46 NONLINEAR REPRESENTATION AND SPACES
For the last inequality we have also used that \Zi\ + \Z2\ = \Z\. We obtain in the same
way for N 0,
\\T21U)\\ENJT%(9)\\E
+ \\T%(f)\\E\\T%(g)\\EN+i (2.99)
CNMZ\ J2 Wfh
WE---
II/^
WE\\9J2 WE---
\\9n2 \\E
(ll/l1ll^+|z|+1llft1llB + ll/«xllBllft.ll^+IZ|+1), ^ 0 -
Let Mi,... ,un e EOQ. Then it follows from inequalities (2.97), (2.98) and (2.99) that
\\1*(I% ® T%)Tn{®Uui)\\EN (2-100)
/ v - ^ \ 3 / 2 - p
CNMZ\ [ 2 ^ I K
WEN+IZ{
I K \\El I K
WE'"
I K
WE)
i
(2-^ IK ll^+|z|+1IK WE-" IK IUJ
CW,n,|Z | ^ 2^ IK li^+|Z| IK 11^ IK IU IK b j
( 5Z IK U*WI+1IK Ik IK WE'" I K IU) , # 1.
Let / i , . . . ,/
n i
, 7ij ,0n25 fi9 D e defined as previously. Statement hi) of Lemma 2.17
gives
II^TO/)®^(ff))ll
B
C-min^
II^U/)^;172!!^^/)!!3/""!!^^!!^).
It follows from the first inequality of Lemma 2.19 that
\\T^l(f)\\EJTTM\\E:1/2\\TTM\t~P (2-102)
Cm,n ( Y, H/'x I I V|
Z l l
"
/ j
'
IIB
•'
»/j-i
WE\\9h llBl+|Z2l life HE ll^
2
ll*r
1 / 2
( £ ll/'i lk
+|Zl l
Wh WE--- WK \\E\\9h b
| Z a l
life l|B llfe2 ll
B
)
3/2
^-
1,3
Inequalities (2.101) and (2.102) give
WTxPzl ® 2 ^ K ( ® ? = i ^ ) l b Cn,|z| min(Q(|Zi|, |Z2|),Q(|Z2|, 1^)), n 2, (2.103)
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