4 1. INTRODUCTIO N operators. S o to avoi d thi s disaste r w e have to avoi d ba d mutua l position s o f Q, R- This goa l i s attaine d b y considerin g rando m decompositio n (wit h respec t t o ran - dom dyadi c lattice ) o f ou r function s an d averagin g procedure . Thi s randomnes s compensates fo r th e degeneracie s o f th e measur e becaus e i t "smoothen s up " th e degeneracies, (no t i n th e strictes t sens e o f thi s word , however) . I n anothe r con - text th e rando m dyadi c lattic e o f course alread y appeare d i n harmonic analysis , i n [22], fo r example . Decompositio n o f functions t o estimat e th e Calder6n-Zygmun d operator i s no t somethin g ne w either se e [11] . Bu t th e combinatio n o f thes e tw o ideas i s wha t allow s u s t o wi n ove r degeneracie s o f measures . Th e machiner y o f this i s represente d below alon g wit h tw o application s (mentione d already ) o f thi s technique. We do not cove r here an y relations o f nonhomogeneous harmoni c analysi s wit h the Geometri c Measur e Theory . Th e reaso n i s that, o f course, i n dimension s n 3 this relatio n i s no t ye t established . O n th e plan e th e relation s ar e s o pervasiv e and s o interestin g tha t discussio n o f thi s woul d requir e a n entir e book . Actually , such a boo k exists . Lectur e note s [52 ] o f Pajo t cove r th e geometri c par t o f th e topic i n grea t detail . O n th e othe r hand , th e analyti c par t o f som e recen t result s are onl y sketche d i n thes e lectur e notes . I n th e presen t expositio n w e tr y t o fill this ga p b y presentin g th e self-containe d analyti c approac h t o tw o chose n topics : semiadditivity o f Lipschitz harmoni c capacity, an d two-weight estimate s for certai n singular operators . Le t u s mentio n recen t survey s o n thi s an d relate d topics : [17] , [32], [33] , [68]. Plan: Th e first severa l chapter s ar e devote d t o a sor t o f potential theor y wit h Calder6n-Zygmund signe d kernel s (i n thi s case , Ries z signe d kernels) . Th e non - homogeneous Calder6n-Zygmun d (CZ ) theor y i s developed startin g wit h Chapte r 7. Theorem s 7. 1 and 8. 1 alread y ar e the result s o f the nonhomogeneou s Calderon - Zygmund theory . I n particular, i n Theorem 8.1 , the probabilisti c languag e alread y appears. Th e proof s o f Theorems 7. 1 an d 8. 1 ar e finished i n Chapte r 14 . Th e res t of the exposition is devoted to using the same technique in two-weight problem s fo r certain singula r operators thi s start s i n Chapte r 15 . Fo r th e Hilber t transfor m w e give a necessar y an d sufficien t conditio n o f the two-weigh t boundedness , bu t onl y under a n extra assumptio n o f "smoothness " o f weights. However , i n Chapter 2 2 we give a necessary an d sufficien t conditio n o f the two-weigh t boundednes s o f a thre e member family o f operators, the first member of the family i s the Hilbert transform , and th e secon d an d th e thir d member s ar e certai n simpl e maximal operators . Thi s necessary an d sufficien t conditio n i s obtaine d withou t an y extr a assumption s o n weights. Th e sam e typ e o f criteri a ca n b e obtaine d fo r singula r operator s calle d Martingale transforms . I t i s known that th e Martingale transfor m i s closely relate d to th e Hilber t transform see , fo r example , [1] , [3] . W e d o no t obtai n i t her e fo r Martingale Transforms , bu t th e reade r ca n look a t [50] , [49] for mor e information . Acknowledgements. Thes e ar e th e note s o f 1 0 lecture s give n i n Ma y o f 200 2 at th e NSF-supporte d CBM S conferenc e hel d a t th e Universit y o f North Carolin a at Chape l Hill . I a m ver y gratefu l t o th e NS F fo r th e support , an d t o Professor s Joseph Cim a an d Ale x Masterson fo r th e excellent organization . I am very gratefu l to th e stimulatin g audience . I n th e first chapter s w e follo w closel y th e idea s fro m Tolsa's pape r [64 ] wit h modification s dictate d b y a ) th e lac k o f comple x analysi s for highe r dimensions , an d b ) th e fac t tha t th e magnificien t too l o f Melnikov - Menger's curvatur e (se e [37] , [39] , [52] ) i s "cruell y missing " i n highe r dimensions .
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