Strongly symmetric homeomorphisms on the real line with uniform continuity

TL;DR

This paper studies strongly symmetric homeomorphisms on the real line, focusing on their properties under uniform continuity and their quasiconformal extensions, revealing conditions under which they are preserved under composition and inversion.

Contribution

It demonstrates that uniformly continuous strongly symmetric homeomorphisms on the real line are preserved under composition and inversion, unlike the general case.

Findings

Uniform continuity ensures preservation under composition and inversion.

Barycentric extension of uniformly continuous homeomorphisms induces a vanishing Carleson measure.

Composition and inverse of these quasiconformal homeomorphisms also induce vanishing Carleson measures.

Abstract

We investigate strongly symmetric homeomorphisms of the real line which appear in harmonic analysis aspects of quasiconformal Teichm\"uller theory. An element in this class can be characterized by a property that it can be extended quasiconformally to the upper half-plane so that its complex dilatation induces a vanishing Carleson measure. However, differently from the case on the unit circle, strongly symmetric homeomorphisms on the real line are not preserved under either the composition or the inversion. In this paper, we present the difference and the relation between these two cases. In particular, we show that if uniform continuity is assumed for strongly symmetric homeomorphisms of the real line, then they are preserved by those operations. We also show that the barycentric extension of uniformly continuous one induces a vanishing Carleson measure and so do the composition and the inverse of those quasiconformal homeomorphisms of the upper half-plane.