# Introduction to the theory of distributions, based on the by Israel Halperin By Israel Halperin

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R . where ~ ) . ~ ( 8= ) 6()-'a(6) 6,. and ~ ( 6 =~ -6,. ) Let wherc ~ ( 6 )= 6 ~ ' a ( 6 ) 6 , and 0(6()) = -8,. Then, let S = TP(6,) r(6,) and A = Tr(6,). Consider A-'S = P(6,,). Clearly, P(6(,) commutes with r(62), 6 E pi. Again, ~ ( 6 may ~ )not ~ be -1. but we may complete the argument exactly as in the K+ case, above. We will now produce a canonical form theorem that will reduce the general problem to the two special cases we have just verified. Consider the matrix where IK is a division algebra with involution Since (c")' = C and (cD)' = D'c,' 7.

8) A*~=-Z n*f = P ~ . 2 f j=l 9 J2 Yk Ck=l * - We have a l s o t h a t The important f a c t s about t h e so-called sub-Laplacian A a r e t h a t A i s a hmogeneous d i s t r i b u t i o n of degree -r-2, and t h a t A i s h y p o e l l i p t i c . B. 6. I n f a c t , we Folland [ F o l ] and A. Kaplan [Kap] Let k be t h e f u n c t i o n N 2-r . Then, i n t h e d i s t r i b u t i o n a l sense, k*A = A*k = c6 , where c i s a nonzero c o n s t a n t , and 6 ' i s t h e Dirac measure a t (0,O).

2. L e t n be a p o s i t i v e i n t e g e r . Suppose t h a t 2n K = QN~", where n E C ( C ) , and t h a t 5 = 2n + i n (r) E R\{o}). 5 Then nbn * K where * 5 t12 < c ( n , ~ ) nfu, 1n1 c(n,s)<~(n,n) (lnl-l+ Proof. n It s u f f i c e s t o check t h a t Q * . 4, t h e while. f E cmC(v) 2n+l K We omit t h e d e t a i l s . 3. - Suppose t h a t 5 k R S e t K e q u a l t o Q N ~ (- i n~ t h e s e n s e of Remark 1 . 2 ) , 5 5 and l e t S be t h e s e t o f p o l e s of t h e d i s t r i b u t i o n - v a l u e d f u n c t i o n 2-r 5 - K .