process

194 P. Carrillo RouseThe interesting part of the proposition is that the map is well defined since it isevidently be linear. Let us suppose for the moment that the proposition is true. Underthis assumption, we will define the product in S c .G T /.Definition 4.9. Let f; g 2 S c .G T /, we define a function f g in G T byZ.f g/.x;;0/D f.x; ; 0/g.x; ; 0/ d x ./T x G xandZ.f g/.; t/ D f. ı ı 1 ; t/g.ı; t/tq d s./ .ı/G s./for t ¤ 0.We can enounce our main result.Theorem 4.10. defines an associative product on S c .G T /.Proof. Remember that we are assuming for the moment the validity of Proposition 4.8.Let f; g 2 S c .G T /. We let F WD .f; g/ be the function in .G T / .2/ defined by.f;g/.x;;;0/D f.x;;0/ g.x; ; 0/and.f; g/..; t/; .ı; t// D f.;t/ g.ı; t/for t ¤ 0. Now, from the Leibnitz formula for the derivative of a product it is immediatethat .f; g/ 2 S c ..G T / .2/ /. Finally, by definition we have thatm c ..f; g// D f g;hence, by Proposition 4.8, f g is a well-defined element in S c .G T /.For the associativity of the product, let us remark that when one restricts the productto C 1 c .G T /, this coincides with the product classically considered on C 1 c .G T / (whichis associative, see for example [22]). The associativity for S c .G T / is proved exactly inthe same way that for C 1 c .G T /.It remains to prove Proposition 4.8. We are going to start locally. Let U 2 R p R q R q be an open set and V D U \R p f0gf0g. Let P W R p R q R q ! R p R qthe canonical projection .x;;/ 7! .x; /. We set U 0 D P.U/ 2 R p R q , then U 0is also an open subset, V Š U 0 \ R p f0g and P j V D Id V . We denote also by P therestriction P W U ! U 0 . As in Lemma 3.2 we have a C 1 map PQW U V ! U 0Vwhichin this case is explicitly written asP.x;;;t/D Q.x;;t/:We define PQc W S c . U V / ! S c. U 0/ as follows:Let us prove a lemma.Q P c .F /.x; ; t/ DVZf2R q W.x;;;t/2 U V g F .x; ; ; t/ d: