Generalized dimension distortion under planar Sobolev mappings

Pekka Koskela

University of Jyvaskyla
pkoskela@maths.jyu.fi

Abstract: It is well-known that a planar quasiconformal mapping preserves the class of sets of Hausdorff dimension strictly less than two. In fact, Astala's theorem gives essentially sharp estimates for the dimension distortion. A mapping of exponentially integrable distortion can map a set, say, of dimension one to a set of dimension two. However, estimates for the dimension distortion can be proved in refined scales. As a tool, we describe the setting for general (continuous) Sobolev mappings.