CALDERON'S FORMULA AND A DECOMPOSITION OF L 2 (R") 1 7 used o n th e bi-dis c b y Chan g an d Fefferma n i n 198 0 [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 197 7 articl e o n Hard y space s [21 ] by Coi f ma n an d Weiss and th e 198 0 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 J Q. 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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