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Exceptional Sets for Quasiconformal Mappings in General Metric Spaces II

Behnam Esmayli, Pekka Koskela, Khanh Nguyen

Abstract

A homemorphism between domains in $\mathbb R^n$, $n\ge 2$ is quasiconformal, with its intricate analytic and geometric consequences, if the (pointwise) linear dilatation -- a purely metric quantity -- is uniformly bounded. Gehring proved that it will suffice to verify the uniform bound up to a set of measure zero as long as we can show that the dilatation is finite outside a subset of finite Hausdorff--$(n-1)$ measure. In short, we say that we can allow an exceptional codimension $1$ subset. In the metric setting, it has been proved, roughly speaking, that one can allow an exceptional codimension $p$ subset, $p \ge 1$, if the source space satisfies a $p$-Poincaré inequality. We prove, effectively, the sharpness of the latter claim.

Exceptional Sets for Quasiconformal Mappings in General Metric Spaces II

Abstract

A homemorphism between domains in , is quasiconformal, with its intricate analytic and geometric consequences, if the (pointwise) linear dilatation -- a purely metric quantity -- is uniformly bounded. Gehring proved that it will suffice to verify the uniform bound up to a set of measure zero as long as we can show that the dilatation is finite outside a subset of finite Hausdorff-- measure. In short, we say that we can allow an exceptional codimension subset. In the metric setting, it has been proved, roughly speaking, that one can allow an exceptional codimension subset, , if the source space satisfies a -Poincaré inequality. We prove, effectively, the sharpness of the latter claim.

Paper Structure

This paper contains 35 sections, 33 theorems, 112 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Removability of sets in metric quasiconformality
  3. Main results
  4. Comparison to non-self-similar Sierpiński carpets
  5. Preliminaries
  6. Ahlfors regularity
  7. Absolute continuity and upper-gradients
  8. Poincaré inequality
  9. Pointwise inequalities
  10. Coarea integration
  11. Pencil of curves
  12. Poinaré inequality on $\mathbb R^2$
  13. Poincaré inequality on the bow-tie
  14. The outline of our proof of the $p$-Poincaré inequality
  15. The Space $({\mathbf{X}},d,\mu)$
  16. ...and 20 more sections

Key Result

Theorem 1.1

Let $f \colon \Omega \to \Omega'$ be a homeomorphism between domains in $\mathbb{R}^n, n\ge 2$. Suppose that $E \subset \Omega$ has $\sigma$-finite ${\mathcal{H}}^{n-1}$-measure, $h_f(x) <\infty$ at every $x \in \Omega \setminus E$, and for some $H < \infty$, $h_f(x) \le H$ almost everywhere on $\Om

Figures (13)

  • Figure 2.1: Coarea integration.
  • Figure 2.2: Truncated cone and the maximal function.
  • Figure 2.3: The pencil of curves from $x$ to $y$ in proving the $1$-Poincaré inequality on $\mathbb R^2$.
  • Figure 2.4: The pencil of curves from $x$ to $y$ is forced to "pinch" and lose volume, resulting in a weaker Poincaré inequality.
  • Figure 3.1: Part of the space $\mathbf X$ (shaded).
  • ...and 8 more figures

Theorems & Definitions (69)

  • Theorem 1.1: Kallunki-Kosk-2003
  • Theorem 1.2: bal-kosk-rogKosk-wild
  • Theorem 1.3
  • Remark 1.4
  • Definition 2.1: Kosk-wild
  • Lemma 2.2
  • Lemma 2.3: HKST:15
  • Corollary 2.4
  • Lemma 2.5
  • Lemma 2.6: Haj:Ko:met, Theorem 3.3
  • ...and 59 more