1.1. DEFINITIONS AND BASIC PROPERTIE S 11
(OlJ
But E is complete, so YlZiifi ^ O converges to an element {fi}iZ0 G E. We
have to prove that each fi belongs to Sf, and that suppTfi C 5(0, 2Z+1).
By Lemma 1.1.7 we have for any x
sup \f?\x + z)\CMr,df?\x),
\z\2~*
or equivalently, for any x
f.W(^M|
" n inf A/f
It follows that for any R 1
(1.1.16) | y f (s) C inf Mr4f\l,fW \x + z).
\z\2
2~iN^
sup \ff\x)\c( f Mr,df?\xYdx
xEB(0,R) \JB(0,R)
l/r
(LL17)
C(1 +
R)d
inf (f
xeB(0,R)\JEB{0,2R)
Mrtdf?\x + zy
d^1/r
xeB(0,R)\J (l +
|z|)rd
C(l +
R)N'r+d
inf
Mr,d{Mr4f[l)){x).
xeB(0,R)
As in the proof of Lemma 1.1.4
{1 +
RrWr+d)
s u p
\fM(x)\\\{6ojx(B{0,R))}r=o\\E
xEB(0,R)
C2iN/r\\{5ojMr,d(Mr,dfll))}°L0\\E
C2
J
W
r
-
£
+)||{/ f }°i
0
Since \\{6OJX(B(0, R))}°°=0\\E \\{5ojx(B{0, 1))}£
0
||
B
0 for i? 1, this proves
that
(1.1.18) sup(l +
\x\)-Wr+d\f?\x)\
C2 W'-
e
+)||{/j'
)
}51
0
||
£
,,
X
and consequently
oo oo
SUP(1 + \X\)-W'+* £ |/ f (X)\ C&N*-^ (Yl
\E'
\\r'\\YL{E))1/K\\YL(E))il)\\f
1 = 1
It follows that J2Zifi 0 ) = Mx) w i t h
convergence in S', and uniformly on
compact sets. Thus supp J7/* C £(0,2*+1), / - £ £ o /, G r L ( £ ) , and J2Zi f{l)
converges to / in YL(E).
PROPOSITION
1.1.10. Le£ satisfy the conditions in Definition 1.1.6. Tften
£/ie space
^(JE?)
is continuously embedded in S''.
PROOF .
Let {/;}°^0 ^ ^
an^
suppose that (1.1.10) and (1.1.11) are satisfied.
We claim that E ° ^
0
/i converges in the sense of S'.
We first observe that in order to prove that YlZo fi converges in
Sf,
it is
sufficient to prove that X ^ o
2~1T
fi converges in
Sf
for some a 0. In fact, we shall
see that if YlZo fi converges in S', then X ^ o
22J/z
converges in «S' for any cr G R.
In order to prove this we choose CQ° functions (pi such that supp^o C 5(0,2),
s u p p ^ C £(0,2
i + 1
) \ ^ ( O ^ -
1
) , i = 1, 2, ... , and ^(£) = 1 on supp^/*. Set
*i =
EZo22ia^2i
and $
2
=
E*"o2(2i+1)(V2z+i.
Then $
x
and $
2
belong to C°°
and $ i ^ , ^ 2 ^ belong to 5 for any ip in S.
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