42 NONLINEAR REPRESENTATION AND SPACES
The inequalities ||AfM|/i||a2|||L2 \\h\\L*\\M^\\Lv„ q = 6(3 - 2p)"\ U / J ^ C
||(/i,0)||M (cf. (2.61)) and ||MMa2||LS/, C E
M
, II^MMa2||L2 give
£ I I ^ M I / I I M L * C||(/i,0)||M||a2||Di. (2.83)
Since
||(/,0)||
M B
|k||
B n
, ||ai||Dn |k||
B n
it follows from (2.81), (2.82) and (2.83) that
T2x{ul®u2)\\E C(||ui||
B
||«
2
|b
i
+ I M I B J M I E ) ,
which proves the first statement of the lemma in the case where N = 0.
Let u = (/, /, a) Eoo and |/x| 1. It follows from (2.60a) of Theorem 2.12 and from
the inequality
||(1 + \x\y-WaWv 11(1 +
W)a||£1/2||a-||3L/22-p
I N I ^ I M ^
2
- ' ,
that
l|MM|/||a|||L, sup ((1 + \x\)3/2-'\f(x)\)\\(l + M ) ' - 1 ' 2 ^ (2.84)
P -
1
^ ! ! i|3/2-
C\\UM\MM\PD;'
\\*\\r~p
p
M i .
It follows from Lemma 2.16 that, for X II, ||Tjr(tti g U2)\\E is bounded by a sum of
terms of the form
CN\\Mlt\Mllld^a1\\Mllad^a2\\\Lp=CNI(n,n1,ii2,v1,va), p =
6(5-2p)~1,
(2.85)
where |/x| 1, \vi\ + ]i/2| N, \[i\\ \ui\, |/L*2| \v2\ and of terms of the form
CWJ(/x,^i,
\X2,VX,V2)
(2.86)
= CviWM^M^hWM^a^hi + H M j M ^ / a l l M ^ a x l H ^ ) ,
where \(i\ 1, \ui\ + |n | N, \m\ + \/j,2\ N.
Let first |ya| |i/2| in (2.85). Then (2.82) gives
\\Mtl\Mllld^a1\\M^a2\\\LP CM\\Mllld^a1\\D\\Mfl2d^a2\\Di (2.87)
^CWi\\\ui\\ElJW2\\Elt,2{+1
= I ' ( M , M ) ,
where we have used Theorem 2.9. Since |^i| + |^2| + 1 N + 1, \i/i\ Ny |i/2| + 1
[f ] + 1 N for N 1, Corollary 2.6 gives with N0 = 1
/^lU^D^C^Ol^lUJI^II^ + KII^KII^). (2.88)
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