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The Hausdorff dimension of planar elliptic measures via quasiconformal mappings

Ignasi Guillén-Mola

TL;DR

This work develops a quasiconformal mapping framework to bound the Hausdorff dimension of planar elliptic measures ω_{Ω,A} arising from uniformly elliptic matrices A. By relating L_A-harmonic functions to harmonic ones through a carefully constructed qc map φ, the authors transfer known harmonic-measure dimension results (Makarov, Wolff) to the elliptic setting, obtaining σ-finite gauge bounds and sharp dimension estimates depending only on the ellipticity constant. In the symmetric-det-1 case, they exploit additional regularity to obtain finer results, including Hölder, VMO, DMO, and Sobolev-regularity regimes, with dimension gaps and gauge-containment results that unify and extend prior planar results. The work also develops a detailed treatment of Green functions, harmonic factorization, and the push-forward of measures under qc changes of variables, providing a robust toolkit for understanding elliptic measure in the plane and its delicate interaction with coefficient regularity and boundary geometry.

Abstract

In this paper, we obtain new bounds for the Hausdorff dimension of planar elliptic measure via the application of quasiconformal mappings, with these bounds depending solely on the ellipticity constant of the matrix. In fact, in our case studies, we find a quasiconformal mapping that relates the elliptic measure in a domain to the harmonic measure in its image domain, allowing us to deduce bounds for the dimension of the elliptic measure from the known results on the harmonic side. This extends previous works of Makarov, Jones and Wolff.

The Hausdorff dimension of planar elliptic measures via quasiconformal mappings

TL;DR

This work develops a quasiconformal mapping framework to bound the Hausdorff dimension of planar elliptic measures ω_{Ω,A} arising from uniformly elliptic matrices A. By relating L_A-harmonic functions to harmonic ones through a carefully constructed qc map φ, the authors transfer known harmonic-measure dimension results (Makarov, Wolff) to the elliptic setting, obtaining σ-finite gauge bounds and sharp dimension estimates depending only on the ellipticity constant. In the symmetric-det-1 case, they exploit additional regularity to obtain finer results, including Hölder, VMO, DMO, and Sobolev-regularity regimes, with dimension gaps and gauge-containment results that unify and extend prior planar results. The work also develops a detailed treatment of Green functions, harmonic factorization, and the push-forward of measures under qc changes of variables, providing a robust toolkit for understanding elliptic measure in the plane and its delicate interaction with coefficient regularity and boundary geometry.

Abstract

In this paper, we obtain new bounds for the Hausdorff dimension of planar elliptic measure via the application of quasiconformal mappings, with these bounds depending solely on the ellipticity constant of the matrix. In fact, in our case studies, we find a quasiconformal mapping that relates the elliptic measure in a domain to the harmonic measure in its image domain, allowing us to deduce bounds for the dimension of the elliptic measure from the known results on the harmonic side. This extends previous works of Makarov, Jones and Wolff.
Paper Structure (28 sections, 45 theorems, 195 equations)

This paper contains 28 sections, 45 theorems, 195 equations.

Table of Contents

  1. Introduction and main results
  2. Preliminaries
  3. Notation
  4. Reduction to bounded Wiener regular domains
  5. Small perturbation of the constant case
  6. Quasiconformal mappings
  7. Distortion under quasiconformal mappings
  8. Distortion of gauge Hausdorff measures
  9. Harmonic factorization of A-harmonic functions
  10. A-harmonic functions under quasiconformal mappings
  11. The A-harmonic conjugate in simply connected domains
  12. Harmonic factorization in general domains
  13. Symmetric matrices with determinant 1
  14. Push-forward of the harmonic measure and proof of Theorems \ref{['main thm general domains']} and \ref{['main thm general domains and continuous matrices']}
  15. Proof of Theorem \ref{['main thm general domains']}
  16. ...and 13 more sections

Key Result

Theorem 1.1

There exists a universal constant $C_M>0$ such that $\omega_\Omega \ll {\mathcal{H}}^{\varphi_{1,C_M}}$ for every simply connected domain $\Omega \subset {\mathbb R}^2$.

Theorems & Definitions (94)

  • Theorem 1.1: Makarov
  • Theorem 1.2: Wolff
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Corollary 1.7
  • Definition 2.1: Wiener regular
  • Remark 2.2
  • Lemma 2.3
  • ...and 84 more