By Boris Nikolaevič Apanasov (auth.), Julian Ławrynowicz (eds.)
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Extra resources for Analytic Functions Błażejewko 1982: Proceedings of a Conference held in Błażejewko, Poland, August 19–27, 1982
E. e. in a point at which u and ~ which the relation (14) holds. Yet then, the preceding inequality yields sup s Thus, we have established (19) under the hypothesis (20). e. that u(x+lllxle )' s we have [ ~ Then, if ~(x+lllxle) lu(xl-u(x+lllxle) III xl s] p luk(xl-uk(x+lllxle s ) [lllxl s =[ ~ k~l[ u(x) > min ui(x+lllxle) =u(x+lllxle s )' 1~~q s lu(x)-uk(x+lllxle s )I p III xl ] q ]p = l~(x)-~(x+lllxles) lllxl I p ] 52 Petru Caraman and, as above, in the hypothesis (20), we obtain (19) in this case as well.
M;r" y 1 Consequently, = M;r" M;r, :SOM;r", as desired. is rectifiable, and p E F°D. (r 1) J Cds < 00, so i t remains to prove y the opposite inequality. , e; > 0 and r (x) = d (x, cD U Eo UE ). D. thesis [15J, we establish that p is continuous in t. where 42 Petru Caraman p is and, as in the case of the preceding proposition, we show that bounded in Rn. p EF* (r 1)' Thus we have only to prove that least that ! 2, we obtain or at Indeed, arguing as in the case ! [x+e:r (x) y ]dm(y) ds (x) Y1 B 1+e:!
1 Finally, since u*/h is admissible for cap (E ,E ,D) with h ~1 p 0 1 and taking into account (15), we obtain cap (E ,E ,D) ~ 1 fl~u*IPdm ~ -L fpPdm ~ fpPdm. 1 p 0 hP D hP Since p all such is an arbitrary function of p, cap (E ( 17) p FO~(rl)' taking the infimum over we deduce that 0' E l' D) ~ MO~r P - M r 1 - p' E . which, together with (12), yields (1) if (11) holds for When E1 o satisfies the condition (11), we repeat the above argument for El instead of Eo and finally consider the function v=l -u*/h, which is admissible for cap (E ,E ,D).