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)