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

Abstract

We present a new characterization of Muckenhoupt $A_{\infty }$-weights whose logarithm is in $\mathrm{VMO}(\mathbb{R})$ 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.

arXiv commentIn the new version, we adjusted some proofs of the main theorem and deleted the appendix. This version is accepted by PAMS