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On an integral formula of Berndtsson related to the inversion of the Fourier-Laplace transform of \(\overline{\partial}\)-closed \((n,n-1)\)-forms. (English) Zbl 1085.32002

In a previous article [Z. Anal. Anwend. 17, No. 4, 907–915 (1998; Zbl 0924.43004)], motivated by B. Berndtsson’s treatment [Several complex variables, Proc. Mittag-Leffler Inst., Stockholm/Swed. 1987–88, Math. Notes 38, 160–187 (1993; Zbl 0786.32003)] of the Pólya-Martineau theorem, the author established (for a compact, convex set \(K\) in \(\mathbb{C}^n\)) a linear isomorphism between the space of \(\overline{\partial}\)-cohomology classes of \((n,n-1)\) forms in the complement of \(K\) and the space of entire functions \(f\) of exponential type with the property that for every positive \(\epsilon\) there exists a constant \(C_\epsilon\) such that \(| f(z)| \leq C_\epsilon \exp(\epsilon| z| + \sup_{\zeta\in K}\operatorname{Re}\langle \zeta,z\rangle)\). In this article, the author reworks Berndtsson’s proof by deriving the key integral formula through a direct computation.

MSC:

32A26 Integral representations, constructed kernels (e.g., Cauchy, Fantappiè-type kernels)
32A15 Entire functions of several complex variables
42B10 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
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References:

[1] Berndtsson, B.; Fornaess, J. E., Weighted integral formulas, Several Complex Variables, 160-187 (1993) · Zbl 0786.32003
[2] Hatziafratis, T., Note on the Fourier-Laplace transform of \(\bar{\partial } \)-cohomology classes, Zeitschrift für Analysis und ihre Anwendungen, 17, 907-915 (1998) · Zbl 0924.43004
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