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Journal of Lie Theory 28 (2018), No. 1, 245--263
Copyright Heldermann Verlag 2018

Cohomological Laplace Transform on Non-convex Cones and Hardy Spaces of ∂-cohomology on Non-convex Tube Domains

Simon Gindikin
Dept. of Mathematics, Rutgers University, 110 Frelinghysen Road, Piscataway, NJ 08854-801, U.S.A.

Hideyuki Ishi
Graduate School of Mathematics, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 242-8602, Japan


We consider a class of non-convex cones $V$ in $\mathbb{R}^n$ which can be presented as (not unique) union of convex cones of some codimension $q$ which we call the index of non-convexity. This class contains non-convex symmetric homogeneous cones studied in D'Atri-Gindikin [{\it Siegel domain realization of pseudo-Hermitian symmetric manifolds}, Geom.\ Dedicata {\bf 46} (1993) 91--125] and Faraut-Gindikin [{\it Pseudo-Hermitian symmetric spaces of tube type}, in: Topics in Geometry, Progr.\ Nonlinear Differential Equations Appl. {\bf 20} (1996) 123--154]. For these cones we consider a construction of dual non-convex cones $V^*$ and corresponding non-convex tubes $T$ and define a cohomological Laplace transform from functions at $V$ to $q$-dimensional cohomology of $T$ using the language of smoothly parameterized \u{C}ech cohomology. We give a construction of Hardy space of $q$-dimensional cohomolgy at $T$.

Keywords: Non-convex cone, Laplace transform, Paley-Wiener Theorem, symmetric cone, cohomology, Hardy norm.

MSC: 32F10, 32C35, 42B30.

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