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.
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