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 .