Analytic Functions Błażejewko 1982: Proceedings of a by Boris Nikolaevič Apanasov (auth.), Julian Ławrynowicz (eds.) PDF

By Boris Nikolaevič Apanasov (auth.), Julian Ławrynowicz (eds.)

ISBN-10: 3540127127

ISBN-13: 9783540127123

ISBN-10: 3540386971

ISBN-13: 9783540386971

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Additional info for Analytic Functions Błażejewko 1982: Proceedings of a Conference held in Błażejewko, Poland, August 19–27, 1982

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We obtain In the hypotheses of the preceding proposition, u is Fran the preceding proposition and lemna, we deduce the following COROLLARY. ~ obtain for sets) an equivalent definition (3) cap (C 'C ,0) p O 1 (CO,c1eD closed disjoint £i cap (C ,C"D) P 0 where the functions u admissible for cap (C ,C ,D) p O 1 ~ supposed to 35 n R About the Equality between the p-Module and the p-Capacity in be ACL in 0, not only in PROPOSITION 2. closed If 0 0 - (Co UC ) , 1 Co'C, C. D ~ ~ disjoint is bounded, then ~, cap p (C 'C ,0) = inf f IvulPdm, O 1 u 0- (C UC ) o 1 where the infimum is taken ~ all u EC' in 0 the boundary values on Co and 2£ 1 C 1 0 - (Co UC,) and with (our paper [7), Lemma 10).

X)] x+1; y ~ XED k (m are said to be = 1 ,2, ... ) such that Then (1) holds. Using the notation of Lemma 3, clearly, the arc families Dk k n Ei of E ,E ~ two sets such that o 1 D with Dk n Ei 'I ~ (i = ~ 0, where the infimum is taken over all Dk~ 1,2, ••• ) Em m THEOREM 1. (11) if of all continua in to contain the complement of ~ ball then m y; 1, wn B ! J...! 1 (c) of Vaisala's paper [22J, we have Mpr E M rk k P +E M r m O P m +E M r q 1 P q' where r and r O m =r 1 =¢ q yields M r O =Mp r q1 =0 pm (m, q = 1 ,2, ••• ) • Then, taking into account Lemma 3, it follows that we may suppose, without loss of generality, that D itself is a domain, since otherwise, we can establish (11) for each component of D, separately.

E. the corresponding the p-capacity in the cases (iii) and (iii') inequality with q '1t ==), we have, to suppose for instance, that all (k =1,2, ••• ) are locally Lipschitzian with a fixed Lipschitz constant K < =, or at least that the set of all these constants is bounded (which is equivalent). Let q(x,y) = Ix-yl/(1+lx 2 1) 1/2(1+IYI2) 1/2 tance between inf{q(x,y) x and y. If : x EEl' y EE }. 2 PROPOSITION 5. 3) . This proposition means that if ro is the family of all bounded curves, which are not rectifiable, then M r = o.

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Analytic Functions Błażejewko 1982: Proceedings of a Conference held in Błażejewko, Poland, August 19–27, 1982 by Boris Nikolaevič Apanasov (auth.), Julian Ławrynowicz (eds.)


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