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Analytic Capacity and Measure by J. Garnett

By J. Garnett

Publication via Garnett, J.

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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. 2: Let bounded set E h(t) and ~(E) H(t) be measure functions.

O . Thus a necessary cond it ion that Tf oV l' 0. is be bounded with respect to the norm that the linear f unctional IlgIIE • Using the Hahn-Banach and Riesz representation theorems, we see that this condition is also suffi cient. t i sfy 0-(z) for Let E be a compact set and let f(<>:t) " O. z , E There is a measure be analytiC f iJ. on if and only If there is a constant E such that such -53- for all g analytic near When this is the case we may take E. 6) is necessary. Proof: Tf Then the linear functional has an extension to and this extension is represented by a measure "Where z ¥ E.

Tion. such that fez) = ~(z) Let f I! A(E,l) Then there is a unique measure almost everywhere. 4. 9 below). vin's theorem [40 J does provide 6. necessary and sufficient condition f or a function to be a Cauchy tr ansform. 3(b) as simple corollarie s. theorem al~o has f(z) = 0-(z), z "E Suppose V Th i s be a regular ne ighborh ood of where is a measure on fJ. and let E g E. be analyt ic on Let V. (O . Thus a necessary cond it ion that Tf oV l' 0. is be bounded with respect to the norm that the linear f unctional IlgIIE • Using the Hahn-Banach and Riesz representation theorems, we see that this condition is also suffi cient.

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