Sharp nonremovability examples for H\"older continuous quasiregular mappings in the plane

TL;DR

This paper constructs specific nonremovable sets for H"older continuous quasiregular mappings in the plane, demonstrating the sharpness of known removability criteria and providing extremal distortion examples.

Contribution

It improves previous results by constructing nonremovable sets with finite positive Hausdorff measure and introduces a precise extremal quasiconformal mapping with enhanced H"older regularity.

Findings

Constructed nonremovable sets with finite positive Hausdorff measure.

Developed a planar quasiconformal map with H"older exponent exceeding 1/K.

Showed the sharpness of the Hausdorff dimension threshold for removability.

Abstract

Let $\alpha\in(0,1)$, $K\geq 1$, and $d=2\frac{1+\alpha K}{1+K}$. Given a compact set $E\subset\C$, it is known that if $\H^d(E)=0$ then $E$ is removable for $\alpha$-H\"older continuous $K$-quasiregular mappings in the plane. The sharpness of the index $d$ is shown with the construction, for any $t>d$, of a set $E$ of Hausdorff dimension $\dim(E)=t$ which is not removable. In this paper, we improve this result and construct compact nonremovable sets $E$ such that $0<\H^d(E)<\infty$. For the proof, we give a precise planar $K$-quasiconformal mapping whose H\"older exponent is strictly bigger than $\frac{1}{K}$, and that exhibits extremal distortion properties.