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
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