2 1. INTRODUCTIO N not quantitative , nature . Thi s was a very big road block . Als o the abovementione d nonhomogeneous Tb theore m ha d a horrendously toug h proof, whic h mad e slightl y unrealistic th e hop e t o ge t it s quantitativ e version . Very soon after David-Mattil a an d David's Tb theorems, a sort o f the complet e theory o f Calder6n-Zygmun d operator s i n nonhomogeneou s settin g appeare d i n Nazarov-Treil-Volberg's [44]-[47] . Thes e Tb theorem s agai n concerne d a genera l Calderon-Zygmund operato r rathe r tha n a concrete one (an d thi s turned ou t t o b e essential). Another thin g i s that Nazarov-Treil-Volberg' s Tb theorem s wer e o f quantita - tive nature . Firstly , thi s gav e anothe r (an d mor e streamlined ) proo f o f David' s geometric descriptio n o f removabl e set s fo r bounde d analyti c function s (th e prob - lem o f Painleve , o r Vitushkin) . Secondly , whe n Tols a mad e a decisiv e mov e b y solving Vitushkin' s semi-additivit y conjectur e i n 2001 , his proof wa s based o n thi s quantitative Tb theore m o f Nazarov-Treil-Volberg . The mai n goa l i n wha t follow s i s t o mak e a n expositio n o f th e techniqu e un - derlying th e theor y o f nonhomogeneou s Calderon-Zygmun d operators . W e choose to illustrat e thi s techniqu e b y usin g i t t o solv e two concret e problems . The firs t proble m ha s bee n alread y mentioned . I t concern s Vitushkin' s prob - lem o f semiadditivit y o f analyti c capacity . Thi s proble m wa s solve d b y Tols a i n [64]. Thi s article is forcefully no t self-contained i t contain s man y reference s t o th e nonhomogeneous Tb theore m o f Nazarov-Treil-Volberg. Th e reader will find belo w the self-containe d proo f o f thi s difficul t result especiall y i f th e reade r wil l adap t the expositio n below to the cas e n 2. W e are saying that becaus e we are dealing , as a rule, with the cas e n 3 in what follows . Thus , w e are working with Lipschit z harmonic function s an d Lipschit z harmoni c capacit y (introduce d b y Mattil a an d Paramonov) rathe r tha n wit h bounde d analyti c function s (gradient s o f Lipschit z harmonic functions ) an d Vitushkin' s analyti c capacity . A s i t i s mor e natura l t o work wit h harmoni c function s i n R n tha n o n th e plane , th e expositio n deal s wit h general n. Bu t th e reade r wil l b e abl e t o extrac t th e "analyti c capacity " case , i f necessary. I n thi s cas e ou r approac h differ s slightl y fro m Tolsa' s approach . Thi s i s because onl y fo r n 2 , the notio n o f Menger-Melnikov's curvatur e o f measure (se e [37], [39] , [17], [32], [33] , [64]) i s related wit h Lipschit z harmoni c capacity , an d s o we are oblige d t o wor k withou t thi s fantasticall y beautifu l an d powerfu l tool . On e can d o this fo r an y dimensio n n includin g n = 2. The secon d problem , whic h serve s a s a tes t o f th e strengt h o f th e theor y o f nonhomogeneous Calderon-Zygmun d operators , is the problem of finding necessar y and sufficien t condition s fo r th e boundednes s o f two-weighted Calderon-Zygmun d operators. W e give such necessary an d sufficient condition s in very natural terms, if the operato r i s the Hilber t transform , an d th e weights hav e some mild smoothness . For a certai n mode l Calderon-Zygmun d operator , on e ca n giv e a necessar y an d sufficient condition s o n weights , withou t assumin g anything . W e mea n her e th e so-called Martingal e transform , whic h i s sometimes considere d a s a dyadi c versio n of the Hilber t transform . Notice tha t th e two-weigh t proble m fo r singula r operator s seeme d t o b e ex - tremely difficult adequat e tool s seeme d t o b e no t available . Th e theor y o f non - homogeneous Calderon-Zygmun d operators , a s w e wil l se e below , a t leas t give s considerable hop e i n understandin g suc h two-weight problems . Th e secon d par t o f our expositio n give s several result s t o thi s end .
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