Fourier transform and Hardy spaces of \(\overline\partial\)-cohomology in tube domains. (English. Abridged French version) Zbl 0771.32001

Let \(V\) be a sharp open cone in \(\mathbb{R}^ n\), \(V^*\) be the open dual cone, \(T(V^*)=\mathbb{R}^ n+iV^*\) be the corresponding tube domain in \(\mathbb{C}^ n\), \(L^ 2(V)\subset L^ 2(\mathbb{R}^ n)\) be the subspace of functions with support in \(V,H_ 2[T(V^*)]\) be the Hardy spaces.
The author generalizes the classical Bochner-Paley-Wiener theorem [cf. S. Bochner, Ann. Math., II. Ser. 45, 686-707 (1944; Zbl 0060.243)] asserting that the Fourier transform can be extended in an isometry from \(L^ 2(V)\) onto \(H_ 2[T(V^*)]\) to the case of nonconvex cones, where, instead of Hardy spaces, of holomorphic functions, he considers Hardy spaces of \(\overline\partial\)-cohomology for tube domains with nonconvex basis.
Reviewer: P.Caraman (Iaşi)


32A07 Special domains in \({\mathbb C}^n\) (Reinhardt, Hartogs, circular, tube) (MSC2010)
32A35 \(H^p\)-spaces, Nevanlinna spaces of functions in several complex variables
42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type


Zbl 0060.243