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By J. Garnett

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Extra resources for Analytic Capacity and Measure

Example text

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. Moreover o by fubini's theorem and the def'inition of weak derivative (see [51, p. 14]). Therefore f ~ §2. e. A Characterization of Cauchy Transforms Let f be a locally integrable function.

When this is = Vq(f). riation of also that f v(f) is attained over the grid = VQ(f). 1: Klz - wi. 8: ~ E C(E,l). Let §3. I~I(L); 0 Prove that I~I (J) ; 0 tha,t be a measure on a compa,ct set ~ J if E such that for every straight line L. Prove is a rectifiable curve. riation, and s. Theorem of Havin Throughout this sect ion we fix a. compact set f, supposed, as before, to be analytic on We want to know when there is a meas ure z (E. s a measurable extension at bounded var i ation on C. f It is easier t o attack the problem directly.

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. 4) implies that to the Borel sets. bly additive, bounded extension This can be seen via the Riesz representation theorem, or directly using the usual exhaustion arguments.

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