40 NONLINEAR REPRESENTATION AND SPACES
wherea = 3 / 2 - p + | / ? | .
Estimates (2.72) and (2.74) show that, if
Qn(u) = sup ((1 + |*| + \t\f^\u{t, x)\)
t,x
+ Y, sup ((1 + M -h |t|)(l + ||t| - klD^-^^lti^Ct.o:)!),
ll£ln t,X
then
Qniu) Yl CPtMrOi3/2_p+m(f0Jfi). (2.75)
0\(3\n
It follows from the definition of r
n a
and from Tp0 = da ( ^ *0 J , /3 = (t, a) , that
ro,3/2-p+|/?|(//3,//3) r\p\i3/2-p(f,f),
which inserted into (2.75) gives (with new constant)
Q
n
H c
p
,
n
r
n
,
3
/ 2 - p ( / , / ) . (2.76)
Inserting the result r
n ) 3
/
2
- p ( / , / ) CniP\\(f,f)\\M from Theorem 2.12 into inequality
(2.76) we obtain for some constant Cp^n
Qn(u)Cpn\\(fJ)\\Mn+2. (2.77)
Since S(R 3 ,R 4 ) © 5(R 3 ,R 4 ) is dense in the space M
n + 2
by construction, it follows that
(2.77) is true for (/,/ ) G M
n +
2- This proves the proposition.
It follows directly from definitions (1.5), (1.7) and (1.8) of TX,X G p, that [TX,TY] =
DTx Ty DTy .Tx = T[x,y] on the space of C°° functions. The next lemmas and corollary
prove in particular that Tx, X G p, is a continuous polynomial from E^ to E ^ , which
assures that X »— Tx is a nonlinear representation of p in EQQ.
L e m m a 2.16. Let N 0, Ui = (fi,fi,ai) G E^, i = 1,2, and let p = 6(5 2p) _ 1 . Then
\\1%(UI®U2)\\ENCN( E H A ^ I ^ a x l i a ^ a a l H ^
HkiH2|JV
+ £ ( | | M ^ V i | | ^ 2 a
2
| | |
L 2
+ H A ^ I ^ M i a ^ a x l H ^ ) )
\»\N
\vi\ + \"2\N
and
\\Tti0J(ui®U2)\\EN CN( £ I I M J ^ a x l l ^ a a l H ^
Hl"i| + M+liV+l
+ Yl (l|MM|^/i||^2«2|||L2 + IIA^I^/all^ailH^)).
\fi\N+l
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