Subanalytic Sheaves and Sobolev Spaces

Sheaves on manifolds are perfectly suited to treat local problems, but many spaces one naturally encounters, especially in Analysis, are not of local nature. The subanalytic topology (in the sense of Grothendieck) on real analytic manifolds allows one to partially overcome this difficulty and to define for example sheaves of functions or distributions with temperate growth, but not to make the growth precise. In this volume, one introduces the linear subanalytic topology, a refinement of the preceding one, and constructs various objects of the derived category of sheaves on the subanalytic site with the help of the Brown representability theorem. In particular one constructs the Sobolev sheaves. These objects have the nice property that the complexes of their sections on open subsets with Lipschitz boundaries are concentrated in degree zero and coincide with the classical Sobolev spaces. Another application of this topology is that it allows one to functorially endow regular holonomic D-modules with filtrations (in the derived sense). In the course of the text, one also obtains some results on subanalytic geometry and one makes a detailed study of the derived category of filtered objects in symmetric monoidal categories.