Weak A_\infty weights and weak Reverse Hölder property in a space of homogeneous type

Abstract

In the Euclidean setting, the Fujii-Wilson-type A(infinity) weights satisfy a reverse Holder inequality (RHI), but in spaces of homogeneous type the best-known result has been that A(infinity) weights satisfy only a weak reverse Holder inequality. In this paper, we complement the results of Hytonen, Perez and Rela and show that there exist both A(infinity) weights that do not satisfy an RHI and a genuinely weaker weight class that still satisfies a weak RHI. We also show that all the weights that satisfy a weak RHI have a self-improving property, but the self-improving property of the strong reverse Holder weights fails in a general space of homogeneous type. We prove most of these purely non-dyadic results using convenient dyadic systems and techniques.