8 2. PRELIMINARIE S O N CAPACITIE S By definition , (2.1) 7 + ( £ ) 7 ( £ ) - Or, obviously , set s removabl e fo r Lipschit z harmoni c function s ar e removable for subharmonic Lipschit z harmoni c functions . W e will prov e her e (i n the next 12 5 pages) tha t th e converse i s true. Moreover , w e prove THEOREM 2.1 . There exists a constant A depending only on the dimension n such that (2.2) 7 (E)A1+(E). In the case n = 2 , the attention o f many mathematician s wa s attracte d t o the comparison o f analytic capacit y an d "positiv e analyti c capacity" . Th e capacities 7,7+ ar e very clos e relative s o f these classica l objects , which , abusin g th e conven- tional notation, w e will call T, T+ (usuall y analyti c capacity is called 7, here we use the symbo l T). Definition. Le t E b e a compact se t in C. T{E) : = sup{ lim \zf(z)\ : / G Hol(C\£) , \f(z)\ 1 V . G C \ E , /(OO ) = 0}, z—oo r+(E) : = sup{ lim \z f(z)\ : f(z) = / ^ , / / G M+(E), \f(z)\ 1 Vz G C \ E} . z-+oo J z-Q By definition , (2.3) T + (E)T(E). Parallel t o Theorem 2.1 , we will prov e her e th e following grea t resul t o f X. Tolsa [64] that th e converse to (2.3) is true. Thi s result brough t th e solution of a semiad- ditivity of analytic capacity conjectur e o f Vitushkin an d the conjecture o f Melnikov that ever y se t of positive T support s positiv e measur e o f order 1 (meaning tha t /x(J5(x,r)) r, Vx G C,r 0) and finite Menger's curvature se e [37]. The proo f i n [64] was obliged t o refer man y time s t o the theory o f nonhomo- geneous Calder6n-Zygmun d operators , especiall y t o result s fro m Nazarov-Treil - Volberg [46] , [47]. Ou r exposition, i n thi s respect , ca n be considere d no t onl y as provin g Theore m 2.1 , but also i t migh t tur n ou t to be the first self-containe d exposition o f the proo f o f Tolsa's Theore m 2.2. THEOREM 2.2 . There exists a constant A such that (2.4) T(E) AT+{E). The reader will notice soon that ther e is a great similarit y betwee n 7 and T and between 7+ an d T+. On e can consider Theorem 2.1 to be a correct multidimensiona l analog of Theorem 2.2. Exploring similarit y o f definitions, on e can see that 7 + is defined b y imposing conditions o n potential k * /i, wher e k i s a vector value d Ries z kernel , an d T+ is defined b y imposin g condition s o n potential K * /i, wher e K{z 1 Q = -^- ^ is the (complex-valued) Cauch y kernel. I n both cases , the kernels are Calderon-Zygmund (CZ) kernel s of order n 1, when we are in Mn, n 2. There ar e also some differences . I n one respect, T theory i s more difficul t tha n 7 theor y (w e compare them , o f course, onl y fo r n = 2), because th e restriction on potential k * /i is imposed a.e . i n M2 for 7 theor y (i t follows fro m th e equivalent
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