process

192 P. Carrillo RouseFollowing the lines of the last remark we are going to precise a possible decompositionof our space S c .DX M / that will be very useful in the sequel. Let f.U˛;˛/g˛2a family of X-slices covering X. Consider the open cover of M Œ0; 1 consisting off.U˛ Œ0; 1; ˛/g˛2 together with M .0; 1. We can take a partition of the unitysubordinated to the last cover,f˛;g˛2 :That is, we have the following properties:• 0 ˛; 1,• supp ˛ U˛ Œ0; 1 and supp M .0; 1,• P˛ ˛ C P D 1.Let f 2 S c .D M X/. If we denotef˛ WD f j DU˛V˛ .˛ ı p/ 2 C 1 .D U˛V˛/andf WD f j M .0;1 . ı p/ 2 C 1 .M .0; 1/;then we obtain the following decomposition:f D X˛f˛ C f :Now, since f is conic compactly supported we can suppose, without lost of generality,that• f 2 C 1 c.M .0; 1/, and• that ˛ is compactly supported in U˛ Œ0; 1.What we conclude of all this, is that we can decompose our space S c .D M X/ as follows:S c .D M X / D X˛2ƒ S c .D U˛V˛/ C C 1 c.M .0; 1/: (5)As we mentioned in the introduction, we want to see the space S c .DX M / as a fieldof vector spaces over the interval Œ0; 1, where at zero we talk about Schwartz spaces.In our case we are interested in Schwartz functions on the vector bundle NXM . Let usfirst recall the notion of the Schwartz space associated to a vector bundle.Definition 4.6. Let .E;p;X/be a smooth vector bundle over a C 1 manifold X. Wedefine the Schwartz space S.E/ as the set of C 1 functions g 2 C 1 .E/ such that g is aSchwartz function at each fiber (uniformly) and g has compact support in the directionof X, i:e:, if there exists a compact subset K X such that g.E x / D 0 for x … K.