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

TL;DR

This paper introduces a cohomological Laplace transform for non-convex cones, extending Hardy space theory to ech cohomology on non-convex tube domains, with potential applications in complex analysis and geometry.

Contribution

It develops a novel cohomological Laplace transform for non-convex cones and constructs Hardy spaces of cohomology on associated tube domains, expanding the analytical framework.

Findings

Defined a cohomological Laplace transform for non-convex cones.

Constructed Hardy spaces of ech cohomology on non-convex tube domains.

Extended classical analysis tools to non-convex geometric settings.

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 $V^*$ 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 $T$.