Subanalytic Sheaves and Sobolev Spaces

Sheaves on manifolds are perfectly suited to treat local problems, but many spaces that one naturally encounter, especially in analysis, are not of a local nature. The subanalytic topology (in the sense of Grothendieck) on real analytic manifolds allows the authors 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, the authors introduce the linear subanalytic topology, a refinement of the preceding one, and construct various objects of the derived category of sheaves on the subanalytic site with the help of the Brown representability theorem. In particular, they construct 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 the authors to functorially endow regular holonomic D-modules with filtrations (in the derived sense). In the course of the text, the authors also obtain some results on subanalytic geometry and make a detailed study of the derived category of filtered objects in symmetric monoidal categories.