Conic Sheaves on Subanalytic Sites and Laplace Transform

Abstract

Let $E$ be a $n$ dimensional complex vector space and let $E^{\ast }$ be its dual. We construct the conic sheaves ${\mathcal{O}}_{E_{{\mathbb{R}}^{+}}}^t$ and ${\mathcal{O}}_{E_{{\mathbb{R}}^{+}}}^{\mathrm{w}}$ of tempered and Whitney holomorphic functions respectively and we give a sheaf theoretical interpretation of the Laplace isomorphisms of [10] which give the isomorphisms in the derived category ${\mathcal{O}}_{E_{{\mathbb{R}}^{+}}}^{t\wedge }[n]\simeq {\mathcal{O}}_{E_{{\mathbb{R}}^{+}}^{\ast }}^t$ and ${\mathcal{O}}_{E_{{\mathbb{R}}^{+}}}^{\mathrm{w}\wedge }[n]\simeq {\mathcal{O}}_{E_{{\mathbb{R}}^{+}}^{\ast }}^{\mathrm{w}}$.