Frequency of Sobolev and quasiconformal dimension distortion

Zoltán M. Balogh, Roberto Monti, Jeremy T. Tyson

Research output: Contribution to journalArticlepeer-review


We study Hausdorff and Minkowski dimension distortion for images of generic affine subspaces of Euclidean space under Sobolev and quasiconformal maps. For a supercritical Sobolev map f defined on a domain in Rn, we estimate from above the Hausdorff dimension of the set of affine subspaces parallel to a fixed m-dimensional linear subspace, whose image under f has positive Hα measure for some fixed α > m. As a consequence, we obtain new dimension distortion and absolute continuity statements valid for almost every affine subspace. Our results hold for mappings taking values in arbitrary metric spaces, yet are new even for quasiconformal maps of the plane. We illustrate our results with numerous examples.

Original languageEnglish (US)
Pages (from-to)125-149
Number of pages25
JournalJournal des Mathematiques Pures et Appliquees
Issue number2
StatePublished - Feb 2013


  • Hausdorff dimension
  • Potential theory
  • Quasiconformal mapping
  • Sobolev mapping
  • Space-filling mapping

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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