26 NONLINEAR REPRESENTATION AND SPACES
It follows from the induction hypothesis and (2.12) that
Q1 (y', x, -zV) - Q1(y, x, -zV)QX(X, x, -2V) + ^ ( y , x, -zV)i?2(X, x, -zV), (2.15a)
Q2(y', x, -zV) = #
2
(y, x, -zV)i?x(X, x, -zV) + Q2(y, x, -zV)Q2(X, x, -zV), (2.15b)
^1(y^x,-zV) = Q1(y,x,-zV)i^1(x,x,-z^) + i^1(y,x,-zV)Q2(x,x,-zV), (2.15c)
#
2
(y', x, -zV) = R2(Y, x, -zV)Qi(X,x, -zV) + Q2(Y, x, -zV)#
2
(X, x, -zV), (2.15d)
r(y',x, -z*v) = r(y, x, -zV) r(x, x, -zV). (2.i5e)
Let Di(x,£), D2(x,£) be two polynomials in x,£ G
M3
with degDi = di, i = 1,2, and let
Di(a~1x^a£) = a^D^x,^), a ^ 0, z^ G Z.
It then follows using the Leibniz rule that there is a polynomial D in x and £ such that
£(x, -zV) = £i(x, -zV)£2(z, -*V) and
deg£ = d1+d2,D(a-1x,a£) = aUl+U2D{x^). (2.16)
We apply (2.16) to Qi(Y') in (2.15a), which gives according to (2.13a)
degQ^Y') max(degQ1(y) +degQ1(X),degi^1(y) +degE
2
(X))
max (\Y\ + C(Y) + \X\ + C{X), \Y\ + C{Y) - 1 + \X\ + C{X) + l)
= \Y\ + \X\ + C(Y) + £(X) = \Y'\ + £(y')
and
= a l
y
' l - ^ ' ) Q
1
( y ' , x
)
0 .
This proves that (2.13a) is true for L + 1 and i; = 1. Application of (2.16) to (^(^O*
^ ( y 7 ) , #2(y') r(y ; ) in (2.15b)-(2.15e) proves similarly that (2.13a)-(2.13c) are true for
L + l.
Let Mi(x,£), z = 1,2, be two polynomials in x,£ G
R3.
Using Leibniz rule it follows
that there is a polynomial M(x,£) such that M(x,—iV) = Mi(x, —zV)M2(x, —zV) and
that
degc M = degc M1 + deg^ M2, degx M = degx Mi + degx M2. (2.17)
Formula (2.15e) and relation (2.17) prove (2.14) for L + 1. This proves the lemma.
Lemma 2.8. If u = (/,/,a ) G £f, | p 1, tfien
( £
IIM^(/,/)II
2
M P
+ Yi
\\M?d6*why
c\\u\\El,
0|/3||6|1 \P\1
\S\1
for some constant C depending only on p. Here Mp(x) = x@ = xf 1 ^ x% 3 .
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