arXiv 0905.15022009-05math.CV

CR functions on Subanalytic Hypersurfaces

Debraj Chakrabarti, Rasul Shafikov

备注：: Minor typos fixed. Final version. To appear in the Indiana University Mathematics Journal

英文摘要：

A general class of singular real hypersurfaces, called subanalytic, is defined. For a subanalytic hypersurface M in C^n, Cauchy-Riemann (or simply CR) functions on M are defined, and certain properties of CR functions discussed. In particular, sufficient geometric conditions are given for a point p on a subanalytic hypersurface M to admit a germ at p of a smooth CR function f that cannot be holomorphically extended to either side of M. As a consequence it is shown that a well-known condition of the absence of complex hypersurfaces contained in a smooth real hypersurface M, which guarantees one-sided holomorphic extension of CR functions on M, is neither a necessary nor a sufficient condition for one-sided holomorphic extension in the singular case.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

复流形上的弧分析子分析函数

Arc-analytic subanalytic functions on complex manifolds

研究显示，复流形上在一点处解析的弧分析子分析函数是整个流形上的解析函数。更一般地，实解析CR流形上在非空开集上CR的弧分析子分析函数是整个流形上的CR函数。

2026-03-26math.CV

全局子解析弧对称集

Globally subanalytic arc-symmetric sets

研究证明了全局子解析弧对称集可由连续全局子解析函数表示，该函数在简单正常交集外具有弧解析性质。

2026-02-17math.CV

Control of radii of convergence and extension of subanalytic functions

Let g denote a real analytic function on an open subset U of Euclidean space, and let S denote the boundary points of U where g does not admit a local analytic extension. We show that if g is semialgebraic (respectively, globally subanalytic), then S is semialgebraic (respectively, subanalytic) and g extends to a neighbourhood of cl(U)§as an analytic function that is semialgebraic (respectively, globally subanalytic). (In the general subanalytic case, S is not necessarily subanalytic.) Our proof depends on controlling the radii of convergence of power series G centred at points in the image of…

2001-11-23math.ACmath.AGmath.CV

Tempered subanalytic topology on algebraic varieties

On a smooth algebraic variety over $\mathbb{C}$, we build the tempered subanalytic and Stein tempered subanalytic sites. We construct the sheaf of holomorphic functions tempered at infinity over these sites and study their relations with the sheaf of regular functions, proving in particular that these sheaves are isomorphic on Zariski open subsets. We show that these data allow to define the functors of tempered and Stein tempered analytifications. We study the relations between these two functors and the usual analytification functor. We also obtain algebraization results in the non-proper ca…

2017-03-02math.AGmath.CV

Levi-flat hypersurfaces with real analytic boundary

Let $X$ be a Stein manifold of dimension at least 3. Given a compact codimension 2 real analytic submanifold $M$ of $X$, that is the boundary of a compact Levi-flat hypersurface $H$, we study the regularity of $H$. Suppose that the CR singularities of $M$ are an $\mathcal{O}(X)$-convex set. For example, suppose $M$ has only finitely many CR singularities, which is a generic condition. Then $H$ must in fact be a real analytic submanifold. If $M$ is real algebraic, it follows that $H$ is real algebraic and in fact extends past $M$, even near CR singularities. To prove these results we provide tw…

2008-07-24math.APmath.CV

可构造函数的导数也是可构造的

The Derivative of a Constructible Function is Constructible

本文证明在 tame 实几何中，可构造函数在求导时仍保持可构造性，通过展示导数操作对可构造函数类的稳定性。

2026-04-26math.AGmath.CVmath.LO

更多相关论文收起更多论文

An Edge-of-the Wedge Theorem for Hypersurface CR Functions

The Lewy extension theorem asserts the holomorphic extendability of CR functions defined in a neighborhood of a point on a hypersurface in C^{n+1}. The edge-of-the-wedge theorem asserts the extendability of holomorphic functions defined in wedges in C^{n+1} with edge a maximally real submanifold. In this article we prove under suitable hypotheses the holomorphic extendability to an open set in C^{n+1} of CR functions defined in the intersection of a hypersurface with a wedge whose edge is contained in the hypersurface. Unlike the situation for the classical edge-of-the-wedge theorem, for this …

1999-12-16math.CV

Interpolation and Sampling Hypersurfaces for the Bargmann-Fock space in higher dimensions

We study those smooth complex hypersurfaces W in C^n having the property that all holomorphic functions of finite weighted L^p norm on W extend to entire functions with finite weighted L^p norm. Such hypersurfaces are called interpolation hypersurfaces. We also examine the dual problem of finding all sampling hypersurfaces, i.e., smooth hypersurfaces W in C^n such that any entire function with finite weighted L^p norm is stably determined by its restriction to W. We provide sufficient geometric conditions on the hypersurface to be an interpolation and sampling hypersurface. The geometric condi…

2004-11-13math.CV

Piecewise Weierstrass preparation and division for o-minimal holomorphic functions

Given an o-minimal structure expanding the field of reals, we show a piecewise Weierstrass preparation theorem and a piecewise Weierstrass division theorem for definable holomorphic functions. In the semialgebraic setting and for the structure of globally subanalytic sets and functions we obtain the corresponding results for definable real analytic functions. As an application we show a definable global Nullstellensatz for principal ideals.

2016-10-12math.CVmath.LO

具有超曲面极点的多重复格林函数在特定B-正则域上的连续性

Continuity of pluricomplex Green functions with hypersurface poles on certain B-regular domains

Hanoi National University of Education（河内国家师范大学）；Hanoi-Amsterdam School for the Gifted（河内-阿姆斯特丹天才学校）

本文证明了一类有界(B)-正则域（含所有(B)-正则可凸化域）上具有超曲面极点的多重复格林函数的连续性。

2026-10-02math.CV

Stein compacts in Levi-flat hypersurfaces

We explore connections between geometric properties of the Levi foliation of a Levi-flat hypersurface M and holomorphic convexity of compact sets in M, or bounded in part by M. Applications include extendability of Cauchy-Riemann functions, solvability of the dibar_b-equation, approximability of holomorphic and CR functions, and global regularity of the dibar-Neumann operator.

2006-06-14math.CV

Quasi-normal families of meromorphic mappings sharing hypersurfaces

In this article we prove a sufficient condition of quasi-normality in higher dimension for a family of meromorphic mappings in which each pair of functions of family shares some moving hypersurfaces. We also prove a normality criterion concerning shared moving hypersurfaces.

2019-08-31math.CV

A contractible Levi-flat hypersurface in C^2 which is a determining set for pluriharmonic functions

We construct a real analytic Levi-flat hypersurface M in a neighborhood of an ellipsoid B in C^2 such that the each leaf of the Levi foliation of M is a complex disc, M intersects the boundary of B transversely, and the intersection A of M with B has the following properties: (i) the closure of A is diffeomorphic to the ball in R^3 and has a basis of Stein neighborhoods in C^2, (ii) any real analytic function on A which is constant on each Levi leaf is constant, (iii) any pluriharmonic function in a connected open neighborhood of A and vanishing on A is identically zero.

2005-02-23math.CV

用布朗运动方法研究亚纯映射的第二主定理及具有截断计数函数的超曲面

A Brownian-Motion Approach to the Second Main Theorem for Meromorphic Mappings and Hypersurfaces with Truncated Counting Functions

本文通过布朗运动和随机微积分方法，建立了亚纯映射进入复射影子流形V的第二主定理，并引入了截断计数函数以研究超曲面的分布常数。同时，给出了亚纯映射在复平面到V上共享任意超曲面的唯一性定理。

2026-05-20math.CV

Extension of holomorphic functions defined on singular complex hypersurfaces with growth estimates in strictly pseudoconvex domains of $C^n$

Let $D$ be a strictly pseudoconvex domain and $X$ be a singular analytic set of pure dimension $n-1$ in $C^n$ such that $X\cap D\neq \emptyset$ and $X\cap bD$ is transverse. We give sufficient conditions for a function holomorphic on $D\cap X$ to admit a holomorphic extension which belongs to $L^q(D),$ $q\in [1,+\infty[$, or to $BMO(D)$. The extension is given by mean of integral representation formulas and residue currents.

2018-02-13math.CV

On Normal Forms for Levi-flat hypersurfaces with an Isolated Line Singularity

We prove the existence of normal forms for some local real-analytic Levi-flat hypersurfaces with an isolated line singularity. We also give sufficient conditions for that a Levi-flat hypersurface with a complex line as singularity to be a pullback of a real-analytic curve in C via a holomorphic function.

2014-01-27math.CV

沿某类 Levi-平坦超曲面的转移函数线性化

Linearization of transition functions along a certain class of Levi-flat hypersurfaces

本文研究通过悬挂构造的 Levi-平坦超曲面上转移函数的规范化问题，借助圆周动力学与线性化定理给出获得规范形的充分条件，形成 Arnol'd 定理的几何类比。

2024-05-13math.CVmath.DS

Weighted interpolation from certain singular affine hypersurfaces

We prove that square integrable holomorphic functions (with respect to a plurisubharmonic weight) can be extended in a square integrable manner from certain singular hypersurfaces (which include uniformly flat, normal crossing divisors) to entire functions in affine space. This provides evidence for a conjecture regarding the positivity of the curvature of the weight under consideration.

2014-08-26math.AGmath.CVmath.DG