INTRODUCTION
5
Let us show how we can obtain a representation of / i n terms of a par-
ticular spanning set by a simple discretization of the integral formula (0.10).
We say that a cube Q
cRn
i s a dyadic cube if
Q = Qvk = {* R":
2'vk{
xt 2-
v{kt
+ 1), i = 1, 2, .. . , n}
for som e v e Z an d k = ( ^ , k2, .. . , fcn) Z* . Le t * ? = {(?^ : i/ e
Z, k E Z
n}.
Fo r Q e ^ le t /(£) ) b e th e sid e lengt h o f thi s cub e an d
T{Q) = Q x [^ , /(G) ] c R^
1
. Not e tha t R^
+I
= \J Qe^ T(Q) , where ,
except for a set of measure 0, this is a disjoint union.
We now put
T(Q)
t§-t(P,*f)(y)
and, i f s
Q
^ 0 ,
'T(Q)
°t
t
We then have the decomposition
(o.i2) / w = EvcW«
It i s no t difficul t t o se e tha t th e function s a Q(x) satisf y f a
Q
= 0 an d
Suppa^
c
3Q (wher e cQ i s the dilate of Q wit h side length c time s that
of Q) . Moreover , a
Q
e ° an d satisfie s certai n natura l estimate s (se e
§1 for details) . Observ e that th e coefficient s s
Q
ar e closel y relate d t o th e
Littlewood-Paley function g
l
w e introduced above; e.g.
ifli2=ii*,(/)ii22-
Hence, by (0.3), ||/||
2
« {Y^Qe^
\
SQ\2)1'2
^ ^
s an
iniportan t feature of all the
decompositions w e will discuss that the norm of / i n each of the functio n
spaces we study is equivalent to an expression involving only the magnitudes
of the coefficients i n the representation of / .
In the next section we will discuss in detail a variant of this representation,
and its converse, for /
L2(Rn),
using a compactly supported kernel instead
of the kernel t{dPJdt) . T o exhibit other features of this decomposition we
give a detailed presentation of how these ideas apply to Lipschitz functions.
Ultimately, w e will obtain such a result for all the spaces mentioned in the
first paragraph , by making use of their Littlewood-Paley characterizations (as
described above).
Note tha t in formul a (0.12) th e functions a Q(x) var y with the functio n
/ . Late r (in §6) we will obtain a more canonical decomposition of the form
(0.13)
/ = E / ^ V Q
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