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