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.