본문 바로가기
KERD
논문KCI 등재

A note on zeros of bounded holomorphic functions in weakly pseudoconvex domains in C2

Ly Kim Ha
대한수학회 (The Korean Mathematical Society)|2017.05|대한수학회보 (Bulletin of the KMS)|제54권 제3호|pp. 993-1002

1

조회

0

다운로드

0

인용

0

참고문헌

키워드

pseudoconvex domainsPoincar\'e-Lelong equationzero setfinite area$\bar{\partial}_b$-operatorHenkin solution
주제분류수학

초록

Let $\Omega$ be a bounded, uniformly totally pseudoconvex domain in $\mathbb{C}^2$ with the smooth boundary $b\Omega$. Assuming that $\Omega$ satisfies the negative $\bar\partial$ property. Let $M$ be a positive, finite area divisor of $\Omega$. In this paper, we will prove that: if $\Omega$ admits a maximal type $F$ and the $\rm \check{C}$eck cohomology class of the second order vanishes in $\Omega$, there is a bounded holomorphic function in $\Omega$ such that its zero set is $M$. The proof is based on the method given by Shaw \cite{Sha89}.

Abstract

Let $\Omega$ be a bounded, uniformly totally pseudoconvex domain in $\mathbb{C}^2$ with the smooth boundary $b\Omega$. Assuming that $\Omega$ satisfies the negative $\bar\partial$ property. Let $M$ be a positive, finite area divisor of $\Omega$. In this paper, we will prove that: if $\Omega$ admits a maximal type $F$ and the $\rm \check{C}$eck cohomology class of the second order vanishes in $\Omega$, there is a bounded holomorphic function in $\Omega$ such that its zero set is $M$. The proof is based on the method given by Shaw \cite{Sha89}.

참고문헌 (31)

  1. B. Berndtsson, "Integral formulas for the ∂ ¯∂-equation and zeros of bounded holomorphic functions in the unit ball", Math. Ann, 249(2), pp.163~176 (1980)
  2. A. Bonami, "Solutions de l’equation ¯∂ et z´eros de la class de Nevanlinna dans certains domaines faiblement pseudoconvexes", Ann. Inst. Fourier (Grenoble), 32, pp.53~89 (1982)
  3. J. Bruna, "H¨older and Lp-estimates for the ¯∂-equation in some convex domains with real-analytic boundary", Math. Ann, 296(4), pp.527~539 (1984)
  4. J. Bruna, "Zero varieties for Nevanlinna class in convex domains of finite type in Cn", Ann. of Math. (2), 147(2), pp.391~415 (1998)
  5. H. Cartan, "Differential Forms" (1970), Houghton-Mifflin
  6. IP, Studies in Advanced Mathematics, "Partial Differential Equations in Several Complex Variables, AMS", pp.2001, S. C. Chen
  7. P. L. Duren, "Theory of Hp spaces" (1970), Academic Press
  8. L. Gruman, "The zeros of holomorphic functions in strictly pseudoconvex domains", Trans. Amer. Math. Soc, 207, pp.163~174 (1975)
  9. L. K. Ha, "Tangential Cauchy-Riemann equations on pseudoconvex boundaries of finite and infinite type in C2"
  10. L. K. Ha, "Zero varieties for the Nevanlinna class in weakly pseudoconvex domains of finite", infinite type in C2, ~
  11. L. K. Ha, "Boundary regularity of the solution to the complex Monge-Amp`ere equation on pseudoconvex domains of infinite type", Math. Res. Lett, 22(2), pp.467~484 (2015)
  12. L. K. Ha, "Lp-estimates for the ¯∂-equation on a class of infinite type domains", Internat. J. Math, 25(11), pp.1450106~ (2014)
  13. G. M. Henkin, "Integral representations of functions holomorphic in strictly-pseudoconvex domains and some applications", Math. USSR Sbornik, 7(4), pp.597~616 (1969)
  14. G. M. Henkin, "The Lewy equation and analysis on pseudoconvex manifold", Russian Math. Surveys, 32(3), pp.59~130 (1977)
  15. G. M. Henkin, "H. Lewy’s equation and analysis on a pseudoconvex manifold II", Math. USSR Sbornik, 1, pp.63~94 (1977)
  16. T. V. Khanh, "Supnorm and f-H¨older estimates for ¯∂ on convex domains of general type in C2", J. Math. Anal. Appl, 460(2), pp.522~531 (2013)
  17. J. J. Kohn, "Boundary behaviour of ¯∂ on weakly pseudoconvex manifolds of dimension two", J. Differentail Geom, 6, pp.523~542 (1972)
  18. P. Lelong, "Int´egration sur un ensemble analytique complexe", Bull. Soc. Math. France, 85, pp.239~262 (1957)
  19. P. Lelong, "Fonctions plurisousharmoniques et formes diff´erentielles positives" (1968), Gordon and Breach
  20. J. Noguchi, "Nevanlinna Theory in Several Complex Variables and Diophantine Approximation" (2014), Springer
  21. R. M. Range, "The Carath´eodory metric and holomorphic maps on a class of weakly pseudoconvex domains", Pacific J. Math, 78(1), pp.173~189 (1978)
  22. R. M. Range, "Several complex variables", pp.247~267 (1978), Scuola Norm. Sup. Pisa
  23. R. M. Range, "Holomorphic Functions and Integral Representations in Several Complex Variables" (1986), Springer-Vedag
  24. A. V. Romanov, "A formula and estimates for solutions of the tangential Cauchy-Riemann equation", Math. Sb, 99, pp.58~83 (1976)
  25. W. Rudin, "Function theory in the unit ball of Cn" (1980), Springer-Verlag
  26. M. C. Shaw, "H¨older and Lp estimates for ¯∂ b on weakly pseudoconvex boundaries in C2", Math. Ann, 279(4), pp.635~652 (1988)
  27. M. C. Shaw, "Prescribing zeros of functions in the Nevanlinna class on weakly pseudo-convex domains in C2", Trans. Amer. Math. Soc, 313(1), pp.407~418 (1989)
  28. H. Skoda, "Valeurs au bord pour les solutions de l’op´erateur d”, et charact´erisation des z´eros des fonctions de la classe de Nevanlinna", Bull. Soc. Math. France, 104(3), pp.225~299 (1976)
  29. J. Verdera, "L∞-continuity of Henkin operators solving ¯∂ in certain weakly pseudoconvex domains of C2", Proc. Roy. Soc. Edinburgh Sect. A, 99(1-2), pp.25~33 (1984)
  30. A. Weil, "Sur les th´eor`ems de de Rham", Comment. Math. Helv, 26, pp.119~145 (1952)
  31. G. Zampieri, "Complex Analysis and CR Geometry" (2008), Amer-ican Mathematical Society

관련 논문AI추천

On H\"older estimates for Cauchy transforms on convex domains in $\mathbb{C}^2$Ly Kim Ha대한수학회 · 대한수학회보2019 · 제56권 제4호Solution to $\overline{\partial}$-problem with support conditions in weakly $q$-convex domainsSayed Saber대한수학회 · 대한수학회논문집2018 · 제33권 제2호Boundary Behavior of Holomorphic Discs in Convex Finite Type Domains이강혁영남수학회 · East Asian Mathematical Journal2015 · 제31권 제3호A bound on H\"older regularity for $\overline \partial$-equation on pseudoconvex domains in $\mathbb C^n$ with some comparable eigenvalues of the Levi-form조상현대한수학회 · 대한수학회보2021 · 제58권 제3호A note on the boundary behaviour of the squeezing function and Fridman invariant김혜선, Anh Duc Mai, Thi Lan Huong Nguyen, Van Thu Ninh대한수학회 · 대한수학회보2020 · 제57권 제5호Lp-BOUNDEDNESS OF THE CAUCHY-LERAY-FANTAPPIÈ INTEGRAL ON CERTAIN CLASS OF WEAKLY PSEUDOCONVEX DOMAINS IN C2Ly Kim Ha, Pham Huu Trung대한수학회 · 대한수학회보2024 · 제61권 제6호

이 간행물의 다른 논문 (23건)

Multiplicity of solutions for a class of non-local elliptic operators systemsChuanzhi Baipp. 715-729A construction of two-weight codes and its applicationsTatsuya Maruta, 김선정, 이남영, Yuuki Kageyama, 전은주pp. 731-736Existence theorem for non-Abelian vortices in the Aharony-Bergman-Jafferis-Maldacena theoryMeili Zhu, Ruifeng Zhangpp. 737-746Multiplication operators on weighted Banach spaces of a treeIsaac M. Craig, Robert F. Allenpp. 747-761On the minimum weight of a 3-connected 1-planar graphZai Ping Lu, Ning Songpp. 763-787
원문 보기