By J. Garnett

ISBN-10: 3540060731

ISBN-13: 9783540060734

ISBN-10: 3540380035

ISBN-13: 9783540380030

Publication by means of Garnett, J.

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Te Ga,rabedian function" 1 bl cp( z) = - - + - + 27Ti Z in HcoC \I) ~ite Ilmil such that '1' < <0 - and C 1 likely, this is false. 5: E2 C such that compact. Assume the assertiON in the above problem is true. and there a,re constants there is a measure ~ an i1(z) E C4 such that for any compact set such that =J'E ~t+:(~ 0 € A(E,C3 ) . Show E -28- J dil- = 1'\E) §6. 1 is the following special result on semi-additivity of analytic capacity. 1 (Pommerenke [66J): Let compact connected sets and let K Minkowski sum.

6) is necessary. Proof: Tf Then the linear functional has an extension to and this extension is represented by a measure "Where z ¥ E. ~. 6). E = wi th lies z ¥ ij. " and Ig(Odl-lct;;) ~ ilCz) . 6: with :f(oo) = O. :for all z ~ E Let E be compact and let There is 3. 1 measure i:f and only if there is on f be analytic on E such that M< 00 S\E fez) = (l(z) such that n II \f(~)1 k=l whenever Proof: a l , ... n {E 'k and €. C. lytic near limit of f'unctions of the f'orm n g(z) ~ Lz \'~ k=l - a k E Ilflll < M.

Rhen can only vanish when E of positive E lies on a sufficiently E has zero length. 2 above and the sharpest result here is due to Ivanov [52]. Its intricate proof below is not used later and may be Skipped. Let o 5. 11. 1: (Ivanov) • Let r be a 2- r q:{s} = [, (s) : is real, jer(s) j ::: 1/2. 1) u 12 ~s) - cp(sO) . Is - s 0 I 0 where M depends only on that if is a subset of E r. ds

### Analytic Capacity and Measure by J. Garnett

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