Cohomological Laplace transform on non-convex cones and Hardy spaces of $\bar{\partial }$-cohomology on non-convex tube domains

Abstract

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 by the first author and his collaborators. For these cones we consider a construction of dual non-convex cones $V^{\ast }$ 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 uCech cohomology. We give a construction of Hardy space of $q$-dimensional cohomolgy at $T$.