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Schwartz functions are classically defined on $\mathbb{R}^n$ as smooth functions such that they, and all their (partial) derivatives, decay at infinity even when being multiplied by any polynomial. The space of Schwartz functions is a Fr\'echet space, and its continuous dual is called the space of tempered distributions. A third space that plays a key role in this 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. The study of these 3 spaces has many motivations from various fields in Mathematics. For instance, the space of Schwartz functions is stable under the Fourier transform, and thus this transform can be generalized to the space of tempered distributions.

This theory was formulated on $\mathbb{R}^n$ by Laurent Schwartz in the first half of the $20^{th}$ century, and was extended over the years to various categories, e.g., Nash ($C^\infty$-smooth semi-algebraic) manifolds (first by du Cloux and later by Aizenbud-Gourevitch).

In this talk I will first explain how one can attach a Schwartz space to an arbitrary open subset of $\mathbb{R}^n$. Then, I will show that on ($C^\infty$-smooth) manifolds definable in polynomially bounded o-minimal structures, one can intrinsically define the space of Schwartz functions, as well as the spaces of tempered functions and of tempered distributions. We will see how Lojasiewicz inequality implies that these spaces are "well behaved", in the sense that they form sheaves and co-sheaves on the restricted (Grothendieck) topology. It will also become evident that Miller's theorem asserting that the exponential function is definable in any o-minimal structure that is not polynomially bounded, easily implies that the Schwartz theory is ill-defined on manifolds definable in such structures.

In the second part of the talk I will address the question under what conditions two open subsets of $\mathbb{R}^n$ have isomorphic Schwartz spaces? Such sets are said to be Schwartz equivalent. This question remains open in general, however in a recent joint work with Eden Prywes we used tools from quasiconformal geometry in order to prove the Schwartz equivalence of a few families of planar domains. I will present some of these results and explain some main ideas used in the proofs. In particular we will see that all quasidiscs in the plane are Schwartz equivalent.