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 C° ° 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