44

NONLINEAR REPRESENTATION AND SPACES

Factorization of the right-hand side of (2.91) into a factor with exponent p — 1/2 and a

factor with exponent 3/2 — p and the application of Corollary 2.6 to the terms in the first

factor show that (2.93) is also true for \v\\ |z/2|. This proves the second statement of the

lemma for N 1. The case N = 0 is the same as in the first statement of the lemma.

The first statement of the lemma for N 1 follows from the second by application of

Corollary 2.6 to the factor with exponent 3/2 — p.

Finally statement hi) follows from (2.82), application of estimate (2.83) to one of the

terms ||M/x|/i||a2|||L2 or ||MM|/2||ai|||L2, and application to the other terms of estimate

(2.84). This proves the lemma.

Corollary 2.18. If u G E^ and X G II, then

\\T2X(U)\\EN

^ ( l l t i b J I u l l ^ ^ - ^ l l u l U J I u l U ^ r

1

/

2

, N1,

and

\\TZ(U)\\ENC\\U\\BJU\\EN+I, N0.

We shall need an analogy of Lemma 2.17 and Corollary 2.18 for Ty

= TY+TY,

where

Y Gil', the basis for the enveloping algebra U(p).

Lemma 2.19. Ifui,..., un G E^ and Y G IT, then

i) ||i?(tii®...®un)\\EN cj^ II KlblKII^^ ,

i K K n - 1

for n 1, N 0 and

ii) ||TJJ(tii®..-(8)1^)11^

^-( 2Z^ 11^^-11^7^^,^^^ 11^^^ 11^7^ I T N^^z il^)^ X ^ _ ^ ( ^ 3 M^^-11^^^^, 11^^^ 11^^ XT N^^^ il^)^— 1 X^.

i 1=3 i 1=3

for n 2 and \Y\ + N 1. Here the summation is over all permutation i of (1,... ,n)

and the constant C depends on \Y\,n,N,p.

Proof We prove the first statement by induction. It is true for Y = I, because T\(u) = u.

It follows from Lemma 2.17 that it is also true for Y = X G II. Suppose it is true for

\Y\ L. If Y' = YX, \Y\ L,X G n, then it follows from definition (1.9) of TYX that

with Iq =

®9

J, J = identity in E,

Tvx=

E miq®Tx®In-q-i)rn+ E T2-\lq®Tx®In_q_2)rn, (2.94)

0gn- l 0qn-2

where rn is the normalized symmetrization operator on g E (= EgEg • • • ££, n times).

By the induction hypothesis we have, after reindexation for n 2,

| | 7 r P + 1 ( J , ® Tl ® Jn-,_p)(Tn ®ni=x

UJ)\\EN

Cj2Wun\\E---\Win-i\\E

i

(H««»-illBl|Ix«iJlB +l|riuin_1||B||tin||E ), p=l,2,n-p0.