The boundary correspondence under quasiconformal mappings and VMO-Teichmuller space

TL;DR

This paper introduces a conformally invariant class of vanishing Carleson measures and symmetric homeomorphisms, establishing their mutual generation via quasiconformal mappings to develop a refined VMO-Teichmuller space on the real line.

Contribution

It constructs a new conformally invariant VMO-Teichmuller space by defining vanishing Carleson measures and symmetric homeomorphisms, overcoming previous invariance limitations.

Findings

Mutual generation of measures and homeomorphisms under quasiconformal mappings

Development of a conformally invariant VMO-Teichmuller space

Resolution of invariance issues in previous VMO-Teichmuller spaces

Abstract

In this paper, we introduce a class of vanishing Carleson measures with conformal invariance and corresponding strongly vanishing symmetric homeomorphisms on the real line and prove that they can be mutually generated under quasiconformal mappings. This is motivated by constructing a nice VMO-Teichmuller space on the real line, which completely removes the obstacle of the usual VMO-Teichmuller space that lacks conformal invariance and is repeatedly encountered in the papers [17,22-24].