Maximal functions, $A_ \infty$-measures and quasiconformal maps

Abstract:

In the study of quasiconformal maps, one commonly asks, "Which classes of maps or measures are preserved under quasiconformal maps?", and conversely, "When does the said preservation property imply the quasiconformality of the map?". These questions have been studied by Reimann and Uchiyama with respect to the classes of BMO functions, maximal functions, and ${A_\infty }$-measures. But, both authors assumed additional analytic hypotheses to establish the quasiconformality of the map. In this paper we utilize the geometry of quasiconformal maps to eliminate these auxiliary hypotheses and present results in the cases of maximal functions and ${A_\infty }$-measures.