# Distributions and Nonlinear Partial Differential Equations by Elemer E. Rosinger (auth.)

By Elemer E. Rosinger (auth.)

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Denot- ~ c. v. + t ~ V + T 1 i o icJ , 44 It only remains to prove that t• Y o n T . The r e l a t i o n (19) t = F is a local class. 1). Assume implies ci(si+vi ) Z ieJ where j c I , J finite and c. ~ C 1 . Now, t ~ F i since v i e Y° , w i t h gives ci = 0 , i • J . But V i ¢ J . Then, holds for any t c T , t ~ O (20) ~ V ~ for t tion (12) w i t h • N : V (21) ~ The relations (22) E c i si ¢ J , hence i•J (19) will imply and J Z c. s. E F ieJ i I O ' E c i ei = 0 c E , which i•J t ~ O . 2) x E Rn .

JG,p " presented in Lemmas 1 a n d by ~ • N , x • Rn Lemma 1 If ~ ¢ D ( R n~) Dq(O) then wy,e and satisfies = O, e gG, p v , for a given r ~ NmT, V k ¢ N the condition I r I - = 6 y . For q ~ Nn of the and by = ~((v4)X(x)) • 8(X(x)) " Dqsx(V)(x) , V w ¢ N , x ¢ Rn and B satisfies Lemma 2 If ~ • D(R I) , ~ = 1 given k • N in a n e i g h b o u r h o o d the condition of 0 ~ R1 for a 2 41 Dr~(o) = 0 , V r E N , r -< k then i) S y , q ~ JG ,p n S o ' V G c FF , 2) Sy,q ¢ Vo provided t h a t ]ql G ~ FX , pc ~n , I;l k , < k Proof i) It c a n b e s e e n t h a t The relation 2) It f o l l o w s An i m p o r t a n t Proposition Sy,q Sy,q e S o easily and Fy ~ G results satisfy (14), t h e r e f o r e S y , q e JG,p " easily.

And the ideals IQ(V(p),S ') According to (ii) and (19), the larger IQ(V(p),S ') , p ~ A n , will be. Therefore, maximal ideals, the problem of (V,S') ~ R(P) means to choose V is, the larger as a first approach in securing , with maximal V , will be studied in the present section. An alternative approach to the problem of maximal ideals iQ(v(p),S, ) , P ~ ~n , will be given in chap. 2, §7. And now, several results on the structure of (38) V (V,S') ~ R(P) R(P) • In addition to the relation : V c V° obtained in 2), Remark i, §7, the following two simple results will be useful.