32 NONLINEAR REPRESENTATION AND SPACES

where v G M ^ . We estimate the two last two terms on the right-hand side of this equality.

According to the induction hypothesis

53

\\M^v\\2M

= C:Y, E

I W ^ I I M

(2-40)

H n 1*3 |/w|n

|i/|=n+ l |i/|=n

c; 53

(in(div))2

2

li3

CnCn 2^ \\TPi v\

li3

C'JvfMn+i,

for some constants C'n and C^. For the term with \/i\ = n + 1, and \v\ = n + 1 it follows,

using the same argument which led to (2.37), that

MpP^x&rg, r%(x,0=r%(a-1x,aO, degrlk2n. (2.41)

l,k

This and Lemma 2.8, with a = 0, show that, for |/i| = |z/| = n + 1,

I W t , | |

M

X) \\xidkt%v\\M C^Wrj^vW^, (2.42)

Z,fc l,k

\JJJ\ = \v\ = n -f 1, where r ^ = r(^(x, —iV). It follows from property (2.41) of rj^, the

definition of qM and (2.42) that for \/JL\ — \v\ = n + 1, and suitable constants C, C

n

and

C" that

IIM

M

^H

M

c ^ ( x :

n^MrrMi2M

+

NI2M)

*

i,k xeii

^ ^ ' E ( £

(W^TkMv\\M

+ ||[T£Vffc||M) + N|

M

)

z,fc xe n

Cn ^

«n(7$Mt;)

+ C'J2

\\[TkM,r%MM,

xeu i,k

where r% = r%(x, -zV) .

According to the induction hypothesis this gives with new constants

\\M,d»v\\M cn( 53

(\\TkMv\\Mn+c'53ii[rr,rrfciiM))

(2.43)

xeu i,k

Cn\\v\\Mn+i +C 53

'£\\[TkM,r%]v\\M,

xen i,k