Grassmannian Frequency of Sobolev Dimension Distortion

Zoltán M. Balogh, Pertti Mattila, Jeremy T. Tyson

Research output: Contribution to journalArticle

Abstract

Let (Formula presented.) be a supercritical Sobolev map from a domain (Formula presented.) in (Formula presented.) into a complete separable metric space. For a fixed integer (Formula presented.), (Formula presented.), and (Formula presented.), we estimate from above the Hausdorff dimension of the set of elements (Formula presented.) in the Grassmannian (Formula presented.) (equipped with a Riemannian metric) such that (Formula presented.) has positive (Formula presented.)-dimensional Hausdorff measure. The proof relies heavily on the homogeneous structure of both (Formula presented.) and the Stiefel manifold (Formula presented.) of orthogonal injections of (Formula presented.) into (Formula presented.). A novel feature of the proof is a Morrey–Sobolev-type embedding theorem on the product manifold (Formula presented.), valid for Sobolev maps which factor through the evaluation map (Formula presented.).

Original languageEnglish (US)
Pages (from-to)505-523
Number of pages19
JournalComputational Methods and Function Theory
Volume14
Issue number2-3
DOIs
StatePublished - Oct 31 2014

Keywords

  • Grassmannian manifold
  • Hausdorff dimension
  • Sobolev mapping
  • Stiefel manifold
  • potential theory

ASJC Scopus subject areas

  • Analysis
  • Computational Theory and Mathematics
  • Applied Mathematics

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