  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 $T$.
