相关论文: Separate real analiticity and CR extendibility

We study the problem of holomorphic extension of a smooth CR mapping from a real analytic hypersurface to a real algebraic set in complex spaces of different dimensions.

We study the holomorphic extendability of smooth CR maps between real analytic strictly pseudoconvex hypersurfaces in complex affine spaces of different dimensions.

In this article, we present that the germ of a complex analytic set at the origin in $\mathbb{C}^n$ is regular if and only if the related $L^2$ extension theorem holds. We also obtain a necessary condition of the $L^2$ extension of bounded…

In our recent work [25] we showed that $C^\infty$ CR-diffeomorphisms of real-analytic Levi-nonflat hypersurfaces in $\mathbb C^{2}$ are not analytic in general. This result raised again the question on the nature of CR-maps of real-analytic…

We say that a CR singular submanifold $M$ has a removable CR singularity if the CR structure at the CR points of $M$ extends through the singularity as an abstract CR structure on $M$. We study such real-analytic submanifolds, in which case…

Using complex methods combined with Baire's Theorem we show that one-sided extendability, extendability and real analyticity are rare phenomena on various spaces of functions in the topological sense. These considerations led us to…

We study the real analyticity of a CR mapping on the hypersurface $\h$ defined by $\h=\{z\in\C^n,\Re(z_n)+|z'|^2=0\}$ in model almost complex manifolds. We make use of a method of prolongation for the tangential Cauchy-Riemann equations and…

We consider various notions of holomorphic extendability of complex valued functions defined on subsets of $\mathbf C^n$, including one-sided extendability. We show that in the relevant function spaces, these phenomena of holomorphic…

If $R$ is a real analytic set in $\C^n$ (viewed as $\R^{2n}$), then for any point $p\in R$ there is a uniquely defined germ $X_p$ of the smallest complex analytic variety which contains $R_p$, the germ of $R$ at $p$. It is shown that if $R$…

We discuss a general result of holomorphic extension of a real analytic function $f$ defined on the boundary $\partial D$ of a real analytic strictly convex subset $D\subset\subset \C^n$. We show that this follows from the hypothesis of…

We consider a smooth CR mapping $f$ from a real-analytic generic submanifold $M$ in $\bC^N$ into $\bC^N$. For $M$ of finite type and essentially finite at a point $p\in M$, and $f$ formally finite at $p$, we give a necessary and sufficient…

Let $f$ be a holomorphic function on the strip $\{z\in C: -\alpha<Im z<\alpha\}, \alpha > 0$, belonging to the class $H(\alpha,-\alpha;\epsilon)$ defined below. It is shown that there exist holomorphic functions $w_1$ on $\{z\in C: 0<Im z…

We study propagation of CR extendibility at the vertex $p$ of an analytic sector $A$ contained in a CR manifold $M$. Let $k$ be the weighted vanishing order of $M$ and $\alpha $ the complex angle of $A$ at $p$. Propagation takes place if…

Let M be a connected real-analytic hypersurface in two dimensional complex space, $\mathbb C^2$, containing a connected complex hypersurface E, and let f be a smooth CR mapping sending M into another real-analytic hypersurface M' in…

Using recent development in Poletsky theory of discs, we prove the following result: Let $X,$ $Y$ be two complex manifolds, let $Z$ be a complex analytic space which possesses the Hartogs extension property, let $A$ (resp. $B$) be a non…

Local conditions on boundaries of $C^\infty$ Levi-flat hypersurfaces, in case the boundary is a generic submanifold, are studied. For nontrivial real analytic boundaries we get an extension and uniqueness result, which forces the…

We prove that generic homologically nontrivial $(2n-1)$-parameter family of analytic discs attached by their boundaries to a CR manifold $\Omega$ in $\mathbb C^n, n \le 2$ tests CR functions: if a smooth function on $\Omega$ extends…

Let B be the open unit ball in C^2 and let a, b, c be three points in C^2 which do not lie in a complex line, such that the complex line through a and b meets B and such that <a|b> is different from 1 if one of the points a, b is in B and…

It is known that if f is a continuous function on the complex plane which extends holomorphically from each circle surrounding the origin then f is not necessarily holomorphic. In the paper we prove that if, in addition, f extends…

Let D be a bounded, finitely connected domain in the complex plane without isolated points in the boundary and let f be a continuous function on the boundary bD. Let F be a continuous extension of f to the closure of D. We prove that f…