14

1. A CLASS OF FUNCTION SPACES

and Lemma 1.1.7 shows that \fi+j * ^j(x)\ CMr^fi+j{x). Thus

oo

i=0

oo

=

cj2\\(s+)i({fJ}r=o)\\ic^2-^\\{f3}~=0rE,

i=0 2=0

as claimed. By Lemma 1.1.4 the sums in (1.1.21) are almost everywhere absolutely

convergent, which justifies the operations, and proves the inclusion YL(E) C Y{E).

•

PROPOSITION

1.1.13. Suppose thats+, s- eR,r 0, and let E G S(e+,£_,r).

Then Y(E) c V,L, i.e., the elements ofY(E) are distributions of order at most L,

* / L N m a x { ± - l , 0 } - e + .

PROOF.

It is proved as in Proposition 1.1.12 that $ ^

0

2 "

i L

/ i G L\oc. The

result follows easily. •

In the following two theorems we formulate the basic results for the spaces

YL(E) and Y(E), respectively. Note that by Proposition 1.1.12 for a given E G

S(£+,£_,r) these spaces coincide if e

+

iVmax{^ — 1,0}.

THEOREM

1.1.14. Let s+, S-, r 0, and let E G 5(e+,£:_,r). Then, for any

p such that 0 p oo and — ^ — s+, and any integer M such that M S-,

the following conditions on a function f G Lr,ioc are equivalent {with the usual

modification if'p = oo):

(i) / e YL(E).

(ii) / G Lpjoc, and the functions

(1.1.22) g0(x) = ||/||

M

B(*,i)).

(1.1.23) gi(x) =

2iN^([

\A¥f(x)\»dz) " /orieN,

\JB(x,2- i ) /

satisfy

(1-1-24) ||{ft},~olUo°-

(iii) / 6 Lpjcc, and (1.1.24) is satisfied with go defined by (1.1.22) and

9i{x) =

2iN^£M(f,B(x,2-i),Lp)

fori G N.

(iv) / has a representation

(1.1.25) / = ^2 si,kai,k,

(i,k)e'NoxZN

converging in Lr^\oc, where the functions a^(2

_

*(x + k)) — bi^{x) satisfy

(1.1.26) biik G C f Q - l , ^ ) , and \\b^k\\CM 1,

and the coefficients s^k are such that (1.1.24) is satisfied with

(1.1.27) 9i{x)= J2

\3i,k\x%Ax)

* € N

0

.

kezN

Moreover, the convergence in (1.1.25) holds absolutely almost everywhere, and g —

J2iLo 9i belongs to