By Baruch Z. Moroz

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16) 25 a = m Since u + TnuT -n ~ X (Tmu) d~ (u) . I (p) is an a u t o m o r p h i s m of I (p), one o b t a i n s from (16) an equation: am = f Changing the v a r i a b l e u + TmuT -m Z ~ X' [e Tmue-1 ) d~ (u) . e6e l(p) of i n t e g r a t i o n and f i n a l l y by u ÷ eu) (at f i r s t by u + ue w e can r e w r i t e -I , then by this e q u a t i o n as follows: a where U uT E W(K~ By d e f i n i t i o n , Ik~) (17) = fie I f X' (u~m) d~(u), 1(p) is the Haar m e a s u r e (i (p)) = I° m m on ~ (p) normalised X' Iv) = 0 only w h e n flm and for by the c o n d i t i o n v ~ W(K~ u 6 i (~), Ik'~).

I applied into account to each of the charac- relations (45) and (46). §6. 32) Hypothesis. the e s t i m a t e s obtained a few g e n e r a l considerably improved. 17), remarks. It f o l l o w s that 1+slg(X) ) ( 1 + I I m s[) n/2 /aa(x)b(X)~" lL(s,x)l for and in §4 a n d §5 m a y b e (I) <__ 12n(31~-/~ I Re s >__ ~, X E gr(k). Lemma 1. Let and suppose f(s) be a function holomorphic in the h a l f - p l a n e Re s > that f(s) ~ O for I Re s > ~ , that If(s)l where < B(f) (1+Itl) Z t:= Im s, log f(s) with z > O, B(f) (2) > I, and that = O(Z log(2+n-1)) for Re s > I+~, q > O.

S e S(a,b) (with a r e a l c o n s t a n t c). Proof. It is a s p e c i a l Proposition 2. I O < n ~ ~ , Suppose c a s e of T h e o r e m that p [80, p. 192 - 193]. is of A W type and let X = tr p. If then IL(s,x) I _< (I+q-I) nd(X) i n the s t r i p I in -h < Re s < l+n, of the i d e n t i c a l representation 12 ~(31~[1+s )g(X)b( s , X ) 1 + q - R e where g(x) in and p denotes s (AW 17) the m u l t i p l i c i t y 37 I b(s,x):= Proof. ~2(s,Q) Making (a(x) ~~ I1+s+itpl K pesl pE~2 [ l+s+ I apl+itp 2) use of Lemma 4 we construct with the following (18) 2 ~1 (s,1) two functions and properties: ~1(s,1) = O([Im slC), ~2(s,Q) = O(lIm siC); (19) _1 (1+q-Re s) I~1(s,1)1 for > II+sl s 6 S(a,b) and ,l~2(s,Q) i > IQ+s[-(1+q-Re 2 s) (20) I~ I (b+it) I = l~2(b+it) I = I, Y ~j(a+it,~j) where = [Qj+a+itl I QI = I, Q2 = Q' 71 = -~-q' Y2 a = - q, and define a new function F(s) = L(s,x) = j• j = 1,2 2y I t 6~.