# Elliptic Pseudo-Differential Operators — An Abstract Theory by Heinz O. Cordes By Heinz O. Cordes

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1 is proven. 2) implies c o n t i n u i t y of u over [. 2) is not correct. T h e n we may pick an i n c r e a s i n g sequence {Kj} of c o m p a c t sets w i t h ~ = L ~ K . 2) w i t h ~=~j. 7) (~) l~j (x) I ~ i/j sup , [~I ~ J K. 3 Every compact subset K C~ m u s t be contained familiar conclusion. 3). This is left to the reader. let us shortly discuss the p r o b l e m of topologies spaces 9', E' and S' since it often will be convenient notion of convergence = i, above. 2 is proven. 7) implies that ~j + O in E.

To~) ~, for 5) Show that a temperate d i s t r i b u t i o n has all of its d e r i v a t i v e s temperate d i s t r i b u t i o n s . A l s o show that u £ S' ,a e T implies au ~ S' 34 5. D i s t r i b u t i o n s with compact We a l r e a d y m e n t i o n e d the introduction, by E=E(~). asking a third the topological t h a t + O w h e n e v e r collection follows that ~ C E for use' Proof. f. 2). 1 a l s o can be u n i q u e l y functional over of K O. It f o l l o w s is correct.

There can be o n l y one e x t e n s i o n of v to E, since the e x t e n s i o n of u [ Q = vlD to E was p r o v e n to be unique. This imbedds E' into S'. In o t h e r words, the c o m p a c t l y supported d i s t r i b u t i o n s all are temperate and in p a r t i c u l a r have F o u r i e r transforms. 3. The (inverse) F o u r i e r t r a n s f o r m of a d i s t r i b u t i o n u w i t h c o m p a c t -i support is a f u n c t i o n in T C E . M o r e o v e r u =Fu and u :F u are entire functions of the n complex variables x:(x I .....