Sharp nonremovability examples for Hölder continuous quasiregular mappings in the plane

Abstract

Let $\alpha \in (0,1)$, $K\ge 1$, 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 $\alpha$-Hölder 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 Hölder exponent is strictly bigger than $\frac{1}{K}$, and that exhibits extremal distortion properties.

19 pages, 1 figure