Let n be an integer greater than one. Let D be a bounded domain in real Euclidean space of dimension n. We provide a bird's eye view of the relation between harmonic measure in D, the nature of the set of boundary accessible points of D, and, if n is equal to 2 and D is simply connected, the Riemann mapping of D. We prove new results and give new, easier proofs of known results. We prove in various ways that the set of boundary accessible points of D (which is not necessarily Borel if n is strictly greater than 2) is indeed measurable for harmonic measure. We also establish precisely for which ssets the pullback of harmonic measure under the Riemann mapping is equal to the Lebesgue measure.
Accessible points, harmonic measure, and the Riemann mapping
DI BIASE, FAUSTO;
2010-01-01
Abstract
Let n be an integer greater than one. Let D be a bounded domain in real Euclidean space of dimension n. We provide a bird's eye view of the relation between harmonic measure in D, the nature of the set of boundary accessible points of D, and, if n is equal to 2 and D is simply connected, the Riemann mapping of D. We prove new results and give new, easier proofs of known results. We prove in various ways that the set of boundary accessible points of D (which is not necessarily Borel if n is strictly greater than 2) is indeed measurable for harmonic measure. We also establish precisely for which ssets the pullback of harmonic measure under the Riemann mapping is equal to the Lebesgue measure.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
