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Conic sheaves on subanalytic sites and Laplace transform. (English) Zbl 1239.32009
Sheaf theory is not well suited for the study of analytical objects that are not local, such as tempered distributions. An approach to solve this problem is the theory of ind-sheaves developed in [M. Kashiwara and P. Schapira, Ind-sheaves. Astérisque. 271. Paris: Société Mathématique de France (2001; Zbl 0993.32009)], where also the equivalence of categories is proved between ind-\(\mathbb{R}\)-constructible sheaves on a real analytic manifold \(X\) and sheaves on the subanalytic site \(X_{sa}\). The author of the paper under review works directly in the latter category.
Let \(E\) be an \(n\)-dimensional complex vector space, and let \(E^*\) be its dual. The author constructs the conic sheaves \(\mathcal O^t_{E_{\mathbb{R}^{+}}}\) and \(\mathcal O^W_{E_{\mathbb{R}^{+}}}\) of tempered and Whitney holomorphic functions by taking the Cauchy-Riemann system with values in tempered distributions and Whitney \(\mathcal C^{\infty}\)-functions, respectively. He also gives a sheaf theoretical interpretation of the Laplace isomorphisms of [M. Kashiwara and P. Schapira, J. Am. Math. Soc. 10, No. 4, 939–972 (1997; Zbl 0888.32004)], which induce isomorphisms of the derived categories \(\mathcal{O}^{t\Lambda}_{E_{\mathbb{R}^{+}}}[n]\simeq \mathcal{O}^t_{E^*_{\mathbb{R}^{+}}}\) and \(\mathcal{O}^{W\Lambda}_{E_{\mathbb{R}^{+}}}[n]\simeq \mathcal{O}^W_{E^*_{\mathbb{R}^{+}}}\).

32C38 Sheaves of differential operators and their modules, \(D\)-modules
32B20 Semi-analytic sets, subanalytic sets, and generalizations
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