Displaying similar documents to “Contracting rigid germs in higher dimensions”

An extension theorem for separately holomorphic functions with analytic singularities

Marek Jarnicki, Peter Pflug (2003)

Annales Polonici Mathematici

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Let D j k j be a pseudoconvex domain and let A j D j be a locally pluriregular set, j = 1,...,N. Put X : = j = 1 N A × . . . × A j - 1 × D j × A j + 1 × . . . × A N k + . . . + k N . Let U be an open connected neighborhood of X and let M ⊊ U be an analytic subset. Then there exists an analytic subset M̂ of the “envelope of holomorphy” X̂ of X with M̂ ∩ X ⊂ M such that for every function f separately holomorphic on X∖M there exists an f̂ holomorphic on X̂∖M̂ with f ̂ | X M = f . The result generalizes special cases which were studied in [Ökt 1998], [Ökt 1999], [Sic 2001], and [Jar-Pfl 2001]. ...

A result on extension of C.R. functions

Makhlouf Derridj, John Erik Fornaess (1983)

Annales de l'institut Fourier

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Let Ω an open set in C 4 near z 0 Ω , λ a suitable holomorphic function near z 0 . If we know that we can solve the following problem (see [M. Derridj, Annali. Sci. Norm. Pisa, Série IV, vol. IX (1981)]) : u = λ f , ( f is a ( 0 , 1 ) form, closed in U ( z 0 ) in U ( z 0 ) with supp ( u ) Ω U ( z 0 ) , then we deduce an extension result for C . R . functions on Ω U ( z 0 ) , as holomorphic fonctions in Ω V ( z 0 ) .

The Łojasiewicz numbers and plane curve singularities

Evelia García Barroso, Tadeusz Krasiński, Arkadiusz Płoski (2005)

Annales Polonici Mathematici

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For every holomorphic function in two complex variables with an isolated critical point at the origin we consider the Łojasiewicz exponent ₀(f) defined to be the smallest θ > 0 such that | g r a d f ( z ) | c | z | θ near 0 ∈ ℂ² for some c > 0. We investigate the set of all numbers ₀(f) where f runs over all holomorphic functions with an isolated critical point at 0 ∈ ℂ².

A set on which the local Łojasiewicz exponent is attained

Jacek Chądzyński, Tadeusz Krasiński (1997)

Annales Polonici Mathematici

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Let U be a neighbourhood of 0 ∈ ℂⁿ. We show that for a holomorphic mapping F = ( f , . . . , f ) : U m , F(0) = 0, the Łojasiewicz exponent ₀(F) is attained on the set z ∈ U: f₁(z)·...·fₘ(z) = 0.

On the Rogosinski radius for holomorphic mappings and some of its applications

Lev Aizenberg, Mark Elin, David Shoikhet (2005)

Studia Mathematica

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The well known theorem of Rogosinski asserts that if the modulus of the sum of a power series is less than 1 in the open unit disk: | n = 0 a z | < 1 , |z| < 1, then all its partial sums are less than 1 in the disk of radius 1/2: | n = 0 k a z | < 1 , |z| < 1/2, and this radius is sharp. We present a generalization of this theorem to holomorphic mappings of the open unit ball into an arbitrary convex domain. Other multidimensional analogs of Rogosinski’s theorem as well as some applications to dynamical systems are...

On spaces of holomorphic functions in ℂⁿ

Diana D. Jiménez S., Lino F. Reséndis O., Luis M. Tovar S. (2014)

Banach Center Publications

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Following the line of Ouyang et al. (1998) to study the p spaces of holomorphic functions in the unit ball of ℂⁿ, we present in this paper several results and relations among p ( ) , the α-Bloch, the Dirichlet p and the little p , 0 spaces.

Proper holomorphic self-mappings of the minimal ball

Nabil Ourimi (2002)

Annales Polonici Mathematici

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The purpose of this paper is to prove that proper holomorphic self-mappings of the minimal ball are biholomorphic. The proof uses the scaling technique applied at a singular point and relies on the fact that a proper holomorphic mapping f: D → Ω with branch locus V f is factored by automorphisms if and only if f * ( π ( D f - 1 ( f ( V f ) ) , x ) ) is a normal subgroup of π ( Ω f ( V f ) , b ) for some b Ω f ( V f ) and x f - 1 ( b ) .

On L₁-subspaces of holomorphic functions

Anahit Harutyunyan, Wolfgang Lusky (2010)

Studia Mathematica

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We study the spaces H μ ( Ω ) = f : Ω h o l o m o r p h i c : 0 R 0 2 π | f ( r e i φ ) | d φ d μ ( r ) < where Ω is a disc with radius R and μ is a given probability measure on [0,R[. We show that, depending on μ, H μ ( Ω ) is either isomorphic to l₁ or to ( A ) ( 1 ) . Here Aₙ is the space of all polynomials of degree ≤ n endowed with the L₁-norm on the unit sphere.

On an integral-type operator from Privalov spaces to Bloch-type spaces

Xiangling Zhu (2011)

Annales Polonici Mathematici

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Let H(B) denote the space of all holomorphic functions on the unit ball B of ℂⁿ. Let φ be a holomorphic self-map of B and g ∈ H(B) such that g(0) = 0. We study the integral-type operator C φ g f ( z ) = 0 1 f ( φ ( t z ) ) g ( t z ) d t / t , f ∈ H(B). The boundedness and compactness of C φ g from Privalov spaces to Bloch-type spaces and little Bloch-type spaces are studied

Holomorphic series expansion of functions of Carleman type

Taib Belghiti (2004)

Annales Polonici Mathematici

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Let f be a holomorphic function of Carleman type in a bounded convex domain D of the plane. We show that f can be expanded in a series f = ∑ₙfₙ, where fₙ is a holomorphic function in Dₙ satisfying s u p z D | f ( z ) | C ϱ for some constants C > 0 and 0 < ϱ < 1, and where (Dₙ)ₙ is a suitably chosen sequence of decreasing neighborhoods of the closure of D. Conversely, if f admits such an expansion then f is of Carleman type. The decrease of the sequence Dₙ characterizes the smoothness of f. ...

Converging semigroups of holomorphic maps

Marco Abate (1988)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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In this paper we study the semigroups Φ : + H o l ( D , D ) of holomorphic maps of a strictly convex domain D 𝐂 n into itself. In particular, we characterize the semigroups converging, uniformly on compact subsets, to a holomorphic map h : D 𝐂 n .

The exceptional sets for functions of the Bergman space in the unit ball

Piotr Jakóbczak (1993)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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Let D be a domain in C 2 . Given w C , set D w = z C z , w D . If f is a holomorphic and square-integrable function in D , then the set E D , f of all w such that f ( , w ) is not square-integrable in D w has measure zero. We call this set the exceptional set for f . In this Note we prove that whenever 0 < r < 1 there exists a holomorphic square-integrable function f in the unit ball B in C 2 such that E B , f is the circle C 0 , r = z C z = r .

Converging semigroups of holomorphic maps

Marco Abate (1988)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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In this paper we study the semigroups Φ : + H o l ( D , D ) of holomorphic maps of a strictly convex domain D 𝐂 n into itself. In particular, we characterize the semigroups converging, uniformly on compact subsets, to a holomorphic map h : D 𝐂 n .

Function theory in sectors

Brian Jefferies (2004)

Studia Mathematica

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It is shown that there is a one-to-one correspondence between uniformly bounded holomorphic functions of n complex variables in sectors of ℂⁿ, and uniformly bounded functions of n+1 real variables in sectors of n + 1 that are monogenic functions in the sense of Clifford analysis. The result is applied to the construction of functional calculi for n commuting operators, including the example of differentiation operators on a Lipschitz surface in n + 1 .

Approximation of sets defined by polynomials with holomorphic coefficients

Marcin Bilski (2012)

Annales Polonici Mathematici

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Let X be an analytic set defined by polynomials whose coefficients a , . . . , a s are holomorphic functions. We formulate conditions on sequences a 1 , ν , . . . , a s , ν of holomorphic functions converging locally uniformly to a , . . . , a s , respectively, such that the sequence X ν of sets obtained by replacing a j ’s by a j , ν ’s in the polynomials converges to X.

On extensions of holomorphic functions satisfying a polynomial growth condition on algebraic varieties in 𝐂 n

Jean Erik Björk (1974)

Annales de l'institut Fourier

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Let V be an algebraic variety in C n and when k 0 is an integer then Pol ( V , k ) denotes all holomorphic functions f ( z ) on V satisfying | f ( z ) | C f ( 1 + | z | ) k for all z V and some constant C f . We estimate the least integer ϵ ( V , k ) such that every f Pol ( V , k ) admits an extension from V into C n by a polynomial P ( z 1 , ... , z n ) , of degree k + ϵ ( V , k ) at most. In particular lim k &gt; ϵ ( V , k ) is related to cohomology groups with coefficients in coherent analytic sheaves on V . The existence of the finite integer ϵ ( V , k ) is for example an easy consequence of Kodaira’s Vanishing Theorem. ...