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. i fl i2=ii*,(/)ii 2 2 - Hence, by (0.3), ||/|| 2 « {Y^Qe^ \ S Q\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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