The Hausdorff dimension of planar elliptic measures via quasiconformal mappings

2024/07/30 by Ignasi Guillén-Mola, Guillén-Mola, Ignasi
Mathematics · #28A12 #28A78 #30C62 #31A15 #31B05 #35J08 #35J25. Secondary: 28A25 #Analysis of PDEs (math.AP) #Analytic and geometric function theory #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.2407.21145

openalex publication_date 2024/07/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

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.

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