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].