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Li, Bao Qin

Discrete interpolating varieties in pseudoconvex open sets of \(\mathbb C^n\). (English) Zbl 1118.32013

J. Math. Soc. Japan 58, No. 4, 1185-1196 (2006).
Let \(\Omega\) be an open set in \({\mathbb C}^n\), and \(A(\Omega )\) the ring of holomorphic functions in \(\Omega\). The Hörmander algebra \(A_p(\Omega )\) associated with a plurisubharmonic function \(p:\Omega\to [0,\infty )\) (a Hörmander weight) is defined as \(A_p(\Omega)=\{ f\in A(\Omega ):\,\exists A,B>0:\;| f(z)| \leq Ae^{Bp(z)},\;z\in\Omega \}\), and the weight satisfies the following conditions: (i) all polynomials belong to \(A_p(\Omega )\); (ii) there exist constants \(K_1,\ldots, K_4\) such that \(z\in\Omega\) and \(| \zeta -z| \leq e^{-K_1p(z)-K_2}\) implies that \(\zeta\in\Omega\) and \(p(\zeta)\leq K_3p(z)+K_4\).
The author proves that a necessary and sufficient condition for a discrete variety \(V\) in a pseudoconvex open set \(\Omega\subset {\mathbb C}^n\) to be an interpolating variety for \(A_p(\Omega)\) is that there exists a holomorphic mapping \(f:\Omega\to {\mathbb C}^n\) whose zero set contains \(V\) and whose Jacobian at the points \(\zeta\in V\) is bounded from below by \(\epsilon e^{-Cp(\zeta)}\), for some constants \(\epsilon, C>0\).
Reviewer: Dmitry Kaliuzhnyi-Verbovetskyi (Philadelphia)

Cited in 1 Document

MSC:

32E30 Holomorphic, polynomial and rational approximation, and interpolation in several complex variables; Runge pairs
32A15 Entire functions of several complex variables
32C25 Analytic subsets and submanifolds
46E10 Topological linear spaces of continuous, differentiable or analytic functions

Keywords:

interpolating variety; pseudoconvex set; holomorphic function; weight
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\textit{B. Q. Li}, J. Math. Soc. Japan 58, No. 4, 1185--1196 (2006; Zbl 1118.32013)
Full Text: DOI Euclid

References:

[1] C. A. Berenstein and R. Gay, Complex Analysis and Special Topics in Harmonic Analysis, Springer-Verlag, New York, 1995. · Zbl 0837.30001
[2] C. A. Berenstein and B. Q. Li, Interpolating varieties for weighted spaces of entire functions in \(\bm{C}^n\), Publ. Mat., 38 (1994), 157-173. · Zbl 0820.32002
[3] C. A. Berenstein and D. C. Struppa, Solutions of convolution equations in convex sets, American J. Math., 109 (1987), 521-544. · Zbl 0628.46036
[4] C. A. Berenstein and B. A. Taylor, Interpolation problems in \(\bm{C}^n\) with application to harmonic analysis, J D’Analyse Mathématique, 38 (1981), 188-254. · Zbl 0464.42003
[5] L. Carleson, Interpolation by bounded analytic functions and the corona problem, Ann. of Math. (2), 76 (1962), 547-559. · Zbl 0112.29702
[6] L. Ehrenpreis, Fourier Analysis in Several Complex Variables, Wiley-Interscience, New-York, 1970. · Zbl 0195.10401
[7] R. Gunning, Introduction to Holomorphic Functions of Several Variables, I , Wadsworth, Inc., California, 1990. · Zbl 0699.32001
[8] L. Hörmander, Generators for some rings of analytic functions, Bull. Amer. Math. Soc., 73 (1967), 943-949. · Zbl 0172.41701
[9] L. Hörmander, An Introduction to Complex Analysis in Several Complex Variables, Third Edition, North-Holland, New York, 1990.
[10] J. Horváth, Topological Vector Spaces and Distributions, Addison-Wesley, 1966. · Zbl 0143.15101
[11] J. Kelleher and B. A. Taylor, Finitely generated ideas in rings of analytic functions, Math. Ann., 193 (1971), 225-237. · Zbl 0207.12906
[12] B. Q. Li, Interpolating varieties for entire functions of minimal type, Adv. Math., 194 (2005), 87-104. · Zbl 1072.32002
[13] B. Q. Li and B. A. Taylor, On the Bézout problem and area of interpolating varieties in \(\), Amer. J. Math., 1189 (1996), 989-1010. · Zbl 0863.32002
[14] X. Massaneda, \(A^{-\infty}\) interpolation in the ball, Proc. Edinb. Math. Soc. (2), 41 (1998), 359-367. · Zbl 0891.32003
[15] S. Oh’uchi, Disjoint union of complex affine subspaces interpolating for \(A_p\), Forum Math., 11 (1999), 369-384. · Zbl 0929.32012
[16] M. Ounaies, Zéros d’applications holomorphes de \(\) dans \(\), Ark. Mat., 39 (2001), 375-381. · Zbl 1038.32016
[17] W. Rudin, Function theory in the unit ball of \(\), Springer-Verlag, Berlin, 1980. · Zbl 0495.32001
[18] D. C. Struppa, The fundamental principle for systems of convolution equations, Mem. Amer. Math. Soc., 273 (1981), 1-164.
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