Distortion of Hausdorff measures and improved Painlevé removability for quasiregular mappings

拟正则曲线：奇点的可去性

Quasiregular curves: Removability of singularities

本文证明了欧氏空间有界拟正则曲线的Painlevé定理，通过推广体积形式基本不等式至校准并证明Caccioppoli不等式，建立了定性锐利的奇点可去性结果，并将Bonk-Heinonen定理推广至该设定。

Sharp Examples for Planar Quasiconformal Distortion of Hausdorff Measures and Removability

In his celebrated paper on area distortion for quasiconformal mappings, Astala showed optimal area distortion bounds and dimension distortion estimates for planar quasiconformal mappings. He asked (Question 4.4) whether a finer result held, namely absolute continuity of Hausdorff measures under push-forward by quasiconformal mappings. This was proved in one particular case relevant for removability questions, in joint work of Astala, Clop, Mateu, Orobitg and the author ("Distortion of Hausdorff measures and improved Painlev'e removability for bounded quasiregular mappings", Duke Math J., to ap…

Sharp estimate on the inner distance in planar domains

We show that the inner distance inside a bounded planar domain is at most the one-dimensional Hausdorff measure of the boundary of the domain. We prove this sharp result by establishing an improved Painlevé length estimate for connected sets and by using the metric removability of totally disconnected sets, proven by Kalmykov, Kovalev, and Rajala. We also give a totally disconnected example showing that for general sets the Painlevé length bound $κ(E) \leπ\mathcal{H}^1(E)$ is sharp.

Quasiconformal maps, analytic capacity, and non linear potentials

In this paper we prove that if $ϕ:\C\to\C$ is a $K$-quasiconformal map, with $K>1$, and $E\subset \C$ is a compact set contained in a ball $B$, then $$\frac{\dot C_{\frac{2K}{2K+1},\frac{2K+1}{K+1}}(E)}{\diam(B)^{\frac2{K+1}}} \geq c^{-1} (\frac{γ(ϕ(E))}{\diam(ϕ(B))})^{\frac{2K}{K+1}},$$ where $γ$ stands for the analytic capacity and $\dot C_{\frac{2K}{2K+1},\frac{2K+1}{K+1}}$ is a capacity associated to a non linear Riesz potential. As a consequence, if $E$ is not $K$-removable (i.e. removable for bounded $K$-quasiregular maps), it has positive capacity $\dot C_{frac{2K}{2K+1},\frac{2K+1}{K+1…