process

190 P. Carrillo RouseNow, the spaces S c . U V/ are invariant under diffeomorphisms. More precisely ifF W U ! U 0 is a C 1 diffeomorphism as in Lemma 3.2 then we can prove the nextresult.Proposition 4.2. Let g 2 S c . U 0V/, then Qg WD g ı FQ2 S 0 c . U V /.Proof. The first observation is that Qg 2 C 1 . U V/, thanks to Lemma 3.2. Let us checkthat it has compact conic support. For that, let K 0 U 0 Œ0; 1 be the conic compactsupport of g. We letK D .F 1 Id Œ0;1 / U Œ0; 1;which is a conic compact subset of U Œ0; 1 relative to V , and it is immediate bydefinition that Qg.x; ; t/ D 0 if t ¤ 0 and .x; t ;t/ … K, that is, Qg has compact conicsupport K.We now check the rapid decay property .s 1 /: To simplify the proof we first introducesome useful notation. Writing F D .F 1 ;F 2 / as in Lemma 3.2, we denoteF 1 .x; / D .A 1 .x;/;:::;A p .x; // and F 2 .x; / D .B 1 .x;/;:::;B q .x; //.We also write w D w.x; ; t/ D .A 1 .x;t/;:::;A p .x; t// and D .x; ; t/ D. BQ1 .x;;t/;:::; BQq .x;;t//where BQj is as above, i:e:,Q B j .x;;t/D8ˆ