A∞ TYPE CONDITIONS FOR GENERAL MEASURES IN R1

Abstract

Given two Borel measures $\mu$, $\nu$ in ${\mathbf{R}}^1$, finite on compacts, we deal with the conditions $v\in A_{\infty }(\mu )$ and $\nu \in A_{\infty }^{+}\left(\mu \right)$. We point out several characterizing statements, expressed in terms of maximal functions, $A_p$ classes, reverse Hölder inequalities, exponential type conditions, and their symmetric versions. We use ideas from [1] and [19] and extend their results to this rather general context (in the one-sided case we assume that $\mu$ and $\nu$ are mutually absolutely continuous). We consider an application to the Gehring lemma.