A description of $A_{\infty }$-weights for VMO

TL;DR

This paper characterizes a special class of weights with vanishing mean oscillation in terms of geometric and measure-theoretic properties, providing new insights into their structure and related homeomorphisms.

Contribution

It introduces a novel characterization of $A_{ ext{infty}}$-weights with VMO logarithm using vanishing Carleson measures and doubling weights, linking harmonic analysis and geometric function theory.

Findings

Characterization of $A_{ extinfty}$-weights with VMO logarithm via vanishing Carleson measures.

Description of strongly symmetric homeomorphisms using geometric quantities.

Establishment of connections between weights, measures, and homeomorphisms.

Abstract

We present a new characterization of Muckenhoupt $A_{\infty}$-weights whose logarithm is in $\operatorname{VMO}(\mathbb{R})$ in terms of vanishing Carleson measures on $\mathbb{R}_+^2$ and vanishing doubling weights on $\mathbb{R}$. This also gives a novel description of strongly symmetric homeomorphisms on the real line by using a geometric quantity.