A real-variable construction with applications to BMO-Teichm\"uller theory

TL;DR

This paper develops a real-variable method to construct a specific weight function with BMO properties that is not Muckenhoupt, impacting the study of BMO-Teichmüller theory and chord-arc curves.

Contribution

It introduces a novel real-variable construction of a weight function with BMO but not Muckenhoupt properties, with applications to Teichmüller theory.

Findings

Constructed a doubling weight with BMO log that is not A_infinity.

Applied the construction to BMO-Teichmüller space.

Extended the approach to chord-arc curve analysis.

Abstract

With the use of real-variable techniques, we construct a weight function $\omega$ on the interval $[0, 2\pi)$ that is doubling and satisfies $\log \omega$ is a BMO function, but which is not a Muckenhoupt weight ($A_\infty$). Applications to the BMO-Teichm\"uller space and the space of chord-arc curves are considered.