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 Hölder Inequality (RHI) but in spaces of homogeneous type the best known result has been that $A_{\infty }$ weights satisfy only a weak Reverse Hölder Inequality. In this paper, we compliment the results of Hytönen, Pérez 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 Hölder weights fails in a general space of homogeneous type. We prove most of these purely non-dyadic results using convenient dyadic systems and techniques.

17 pages. v2: introduction updated, Section 8 and some references added, some typos fixed

References in corpus (2)

• The Gehring Lemma in Metric Spaces
• Gaussian heat kernel bounds through elliptic Moser iteration

Cited by in corpus (2)

• A Dyadic Gehring Inequality in Spaces of Homogeneous Type and Applications
• Haar bases on quasi-metric measure spaces, and dyadic structure theorems for function spaces on product spaces of homogeneous type