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