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Zbl 0842.32009
Kytmanov, A.M.; Rea, C.
Elimination of $L\sp 1$ singularities on Hölder peak sets for CR functions. (English)
[J] Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 22, No.2, 211-226 (1995). ISSN 0391-173X

The authors study $L^1_{\text {loc}}$ functions on a CR submanifold $M$ of the euclidean complex space which are CR functions in $M \setminus S$ for a relatively closed $S \subset M$. Under suitable conditions, those functions are restrictions of CR functions on $M$, i.e. $(*)\ L^1_{\text {loc}} (M) \cap CR(M \setminus S) \subset CR(M)$. This can be considered as an analogue of Riemann's theorem on elimination of singularities for holomorphic functions in $\bbfC$, $L^2_{\text {loc}} (\Omega) \cap {\cal O}(\Omega \setminus \{z_0\}) \subset {\cal O}(\Omega)$. A connected submanifold $N \subset M$ is characterized iff $\dim N < \dim M$ and $\dim_{\text{CR}} N = \dim_{\text{CR}} M$. A point $p$ which is not contained in any characteristic submanifold is called minimal. A subset $S \subset M$ is a $C^\lambda$ peak set, $0 < \lambda < 1$, iff there exists a non-constant function $h \in C^\lambda (M) \cap \text{CR}(M)$ with $S \equiv \{h = 1\}$ and $|h|< 1$ on $M \setminus S$. The authors prove: each minimal point $p$ of a CR manifold of class $C^{2,\alpha}$, $0 < \alpha < 1$, has a neighborhood $M$ such that $(*)$ holds for any $C^\lambda$ peak set $S$ in $M$. There is an analogue of $(*)$ for a general complex vector field. A stronger version is given for real analytic manifolds: if the connected real analytic CR manifold $M$ has at least one minimal point then $(*)$ holds for any $C^\lambda$ peak set $S$.
[A.Aeppli (Minneapolis)]

MSC 2000:: *32D20 Removable singularities (several complex variables)
32V05 CR structures etc.

Keywords: CR functions; elimination of singularities

Cited in: Zbl 0889.32016 Zbl 0859.32005

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