Schwartz functions are classically defined on R^n as smooth functions such that they, and all their (partial) derivatives, decay at infinity faster than the inverse of any polynomial. The space of Schwartz functions is a Frechet space, and its continuous dual space is called the space of tempered distributions. A third space that plays a key role in the Schwartz theory is the space of tempered functions – a function is said to be tempered if point-wise multiplication by it preserves the space of Schwartz functions. This theory was formulated on R^n by Laurent Schwartz, later on Nash manifolds (smooth semi-algebraic varieties) by Fokko du Cloux and by Avraham Aizenbud and Dmitry Gourevitch, and on singular algebraic varieties by Boaz Elazar and myself.
The goal of this talk is to present the recently developed Schwartz theory on sub-analytic manifolds. I will first explain how one can attach a Schwartz space to an arbitrary open subset of R^n. Then, I will define (globally) sub-analytic manifolds – loosely speaking these are manifolds that locally look like sub-analytic open sub-sets of R^n (I will explain what are these too) and have some ”finiteness” property. Model theorists may think of definable manifolds in R^n. I will prove that one can intrinsically define the space of Schwartz functions (as well as the spaces of tempered functions and of tempered distributions) on these manifolds, and prove
that these spaces are ”well behaved” (in the sense that they form sheaves and co-sheaves on the Grothendieck sub-analytic topology). Along the way we will see where sub-analyticity is used, and why this theory is ill-defined in the category of smooth (not necessarily sub-analytic) manifolds. Mainly, some ”polynomially bounded behaviour” (that holds in the sub-analytic case thanks to Lojasiewicz’s inequality) is required. As time permits I will describe some possible applications.