Weak $$A_\infty $$ Weights and Weak Reverse Hölder Property in a Space of Homogeneous Type

Abstract

In the Euclidean setting, the Fujii–Wilson-type $$A_\infty $$ weights satisfy a reverse Holder inequality (RHI), but in spaces of homogeneous type the best-known result has been that $$A_\infty $$ 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 $$A_\infty $$ 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.