
Journal of Lie Theory 28 (2018), No. 1, 245263 Copyright Heldermann Verlag 2018 Cohomological Laplace Transform on Nonconvex Cones and Hardy Spaces of ∂cohomology on Nonconvex Tube Domains Simon Gindikin Dept. of Mathematics, Rutgers University, 110 Frelinghysen Road, Piscataway, NJ 08854801, U.S.A. gindikin@math.rutgers.edu Hideyuki Ishi Graduate School of Mathematics, Nagoya University, Furocho, Chikusaku, Nagoya 2428602, Japan hideyuki@math.nagoyau.ac.jp [Abstractpdf] We consider a class of nonconvex 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 nonconvexity. This class contains nonconvex symmetric homogeneous cones studied in D'AtriGindikin [{\it Siegel domain realization of pseudoHermitian symmetric manifolds}, Geom.\ Dedicata {\bf 46} (1993) 91125] and FarautGindikin [{\it PseudoHermitian symmetric spaces of tube type}, in: Topics in Geometry, Progr.\ Nonlinear Differential Equations Appl. {\bf 20} (1996) 123154]. For these cones we consider a construction of dual nonconvex cones $V^*$ and corresponding nonconvex 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: Nonconvex cone, Laplace transform, PaleyWiener Theorem, symmetric cone, cohomology, Hardy norm. MSC: 32F10, 32C35, 42B30. [ Fulltextpdf (331 KB)] for subscribers only. 