Laurent-Thiébaut C, Shaw M C. On the Hausdorff property of some Dolbeault cohomology groups. Math Z, 2013, 274: 1165-1176
[12] Li X S, Shaw M C. The

$\bar{\partial}$ -equation on an annulus with mixed boundary conditions. Bull Inst Math Acad Sin, 2013, 8: 399-411
[13] Lions J L, Magenes E.Non-Homogeneous Boundary Value Problems and Applications: Volume I. New York: Springer-Verlag, 1972
[14] Shaw M C. The closed range property for

$\bar{\partial}$ on domains with pseudoconcave boundary//Ebenfelt P, Hungerbühler N, Kohn J, et al. Complex Analysis. Basel: Birkhäuser, 2010: 307-320
[15] Shaw M C. Duality between harmonic and Bergman spaces//Barkatou Y, Berhanu S, Meziani A, et al. Geometric Analysis of Several Complex Variables and Related Topics. Providence, RI: Amer Math Soc, 2011: 161-171
[16] Shaw M C. Topology of Dolbeault cohomology groups//Feehan P, Song J, Weinkove B, Wentworth R. Analysis, Complex Geometry,Mathematical Physics: In Honor of Duong H. Phong. Providence, RI: Amer Math Soc, 2015: 211-225
[17] Straube E J.Lectures on the

$L^2$ -Sobolev theory of the

$\bar{\partial}$ -Neumann problem. ESI Lectures in Mathematics and Physics. Zürich: European Mathematical Society (EMS), 2010
[18] Shi Z M, Yao L D.Existence of solutions for

$\bar{\partial}$ -equation in Sobolev spaces of negative index. arXiv: 2111.09245

Viewed

Full text

Abstract

Cited

Shared

Discussed

Just accepted

Online first

Just accepted

Online first

[1]	彭茹, 范亚庆. TOEPLITZ OPERATORS FROM HARDY SPACES TO WEIGHTED BERGMAN SPACES IN THE UNIT BALL OF Cⁿ[J]. 数学物理学报(英文版), 2022, 42(1): 349-363.
[2]	邓方文, 欧阳才衡, 彭茹. SOME QUESTIONS REGARDING VERIFICATION OF CARLESON MEASURES[J]. 数学物理学报(英文版), 2021, 41(6): 2136-2148.
[3]	Ruhan ZHAO. ON F(p,q,s) SPACES[J]. 数学物理学报(英文版), 2021, 41(6): 1985-2020.
[4]	彭茹, 邢晓蕾, 江良英. POINTWISE MULTIPLICATION OPERATORS FROM HARDY SPACES TO WEIGHTED BERGMAN SPACES IN THE UNIT BALL OF Cⁿ[J]. 数学物理学报(英文版), 2019, 39(4): 1003-1016.
[5]	任广斌, Uwe KAHLER. PSEUDOHYPERBOLIC METRIC AND UNIFORMLY DISCRETE SEQUENCES IN THE REAL UNIT BALL[J]. 数学物理学报(英文版), 2014, 34(3): 629-638.
[6]	江良英, 欧阳才衡. COMPACT DIFFERENCES OF COMPOSITION OPERATORS ON HOLOMORPHIC FUNCTION SPACES IN THE UNIT BALL[J]. 数学物理学报(英文版), 2011, 31(5): 1679-1693.