1.1. DEFINITIONS AND BASIC PROPERTIES 7
DEFINITION
1.1.2. If E e S(e+, £_, r), we denote by Ed-ls the space of sequences
of constants {si k}(i
k)eN0xzN s u c n
that {fi}iZo ^ ^ if the functions /^ are defined
by
fi = E sWXi,k i E N ° *
kezN
The space is normed by
\\{si,k}(i,k)eTS!0xZN\\Edis
= ||{/i}£oll^-
The following two lemmas are easy consequences of Definition 1.1.1, and will
play an important part in the following. The first one illustrates the importance of
condition (b) in the definition. The role of the maximal theorem is well formulated
by M. Frazier and B. Jawerth in [20], p. 228, and the lemma is essentially Lemma
3.1 in [20]; see also [21].
LEMMA
1.1.3. LetEeS(s+,S-,r),letbl,andsetXi,k=x(Qi,k{1b)) Then
there is a constant C such that for all sequences {si^} £ ^dis
II f
v~^
~ 1 °° II II f
\~^
1 °°
d-1-3)
l|{z.fcez« •i***)JhE ±
c
|||L
f e 6 Z
, «.**,* ta
Moreover, if a sequence
{Ujk}(i,k)e'NoxZN ^s
defined by t^k Yli
si,i
where the sum
is taken over all I such that Qij(b) D Qi,k, then
(1-1-4) ||{^}||
B d i 8
C||K
f c
}lb
d i s
-
PROOF.
By the lattice property of E we can assume that all s^k are nonnega-
tive. Set fi(x) = J2ieZN si:iXi,i(x) for i = 0, 1, .... Let x e Qi,k- Then
E
U,iXi,i(x)
= U,k = E
Sj»l
- E
Si 1
leZN
Qi,i(b)DQi,k Qi,i(b)3x
(1.1.5)
=
E
s^iAx)
^ E
Si i
iezN
QitiO)nQi,k&
^2
Si 1
- E
Si 1
'
Qi,i(b+2)DQi,k
Qi,lCB(x,p2-i)
where p = \{b + 3)y/N. It follows that
(L1'6)
(E
l e Z
- tW^Y ^ {T,leZN *i,lXi,l(*)y Cr E
Sh
Q M C B M - ' )
Cr2iN f Myy dy CrffMrMxY CMrtdMxY,
JB(x,p2-i)
where Cr = 1 for 0 r 1, and Cr depends on r and the number of nonzero terms
in the sum, i.e., on b and N, for r 1. The last inequality is true for all d 0, with
the constant C in addition depending on d, because of the fact that for i 0 the
radii of the balls appearing in the integral are bounded by p. The lemma follows
from (b) in Definition 1.1.1.
The next lemma shows the role of the constant £+. Denote by Sij, i,j G Z, the
Kronecker symbol, i.e., 5{j = 0 if i =^ j , 5^ = 1, so that {5ojfj}f^0 = {fo, 0,0,...}.
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