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Quasiconformal and Sobolev distortion of dimension

Jeremy T. Tyson

Abstract

We review a selection of the literature on the distortion of metric notions of dimension under quasiconformal, quasisymmetric, and Sobolev mappings. Our story begins with Gehring's landmark 1973 higher integrability theorem for quasiconformal maps, along with its implications for the distortion of Hausdorff dimension. Astala's 1994 solution to the planar higher integrability conjecture led to renewed interest in the subject in two dimensions. We continue with results from the 2000s and 2010s on the distortion of dimension by Sobolev maps, including estimates for dimension increase for generic elements in parameterized families of subsets. In the abstract metric setting, Pansu's notion of conformal dimension provides a key quasisymmetric invariant which has been useful in a wide range of applications. We briefly review relevant facts about conformal dimension, highlighting results of interest in the Euclidean setting. We conclude with recent work of the author in collaboration with Chrontsios Garitsis and with Fraser, extending the previous theory to interpolating dimensions and providing new insight into both quasiconformal classification and conformal dimension.

Quasiconformal and Sobolev distortion of dimension

Abstract

We review a selection of the literature on the distortion of metric notions of dimension under quasiconformal, quasisymmetric, and Sobolev mappings. Our story begins with Gehring's landmark 1973 higher integrability theorem for quasiconformal maps, along with its implications for the distortion of Hausdorff dimension. Astala's 1994 solution to the planar higher integrability conjecture led to renewed interest in the subject in two dimensions. We continue with results from the 2000s and 2010s on the distortion of dimension by Sobolev maps, including estimates for dimension increase for generic elements in parameterized families of subsets. In the abstract metric setting, Pansu's notion of conformal dimension provides a key quasisymmetric invariant which has been useful in a wide range of applications. We briefly review relevant facts about conformal dimension, highlighting results of interest in the Euclidean setting. We conclude with recent work of the author in collaboration with Chrontsios Garitsis and with Fraser, extending the previous theory to interpolating dimensions and providing new insight into both quasiconformal classification and conformal dimension.
Paper Structure (11 sections, 25 theorems, 23 equations, 3 figures, 1 table)

This paper contains 11 sections, 25 theorems, 23 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Basic definitions and notation
  3. Acknowledgements
  4. Quasiconformal mappings: history and background
  5. References
  6. Quasiconformal dimension distortion: the story begins
  7. Sobolev dimension distortion: a new perspective
  8. Conformal dimension
  9. References
  10. Sobolev and QC distortion of dimension interpolants
  11. References

Key Result

Theorem 3.1

For each $K \ge 1$ and $n \ge 2$, there exists $p = p(n,K) > n$ so that if $f:\Omega \to \Omega'$ is a $K$-quasiconformal mapping between domains in ${\mathbb{R}}^n$, then $f \in W^{1,p}_{{loc}}(\Omega:{\mathbb{R}}^n)$.

Figures (3)

  • Figure 1: The von Koch curve
  • Figure 2: (a) The Sierpiński gasket $\mathop{\mathrm{SG}}\nolimits$; (b) a QS deformation of $\mathop{\mathrm{SG}}\nolimits$, given as the invariant set for a new self-similar IFS.
  • Figure 3: Examples of multi-dimensional polynomial sequence sets $E_{p,2}^\alpha$ (left) and $E_{p,3}^\alpha$ (right)

Theorems & Definitions (45)

  • Theorem 3.1: Bojarski, $n=2$; Gehring, $n \ge 3$
  • Theorem 3.2: Gehring--Väisälä, $n=2$; Gehring, $n \ge 3$
  • Example 3.3
  • Example 3.4
  • Definition 3.5
  • proof : Proof of \ref{['eq:qc-haus-dim-distortion-b']}
  • Theorem 3.6: Astala
  • Theorem 3.7: Smirnov
  • Theorem 3.8: Prause--Smirnov
  • Proposition 3.9
  • ...and 35 more