process

A Schwartz type algebra for the tangent groupoid 191are bounded in K . For the expressions k@ ı z j .z/k, we proceed first by developing asin Lemma 3.2, that is, @Bj j .x;;t/D@ .x; 0/ C hj .x; t/ :Now, since we are only considering points in K , it is immediate that we can findconstants C j >0such thatk@ ı z j .z/k C j kk m ı:In the same way (remember that F is a diffeomorphism) we can have constants C i >0such thatk i .!;;t/kC i kk:Putting all together, and using the property .s 1 / for g, we get bounds C>0such thatand this concludes the proof.kk k k@˛zQg.z/k C;Remark 4.3. We can resume the last invariance result as follows: If .U; V/ is aC 1 pair diffeomorphic to .U; V / with U E, an open subset of a vector spaceE, and V D U \ E, then S c .D U V/ is well defined and does not depend on the pairdiffeomorphism.With the last compatibility result in hand we are ready to give the main definitionin this work.Definition 4.4. Let g 2 C 1 .D M X /.(a) We say that g has compact conic support K, if there exists a compact subsetK M Œ0; 1 with K 0 WD K \ .M f0g/ X (conic compact relative to X)such that if t ¤ 0 and .m; t/ … K then g.m; t/ D 0.(b) We say that g is rapidly decaying at zero if for every X-slice chart .U;/ andfor every 2 Cc1.U Œ0; 1/, the map g 2 C 1 . U V/ given byg .x;;t/D .g ı ' 1 /.x;;t/ . ı p ı ' 1 /.x;;t/is in S c . U V /, where p is the projection p W DM X.x;;0/7! .x; 0/, and .m; t/ 7! .m; t/ for t ¤ 0.! M Œ0; 1 given byFinally, we denote by S c .DX M / the set of functions g 2 C 1 .DX M / that are rapidlydecaying at zero with compact conic support.Remark 4.5. (a) It follows from the definition of S c .DX M / that the latter containsCc1.DM X/ as a vector subspace.(b) It is clear that Cc1.M .0; 1/ can be considered as a subspace of S c.DX M / byextending by zero the functions at NX M .