Note on the Fourier-Laplace transform of \(\overline{\partial}\)-cohomology classes
This paper derives from \textit{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, which says that the Fourier-Laplace transform establishes a bijection between the set of analytic functionals carried by a compact, convex set \(K\) in \(\mathbf{C}^n\) and the space \(A_K(\mathbf{C}^n)\) of entire functions \(f\) of exponential type with the property that for every positive \(\epsilon\) there exists a constant \(C_\epsilon\) such that \(| f(z)| \leq C_\epsilon \exp(\epsilon| z| + \sup_{\zeta\in K}\Re\langle \zeta,z\rangle)\). Moreover, there is an explicit formula for the inverse transform. The author's main observation is that by thinking of integrating differential forms on an arbitrary smooth surface surrounding \(K\), one can recast the theorem as establishing a linear isomorphism between \(A_K(\mathbf{C}^n)\) and the space \(H^{(n,n-1)}(\mathbf{C}^n \setminus K)\) of \(\overline{\partial}\)-cohomology classes of \((n,n-1)\) forms in the complement of \(K\).
scientific article; zbMATH DE number 1270169
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