Support functions for convex domains of finite type

Diederich, Klas; Fornæss, John Erik

doi:10.1007/pl00004683

Cited by 53 publications

(72 citation statements)

References 13 publications

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“…For them a good C ∞ -family S(z, ζ) of support functions on D, holomorphic in z ∈ D and C ∞ in ζ chosen in a suitable neighborhood of ∂D together with the corresponding Leray section s(z, ζ) is needed. We obtain it from the family S(z, ζ) constructed in [5]. The necessary estimates for S and s are contained in [4].…”

Section: K Diederich and E Mazzillimentioning

confidence: 99%

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Zero varieties for the Nevanlinna class on all convex domains of finite type

Diederich¹,

Mazzilli²

2001

Nagoya Mathematical Journal

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Abstract. It is shown, that the so-called Blaschke condition characterizes in any bounded smooth convex domain of finite type exactly the divisors which are zero sets of functions of the Nevanlinna class on the domain. The main tool is a non-isotropic L 1 estimate for solutions of the Cauchy-Riemann equations on such domains, which are obtained by estimating suitable kernels of BerndtssonAndersson type. §1. Introduction and resultsIn their article [2] J. Bruna, Ph. Charpentier and Y. Dupain did a large step towards a complete characterization of the zero sets of the Nevanlinna class on all convex domains of finite type in C n . The purpose of this article is to complete their result.For the convenience of the reader we recall at first briefly the following definitions and facts:If D ⊂⊂ C n is a C ∞ -smoothly bounded domain and ρ a smooth defining function for D, then the Nevanlinna class N (D) isHere ∂D ε = {z : ρ(z) = −ε} for ε > 0 small enough and dσ ε (z) denotes the Euclidean surface measure on ∂D ε . Furthermore, if X ⊂ D is a complex analytic hypersurface with irreducible decomposition

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Section: K Diederich and E Mazzillimentioning

confidence: 99%

“…Let D ⊂⊂ C n be a smooth convex domain of finite type m and ρ a convex defining function of D in a neighborhood U of ∂D. Then the function S(z, ζ) ∈ C ∞ (D × U ), holomorphic in z, constructed in [5], has the following property :…”

Section: More Precisely Proposition 41 From [4] Saysmentioning

confidence: 99%

Zero varieties for the Nevanlinna class on all convex domains of finite type

Diederich¹,

Mazzilli²

2001

Nagoya Mathematical Journal

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“…The key point is the reproducing kernel with right estimate matching quasimetric on ∂D. For the case of convex domains of finite type we use the holomorphic support function with best possible non-isotropic estimates constructed by Diederich-Fornaess [4].…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Estimates on the mean growth of $H^p$ functions in convex domains of finite type

Cho

2003

Proc. Amer. Math. Soc.

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“…This is a consequence of a result of McNeal on the equivalence of linear type and D'Angelo type for convex domains. (See [DF2], for instance). Theorem 3.6 below implies that no open piece of the boundary of the classical Kohn-Nirenberg domain can be mapped by a non-constant holomorphic map into any connected compact smooth algebraic hypersurface in C n , that is locally holomorphically convexfiable.…”

Section: Hypersurfaces Not Embeddable Into Certain Real-algebraic Hypmentioning

confidence: 99%

Non-embeddable real algebraic hypersurfaces

Huang

Zaitsev

2013

Math. Z.

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We study various classes of real hypersurfaces that are not embeddable into more special hypersurfaces in higher dimension, such as spheres, real algebraic compact strongly pseudoconvex hypersurfaces or compact pseudoconvex hypersurfaces of finite type. We conclude by stating some open problems. IntroductionThis paper is motivated by the following general problem:Given a real hypersurface M in a complex manifold X, when can it be (holomorphically) embedded into a more special real hypersurface M ′ in a complex manifold X ′ of possibly larger dimension? More specifically, which strongly pseudoconvex hypersurfaces can be embedded into a sphere?By a holomorphic map (resp. embedding) of M into M ′ , we mean a holomorphic map (resp. embedding) of an open neighborhood of M in X into X ′ , sending M into M ′ . In particular, it follows that a hypersurface holomorphically embeddable into a sphere S 2N −1 := { j |z j | 2 = 1} ⊂ C N is necessarily strongly pseudoconvex and real-analytic. However, not every strongly pseudoconvex real-analytic hypersurface can be even locally embedded into a sphere, as was independently shown by Forstneric [For1] and Faran [Fa]. These results, showing that such hypersurfaces in general position are not embeddable into spheres, were more recently further extended and strengthened by Forstneric [For2] showing that they also do not admit transversal holomorphic embeddings into a hyperquadric(By a transversal embedding F we mean one not sending the tangent space) Explicit examples of non-embeddable strongly pseudoconvex real-analytic hypersurfaces were given by the second author [Z2] along with explicit invariants serving as obstructions to embeddability. In Theorem 2.1 below we give an example of a compact strongly pseudoconvex realanalytic hypersurface in C 2 that does not admit any holomorphic embedding into a sphere (and more generally any transversal holomorphic embedding into a hyperquadric).

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Support functions for convex domains of finite type

Cited by 53 publications

References 13 publications

Zero varieties for the Nevanlinna class on all convex domains of finite type

Zero varieties for the Nevanlinna class on all convex domains of finite type

Estimates on the mean growth of $H^p$ functions in convex domains of finite type

Non-embeddable real algebraic hypersurfaces

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