Hatziafratis, Telemachos 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 Ann. Math. Blaise Pascal 11, No. 1, 41-46 (2004). 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. Reviewer: Harold P. Boas (College Station) 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 Keywords:entire function; analytic functional Citations:Zbl 0924.43004; Zbl 0786.32003 PDFBibTeX XMLCite \textit{T. Hatziafratis}, Ann. Math. Blaise Pascal 11, No. 1, 41--46 (2004; Zbl 1085.32002) Full Text: DOI Numdam EuDML 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 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.