J. Garnett's Analytic Capacity and Measure PDF

By J. Garnett

ISBN-10: 3540060731

ISBN-13: 9783540060734

ISBN-10: 3540380035

ISBN-13: 9783540380030

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L K. containing the origin. has a simple pole ~ is univalent. t "'. i'\E)'::;: i'\T) :: i'\K). 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~

Ble number of lines. pproximate identity introduced above, and write Xp * XP E: 00 C. ,. Let f f(z)dz R € ~ -1 rr f P is the Cauchy transform of M -2 dxdy . 5). • ,R n ~~, then \' I L j Consequently Let R €: IhJ) 1 S v( f) . l Let be so small thllt nd let U5 is the o-neighborhood of dR. lp l(u be a weak star limit of the o/ 2 ) Then choose g E = V(f p 'U 5/ 2 ) < 8/2 if CoCR) £/2, where such that P < 6/2. 3, v(f,u o) < hi) . l fez) is supported on E. s. 3. By Corollary -47- There is an alternate proof, which we merely outline.

G € In Chapter IV there is an example of a totally disconnected K ana a function f E C(K,l) such that f f 0 but E Prove that there is a totally disconnected compact set A{E,l) such that <<0 but is not a Cauchy transform. 2 fails. One strategy is outlinea· below, but there coula be a simpler construction. :: is a. regular neighborhood of lin) with En 'V(Ln) > 0, sufficiently close to n oV has length exceeding n. C z )j ds > C where c :is a constant independent of totally disconnected compact set n.

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Analytic Capacity and Measure by J. Garnett

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