CALDERON'S FORMULA AND A DECOMPOSITION OF L
2(R")
17
used o n th e bi-dis c b y Chan g an d Fefferma n i n 1980 [16]. Oddl y enough ,
as fa r a s w e know, th e formul a i n it s simples t for m (a s i n Theore m (1.4),
in R
n
wit h homogeneous dilations only) was not presented until Uchiyama's
1982 paper [84 ] on constructively decomposin g BMO. Our presentation, es-
pecially Theore m (1.4), follow s Uchiyama' s quit e closely . Th e notio n o f a
molecule occur s i n th e 1977 articl e o n Hard y space s [21] by Coi f ma n an d
Weiss and th e 1980 article [79 ] by Taibleson an d Weiss . Smoot h molecule s
were introduced, i n a particular case , by Wilson [86] . Thi s idea als o occur s
implicitly i n Uchiyama' s wor k whe n h e show s that a molecul e ca n be writ -
ten a s a weighted su m o f dilation s o f atoms ; i.e. m
Q
= X^o^~^"
+ 1 )
^ ,
where b
Q
. i s smooth and supported i n 2
JQ.
Fro m this, Uchiyama implic -
itly obtain s th e boundednes s resul t describe d abov e fo r Calderon-Zygmun d
convolution operators from a n atomic, rather than molecular, version of The-
orem (1.14). Th e general definition o f a smooth molecule was first presented
in [36].
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