2. PRELIMINARIE S O N CAPACITIE S 9 definitions o f Lipschitz harmoni c capacities ) an d onl y i n R 2 \ E fo r T theory . Thi s difference i s not a s trivial a s one coul d think . Fo r example , i f we introduc e C+(E) : = sup{|(A/ , 1)| : |V/(x)| 1 , \fx e R n \ E, Af = » e M+(E)} , we hit th e followin g unsolve d Problem. I s C+{E) x 7+(E' ) wit h constant s independen t o f E, n 3? In principle , thi s coul d hav e bee n a proble m fo r n = 2 a s well . Th e reaso n why the differenc e betwee n a.e . restrictio n an d "outsid e o f E restriction " turn s ou t to b e inessentia l fo r n = 2 is th e following . Fortunately , t o prov e Theore m 2. 2 i t is sufficien t t o prov e i t (wit h constan t independen t o f E) onl y fo r E fro m a ver y special class , namely , onl y fo r E tha t ar e a finit e unio n o f smoot h curve s o n th e plane (an d thes e ar e undoubtedl y th e set s o f zer o plana r Lebesgu e measure) . Fo r n 3 this remar k ma y no t work , an d th e proble m remains . O f course , al l o f thi s is a problem onl y i f Lebesgue measur e o f E i s positive . We just explaine d why T theory is worse than 7 theory. Bu t i n another respect , 7 theor y i s wors e tha n T theory , becaus e fo r th e Cauch y potentia l (dealin g wit h T) w e ca n us e Comple x Analysis , an d fo r Ries z potential s (dealin g wit h 7 ) i t i s unavailable (a t leas t fo r n 3). Thi s creates two serious complications fo r adaptin g Tolsa's theore m fo r th e proo f o f Theorem 2.1: 1) Simpl e localizatio n estimate s fo r th e potentia l shoul d b e obtaine d withou t Complex Analysis . Thi s i s done i n Chapte r 3 . 2) Suitabl e analog s o f Menger' s curvatur e an d al l magi c formula e ar e "cruell y missing" b y the expression of Guy David in higher dimensions (se e [20]) . An d thes e tools were essentially used by Tolsa in proving Theorem 2.2 . W e have to circumven t this difficult y too . Thi s i s done i n Chapte r 5 . We want t o discus s a bit th e genesi s o f the problems . Th e problem s hav e bee n in the focus of attention of many analysts recently. Ther e is a fantastically beautifu l geometric sid e o f the stor y involvin g P . Jones ' travelin g salesma n proble m an d it s analogs, bu t w e d o no t touc h thi s her e a t all ! W e refe r th e reade r t o th e concis e introduction t o th e geometri c sid e o f the stor y i n the boo k o f H. Pajo t [52] . The analytica l par t o f th e stor y wa s starte d b y th e articl e o f M . Chris t [5] . He proved Theore m 2. 2 fo r so-calle d Ahlfor s regula r sets . Th e struggl e wit h extr a regularity brough t th e specia l Tb theore m o f G . Davi d an d P . Mattil a [18 ] an d David [19 ] and th e series of works by Tolsa. A t the sam e time, general approac h t o nonhomogeneous Calderon-Zygmun d theor y i n Nazarov-Treil-Volberg' s [43]-[47 ] turned ou t t o b e possible , and , a s w e will see , useful . Thi s approac h doe s no t us e extra regularit y o f measures. Th e reade r ca n hav e a brief bu t ver y good accoun t o f this in reviews of David [17 ] and Mattila [32] , [33]. Briefl y speaking, the outcome of the approach developed in [43]-[47 ] is that estimate s are simpler and the statement s are mor e flexibl e an d giv e rise t o quit e genera l nonhomogeneou s Tb theorem s wit h bounds. Thes e bound s wer e use d b y Tols a i n [64 ] t o prov e Theore m 2.2 , whic h crowned al l of the previou s efforts . As we have alread y said , ou r goa l her e wil l b e t o giv e simultaneousl y th e self - contained proo f o f Theore m 2. 2 an d o f it s Lipschit z harmoni c analo g i n Theore m 2.1. On e o f the byproduct s o f al l o f thi s i s tha t o n th e plan e al l o f ou r capacitie s are equivalent : 7 x r x 7+ x r + .
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