24 NONLINEAR REPRESENTATION AND SPACES

This inequality gives using that

supsGRs2(n_z)(l

—

(f(s))2

1, n /,

| | ( l - A r / 2 ( l - r

T

) / | | 2

H

r 2 ( " - ' ) | | ( l - A i / 2 / | |

f f

, nl,r0. (2.11)

Inequality (2.11) proves statement iii).

Finally we have, since r i— • pT G L°°(E) is differentiable,

(1 - ±)n/2-^rTf = -] A"A/rV'(A/r)d(eA/),

where

pf

is the derivative of (p. If n 0, I 0 and / belongs to the domain of (1 —

A)*/2,

it then follows that

\\(l-Ar'2±rTf\\l

= r-

2

jf

A2("-')(A/r)2(¥'(A/r))2A2'd||eA/||2H

r2*"-'-1) sup (S2("-!+1)(^'(S))2) ||(1 - A)l'2f\\%.

By the definition of p we have p'(s) = 0 for |s| 1 and \s\ 2. The last inequality then

gives

||(1 - ST^IV/II? , Cntl r^-'-^IKl - A)'/2/!!2,, n,I 0,

where Cn,/ = supsGE (s2^n_z+1^(y?/(s))2) oo. This proves the statement iv).

We are now prepared to prove the existence of a smoothing operator for the sequence

of Hilbert spaces defined by (1.6).

Theorem 2.5. There exists a C°° one-parameter family TT G L^E^Eoo), r 0, such

that if f G Ei, I 0, then statements i)-iv) of Theorem 2.4 hold with Hn and Hi replaced

by En and Ei respectively.

Proof According to Theorem 2.2, Eny n 0, is the Hilbert space of n-differentiable vectors

of the unitary representation g i— • Vg of Vo in E. It follows then from Theorem 2.4 that

there exists a C°° one-parameter family with the announced properties.

The existence of a smoothing operator guarantees that the norms || • \\E satisfy a

convexity property, which we make explicit now:

Corollary 2.6. Let 0 n^ n n\, n\^ri2. Then

ll/lk ^ Cm.nJI/llJp ll/llf^, / e Eni.

Moreover if N$ rii N, i = 1,2, and n\ + n± NQ + TV, then

ll/lkjlfflk, ^ C A r O l / b ^ y i ^ + l l / l l ^ i y ^ ) , f,g€EN.