By J. Garnett

ISBN-10: 3540060731

ISBN-13: 9783540060734

ISBN-10: 3540380035

ISBN-13: 9783540380030

Publication via Garnett, J.

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Extra info for Analytic Capacity and Measure

Sample text

Functional f -. f' (00) is linear. The set A(E,l) is convex and the So there is a unique Ahlfors function if and only if each extremal function is an extreme point of this convex set. Fisher obtained uniqueness by showing any extremal function is an extreme point. 2: If En Ahlfors functions for If each OEn are compact sets decreas ing to En E, we know' the converge to the Ahlfors function for E. rabedian functions for En converge? bedian function for for a discussion of this problem. measure * If so, the limit is a natural E.

Connected, we can assume t he curve of E and EJ. n R 11: ~ (,I, 6j =:: min(t(I'j),5/2). if ~ 1' . J # k. ). J Ej J by discs 6( ~ JO), ~ E E. L. + 1 OJ discs. ) < 4(1 + E)K(E) • J - Clearly there is a similar result if each component of the plane into at most In this nature 1s necessary. components. irwi se disjoint open discs 6(a . ) J J J -60:E r. , and thus infinite ro. l Hausdorff measure, but it has finite painleve length. Of course origin. depends only on the behavior of ~ h et) near the lhe next lemma follows directly from the definition.

An application is the explicit computation of set with a 0: ~(w) Then Iwl < 1 extends to be meromorphic on for a linear ~(E) E. 2~ Let E be a subset of the real axis. "r'\E) where Proof: £(E) For £(E)/4 is the inner arc length of E E. an interval this was proved in §l. •• + En Then First assume {Ej}~=l' is a closed interval of length E Then £(E). 2, 11m f(z)1 E. =m iE 2 f dt t'="Z E and dt (t - :x) 2 + y2 ' z x + iy ~ so that Then Re ef(z) > 0 IF(z) du ---2 1 +u -OQ 7r ~'2 and F(Z) satisfies I~

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Analytic Capacity and Measure by J. Garnett


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