Removable singularities of CR-functions

Jöricke, Burglind

doi:10.1007/bf02386112

Cited by 22 publications

(22 citation statements)

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“…It is familiar that holomorphic fnnctions defined on the complement extend through totally real submanifolds. In 1988, B. JSricke [13] discovered the remarkable phenomenon that any compact subset of a totally real disc embedded in a strictly pseudoconvex boundary is removable for CR functions. Our main result shows that this holds true without any assumption on pseudoconvexity.…”

Section: For Cr Functions) If Any Continuous Ctl Function F On Od\k Amentioning

confidence: 99%

“…In order to construct an analytic extension of f to a one-sided neighborhood IV attached to a neighborhood of K in OD, we shall employ Bishop discs. In [13], a convenient family is constructed explicitly. With regard to the non-pseudoconvex case, we shall instead apply the powerful existence theorem of E. Bedford and W. Klingenberg [3] that every generic two-sphere contained in a strictly pseudoconvex boundary can be filled by a Leviflat three-ball.…”

Section: The Strictly Pseudoconvex Casementioning

confidence: 99%

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Totally real discs in non-pseudoconvex boundaries

Porten

2003

Ark. Mat.

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Abstract. Let D be a relatively compact domain in C 2 with smooth connected boundary OD. A compact set KCOD is called removable if any continuous CR function defined on OD\K admits a holomorphic extension to D. If D is strictly pseudoconvex, a theorem of B. J6ricke states that any compact K contained in a smooth totally real disc SCOD is removable. In the present article we ,show that this theorem is true without any assumption on pseudoconvexity.

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Section: For Cr Functions) If Any Continuous Ctl Function F On Od\k Amentioning

confidence: 99%

Section: The Strictly Pseudoconvex Casementioning

confidence: 99%

Totally real discs in non-pseudoconvex boundaries

Porten

2003

Ark. Mat.

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“…The point O is not removable for the integrable CR functions on S, i.e., for a CR function f on S \{O}, the inclusion f ∈ L 1 (S) does not guarantee in general that f is a CR function on all of S. In the case of smooth hypersurfaces S, removable singularities of integrable CR functions are studied in [Jör88], [Kyt89], [Sto93], [KR95]. The point O is not removable for the integrable CR functions on S, i.e., for a CR function f on S \{O}, the inclusion f ∈ L 1 (S) does not guarantee in general that f is a CR function on all of S. In the case of smooth hypersurfaces S, removable singularities of integrable CR functions are studied in [Jör88], [Kyt89], [Sto93], [KR95].…”

Section: Example Of a Non-representable Cr Functionmentioning

confidence: 99%

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Kytmanov¹,

Myslivets²,

Тарханов

2002

Math. Nachr.

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“…

REMOVABLE SINGULARITIES FOR INTEGRABLE CR FUNCTIONSbecome classical and easy in case n = 0, i.e.

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confidence: 99%

On removable singularities for integrable cr functions

Merker¹,

Porten²

1999

Indiana Univ. Math. J.

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The subject of removable singularities for the boundary values of holomorphic functions of several variables has been intensively studied in recent years, see works: [1], [14], [17], [18], [19], [21], [23], [24], [25], [26], [28], [32], [34], and the excellent surveys [11], [40].The present paper is devoted to a further step into this direction. Its main intention is to illustrate the way how, on a CR-manifold M of arbitrary codimension, removable singularity theorems may be understood in terms of the analysis of CR-orbits of M.Let M be a locally embeddable CR manifold,We give various conditions in order that Φ is L 1 -removable, i.e. REMOVABLE SINGULARITIES FOR INTEGRABLE CR FUNCTIONSbecome classical and easy in case n = 0, i.e. M is an open set Ω in C m , m ≥ 2. Finally, there is obstruction to removability in case N is not generic at any point, similar to N being a complex hypersurface in Ω if n = 0.Remark. After the reduction to M being globally minimal, the assumption that M\Φ is globally minimal too is essential and cannot be dropped (see below).Remark. Theorem 3 is treated in [29] and the following will appear in [33]: (iv) If m ≥ 2, codim M N = 1 and N is generic, then every closed set K ⊂ N which does not contain any CR orbit of N is L 1 -removable.Finally, using the theory of CR orbits, we extend also [24].Theorem 5. Let M be C 2,α globally minimal. Then every C λ peak set S satisfies H d (S) = 0 and is L 1 -removable.Now, we explain the terminology, compare our results to the codimension one case and give some motivations.The general feature of our work is that L p -removablity is linked with W-removability, i.e. with envelope of holomorphy results.a. About the methods. Of course, different approaches for proving (L p , ∂ b )-removability are conceivable, and the distinguished role of holomorphic hulls in our context is not clear yet.For instance, one could consider the problem from the viewpoint of the general theory of removable singularities for solutions of linear partial differential operators. Let Ω ⊂ R n be a domain, let K ⊂⊂ Ω be a compact and let P = |β|≤e a β (x)∂ β x be such an operator. K is called (L p , P )-removable if any u ∈ L p (Ω) with P u ≡ 0 on Ω\K satisfies P u ≡ 0 on Ω (in the distributional sense). Of course, the notion makes sense by replacing L p (Ω) with other differentiability classes, e.g. C 0 (Ω), C k (Ω), or even D ′ (Ω).Harvey and Polking [14] have proved removable singularity theorems for general P , but they give results only in case p > 1. Indeed, their main theorem 4.1 [14], states that K is (L p , ∂ b )-removable if the Hausdorff measure H n−p ′ (K) < ∞, p ′ = p/(p − 1), e = deg P . The authors further point out that this result cannot be improved in terms of Hausdorff measures in the class of all first order differential operators. Especially information about L 1 -removability is never available on this level (more precisely, in our special setting results for p < e/(e − 1) cannot be derived from the above theorem).In fact, one of the main argument here (cf. [14],...

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Removable singularities of CR-functions

Cited by 22 publications

References 8 publications

Totally real discs in non-pseudoconvex boundaries

Totally real discs in non-pseudoconvex boundaries

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On removable singularities for integrable cr functions

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