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A pure smoothness condition for Radó’s theorem for \(\alpha \)-analytic functions. (English) Zbl 1374.35116

Summary: Let \(\Omega\subset\mathbb{C}^n\) be a bounded, simply connected \(\mathbb{C}\)-convex domain. Let \(\alpha\in\mathbb{Z}_+^n\) and let \(f\) be a function on \(\Omega\) which is separately \(C^{2\alpha_j-1}\)-smooth with respect to \(z_j\) (by which we mean jointly \(C^{2\alpha_j-1}\)-smooth with respect to \(\operatorname{Re} z_j\), \(\operatorname{Im}z_j\)). If \(f\) is \(\alpha \)-analytic on \(\Omega\setminus f^{-1}(0)\), then \(f\) is \(\alpha\)-analytic on \(\Omega\). The result is well-known for the case \(\alpha_i=1\), \(1\leq i\leq n\), even when \(f\) a priori is only known to be continuous.

MSC:

35G05 Linear higher-order PDEs
30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
32A99 Holomorphic functions of several complex variables
32U15 General pluripotential theory
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References:

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