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

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 \(\epsilon, C>0\).


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
Full Text: DOI Euclid


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