6533b862fe1ef96bd12c61ed

RESEARCH PRODUCT

Dimension gap under conformal mappings

Stanislav HenclTomi NieminenPekka Koskela

subject

Mathematics(all)General Mathematics010102 general mathematicsMathematical analysista111Hausdorff spaceMinkowski–Bouligand dimensionBoundary (topology)Dimension functionHausdorff dimensionEffective dimension01 natural sciencesConformal mapping010101 applied mathematicsBoundary behaviourPacking dimensionHausdorff dimensionMetric (mathematics)0101 mathematicsMathematics

description

Abstract We give an estimate for the Hausdorff gauge dimension of the boundary of a simply connected planar domain under p -integrability of the hyperbolic metric, p > 1 . This estimate does not degenerate when p tends to one; for p = 1 the boundary can even have positive area. The same phenomenon is extended to general planar domains in terms of the quasihyperbolic metric. We also give an example which shows that our estimates are essentially sharp.

yearjournalcountryeditionlanguage
2012-06-01Advances in Mathematics
10.1016/j.aim.2012.03.018http://juuli.fi/Record/0248645912