32 NONLINEAR REPRESENTATION AND SPACES
where v G M ^ . We estimate the two last two terms on the right-hand side of this equality.
According to the induction hypothesis
53
\\M^v\\2M
= C:Y, E
I W ^ I I M
(2-40)
H n 1*3 |/w|n
|i/|=n+ l |i/|=n
c; 53
(in(div))2
2
li3
CnCn 2^ \\TPi v\
li3
C'JvfMn+i,
for some constants C'n and C^. For the term with \/i\ = n + 1, and \v\ = n + 1 it follows,
using the same argument which led to (2.37), that
MpP^x&rg, r%(x,0=r%(a-1x,aO, degrlk2n. (2.41)
l,k
This and Lemma 2.8, with a = 0, show that, for |/i| = |z/| = n + 1,
I W t , | |
M
X) \\xidkt%v\\M C^Wrj^vW^, (2.42)
Z,fc l,k
\JJJ\ = \v\ = n -f 1, where r ^ = r(^(x, —iV). It follows from property (2.41) of rj^, the
definition of qM and (2.42) that for \/JL\ \v\ = n + 1, and suitable constants C, C
n
and
C" that
IIM
M
^H
M
c ^ ( x :
n^MrrMi2M
+
NI2M)
*
i,k xeii
^ ^ ' E ( £
(W^TkMv\\M
+ ||[T£Vffc||M) + N|
M
)
z,fc xe n
Cn ^
«n(7$Mt;)
+ C'J2
\\[TkM,r%MM,
xeu i,k
where r% = r%(x, -zV) .
According to the induction hypothesis this gives with new constants
\\M,d»v\\M cn( 53
(\\TkMv\\Mn+c'53ii[rr,rrfciiM))
(2.43)
xeu i,k
Cn\\v\\Mn+i +C 53
'£\\[TkM,r%]v\\M,
xen i,k
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