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On Removable singularities for integrable CR functions
Ecole Normale Superiore Paris.
1999 (English)In: Indiana University Mathematics Journal, ISSN 0022-2518, Vol. 48, no 3, 805-856 p.Article in journal (Refereed) Published
Abstract [en]

Let $M$ be a locally embeddable CR manifold. One defines, for $p\geq 1$, $$L^p_{\rm loc,CR}(M)=\{f\in L^p_{\rm loc}(M)| f {\rm is CR on}\ M\},$$ $$L^p_{\rm loc,CR}(M-E)=\{f\in L^p_{\rm loc}(M)| f {\rm is CR on}\ M-E\},$$ with $E$ a closed subset of $M$. This paper gives conditions under which $L^p_{\rm loc,CR}(M)=L^p_{\rm loc,CR}(M-E)$. In order to state results, let us recall the following definitions. Definition 1: The CR orbit of $p\in M$ is defined as $O_{\rm CR}(M,p)=\{q\in M|$ there exists a piecewise smooth integral curve of $T^cM$ with origin $p$ and target $q\}$. By Sussmann's theorem, $O_{\rm CR}(M,p)$ has the structure of a smooth immersed submanifold of $M$ [see M. S. Baouendi, P. Ebenfelt and L. P. Rothschild, Real submanifolds in complex space and their mappings, Princeton Univ. Press, Princeton, NJ, 1999; MR1668103 (2000b:32066)]. Definition 2: $M$ is globally minimal if $M$ is a single orbit. Let $H^k$ denote $k$-dimensional Hausdorff measure. The main general assumption throughout the paper is that $M$ and $M-E$ are globally minimal ($M$ globally minimal does not imply $M-E$ globally minimal). Among the results of the paper, we state the following theorem: Let $M$ be $C^{2,\alpha}, 0<\alpha<1$, of real dimension $d$ and CR dimension $\geq 1$. Let $E\subset M$ be a closed subset with $H^{d-3}(E)<\infty$. Assume that $M$ and $M-E$ are globally minimal. Then $L^p_{\rm loc,CR}(M)=L^p_{\rm loc,CR}(M-E)$.

Place, publisher, year, edition, pages
1999. Vol. 48, no 3, 805-856 p.
Keyword [en]
removable singularities, submanifolds of CR manifolds, CR orbits
Identifiers
URN: urn:nbn:se:miun:diva-3618Local ID: 5254OAI: oai:DiVA.org:miun-3618DiVA: diva2:28650
Available from: 2008-09-30 Created: 2008-09-30 Last updated: 2009-03-19Bibliographically approved

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  • apa
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