A non-archimedean algebra and the Schwartz impossibility theorem

In the 1950s L. Schwartz proved his famous impossibility result: for every there does not exist a differential algebra in which the distributions can be embedded, where is a linear operator that extends the distributional derivative and satisfies the Leibnitz rule (namely ) and is an extension of the pointwise product on . In this paper we prove that, by changing the requests, it is possible to avoid the impossibility result of Schwartz. Namely we prove that it is possible to construct an algebra of functions such that (1) the distributions can be embedded in in such a way that the restriction of the product to functions agrees with the pointwise product, namely for every f, g is an element of C-1(R) Phi(fg) = Phi(f) circle times Phi (g), (2) there exists a linear operator that extends the distributional derivative and satisfies a weak form of the Leibnitz rule. The algebra that we construct is an algebra of restricted ultrafunctions, which are generalized functions defined on a subset of a non-archimedean field (with ) and with values in . To study the restricted ultrafunctions we will use some techniques of nonstandard analysis.