12 1. A CLASS O F FUNCTIO N SPACES
Suppose that YliLo fi converges in
Sf.
Then YH^Lo^fi also converges in S',
i-e-
Yl^oi^fi^) converges for any ip G S. But then
oo oo
2=0 2= 0
2 = 0
2= 0
2=0 2= 0
Here the last two series are convergent, which proves the convergence in S' of
ET=o **?&
a n d h e n c e o f E2C=o 2ir/i.
We claim that J X o
2~iT/i
converges in S' for a N/r - e+. As in (1.1.18)
(1.1.19) sup(l + | z | ) - W '
+ d
| / ^
We set j A^/r + e+ i, so that i 0. Then
oo oo
^ 2 - - s u p ( l + M)-W'-+
d
)|/
i
(x)|^2-^||{/
j
}f
= 0
||
B
.
2=0 * 2=0
This proves that
oo
(l +
\x\)-Wr+dY,2~i°Mx)
2 = 0
converges uniformly in R^ . The claim, and the continuity of the embedding follow
immediately.
PROPOSITION
1.1.11. Let E satisfy the conditions in Definition 1.1.6. Then
the space Y{E) is complete.
PROOF.
The proof is the same as that of Proposition 1.1.9.
The elements of Y(E) are in general distributions, and not functions, and if
0 r 1 the functions in YL(E) are not in general distributions. However, the
following is true.
PROPOSITION
1.1.12. Suppose thate+, e- e K, r 0, and let E e S(e+,e-,r).
Then YL(E) = Y(E) if e+ Nmax{± - 1,0}.
PROOF.
Let e+
Armax{^
1,0}, and suppose that {/i}^
0
G E satis-
fies (1.1.9). We claim that for # 1
OO -
(1.1.20) (1 + R)-Wr+* J2 / \fi(v)\ dy
CHMZOWE.
i=0
JB(O,R)
If r 1, we have e+ 0, and (1.1.20) follows from Lemma 1.1.4.
Previous Page Next Page