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

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.