By J. Garnett
Booklet by means of Garnett, J.
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Additional info for Analytic Capacity and Measure
1 the case ~: be continuous on the Riemann sphere and analytic f E. off a compact set on of ""Sm 1 Q. F. 6: LSI' f = Assume il f(oo) ~ O. e. if and only if 00. ~ 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.
Assume (1) holds. 1. Now and assume (ii) holds. t this implies R. 2. nd analyt ic on all curves 00. t ~ J for all Then fez) - Q(z). = D\E. D\,E. 2. The set of in D. 2 extends analytically to D. This result is extended somewhat in the next section. kened. We now describe these f I fELIce which are almost everywhere -43- Cauchy transforms. 3) and there is a number M such that whenever Rl , ••. 4) may depend on the grid M f. It It. The infimum of the will be called the total The total variation of the restriction of U will be denoted by v(f ,U) and is defined by with the constraint that only rectangles contained in U are used.
2 ~ ~t~~ = 1 is ero = -'\r' on . 2). easure but ~(E) 'h is a. ratheodory outer measure [38, p. 53], is frequently inf inite, and thus not suitable for an upper -61- bound. We now introduce two other set functions, which, while not additive, will be more Let S(z; 0) use~ul. having its sides parallel to the axes. )} '11 '"---' J J J j=l The next lemma follows easily from the definitions. 3: Now let On z = (p + qi)2 all lines o ~(E) -n-l if and only if ~(E) = , p, q, odd integers, so that x ~ a2- n , y = b2- n , a, b, integers.
Analytic Capacity and Measure by J. Garnett