# Get Analytic Capacity and Measure PDF

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

Example text

Extends t o a omeomorphism h of c( 2Jt::,) -1 k dist(cp ,span[q:>; k f -l}) > o. t By Mergelyan '8 theorem the coordinate generated by ~ on 66. Hence the F. and M. Izl=r I(z h(Z) ~ qJ'(z)/21r1. for h with k > 0 and by HI. We can € 0 < r < I, and ~(z)h(z)dz = 0 • )

1. , then C: ~ ~ o. in the compactly supported continuous functions. 5) and the obviously necessary conditions actually determine the Cauchy transform ~ of ~. ",f(z) = 0 allu let iJ. be a. -39- compactly supported measure such that fez} Then Proof: ~ (If iJ-; 0 almost everywhere. ~ ~(z) Replacing f weakly, and we must show xp Izl < pl. l by approximate identity: fz: as distributions. e . Let = 1, X p be a X p =0 and COO 0 off 6(0,p) Then the convolution D (y f is in p and Ceo, (z) fp f .. xp(z) converges to S f{z - OXp(S)dgdr] f in LlCK) + ill for any compact set K.

F It is easier t o attack the problem directly. inleve length if there is a number U ~ consists of finitely many analytiC Jordan curves surrounding in the usual sense of contour integration. open V ~ E contains regular neighborhood V £ of such that every E such that dV has length at most £. The infimum of such numbers painleve length of E. vinson [45J, is 0. £ is called the slight generaliz8,t ion of Painleve's theorem. ainleve length then there is a measure ~ on E such that K. z ¥E • Cons e qLlently , Proof; of' [Vn 1:=1 such that E z For Let