CHAPTER 1

Introduction

In 1994-199 5 a certain hop e wa s raise d o n th e possibilit y o f describin g remov -

able set s o f bounde d holomorphi c function s (th e so-calle d Painlev e problem , 100

years old ; o r Vitushkin' s problem , approximatel y 5 0 year s old) . Als o ther e wa s

a feelin g tha t th e proble m o f Vitushki n concernin g th e semi-additivit y o f analyti c

capacity (approximatel y 50-year-ol d proble m a s well ) coul d b e solved . Thi s hop e

was base d o n essentiall y tw o things : 1) o n th e impetu s give n t o th e researc h i n

this are a fro m work s o f Melniko v [37 ] an d Melnikov-Verder a [39] , an d 2 ) o n a

certain analyti c approach , whic h wa s gainin g popularit y du e t o th e work s o f Jone s

[25]-[27], Coifman-Jones-Semme s [11]Chris , t [5] , an d Mura i [38 ] fro m th e lat e

80's.

This approac h (which , i n fact , le d t o th e solutio n o f al l thos e problems ) wa s

based on using the real analysis methods to these complex analysis questions. Actu -

ally, it wa s based o n using T1,T6 theorem s fro m th e theor y o f Calderon-Zygmun d

operators. However , i n th e theor y o f Calderon-Zygmun d (CZ ) operators , T1,T6

theorems wer e proved i n the so-calle d homogeneou s spac e setting, whe n the under -

lying measure ha s the doublin g condition . Actually , T1,T6 were first prove d i n th e

Lebesgue measure setting by David-Journe [12], [13], David [15], [16], and David -

Journe-Semmes [14]. Th e generalizatio n o f T1,T6 theorem s fro m th e Lebesgu e

measure settin g eve n t o th e homogeneou s spac e settin g turne d ou t t o b e alread y

quite difficult . Thi s generalizatio n wa s obtaine d b y Chris t i n [5 ] exactl y fo r th e

purpose o f approaching Vitushkin' s conjectures .

This homogeneou s settin g seeme d t o b e s o natura l tha t i t wa s considere d on e

of the cornerstone s o f the theory . However , th e essenc e o f the ne w approac h t o th e

above-mentioned problem s require d gettin g ri d o f this cornerston e entirely . Thi s i s

just becaus e Vitushkin' s conjecture s dea l wit h ver y irregula r set s wit h n o homo -

geneity.

And this "gettin g rid of i s exactly what happened . Independently , an d exactl y

at th e same time, in [61] and i n [43] , the nonhomogeneous T l theore m wa s proved .

Tolsa's proof was tailored to the need of Vitushkin's problems, that i s to Cauch y

singular integra l operato r (i t use d th e notio n o f curvatur e o f measur e relate d t o

the Cauch y kernel) , an d Nazarov-Tfeil-Vol b erg's proo f wa s fo r genera l Calderon -

Zygmund operator s o n nonhomogeneou s spaces . Thi s generalit y turne d ou t t o b e

very essentia l fo r th e approache s t o Vitushkin' s conjecture s tha t follow .

However, nonhomogeneou s T l wa s onl y th e first ste p i n th e righ t direction ,

because Vitushkin' s problem s requir e nonhomogeneou s Tb theorem s rathe r tha n

Tl theorems . Thes e type s o f result s wer e obtaine d b y David-Mattil a [18] an d

David [19]. Th e geometri c descriptio n o f removabl e set s fo r bounde d analyti c

functions (th e proble m o f Painleve , o r Vitushkin ) followed . Bu t semiadditivit y o f

analytic capacit y coul d no t b e derive d fro m thi s resul t becaus e i t ha d qualitative ,

l

http://dx.doi.org/10.1090/cbms/100/01