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 complement 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.
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Acknowledgments
T.A. is supported by an NSF graduate student fellowship. T.H. and O.T. are supported by the European Union through the ERC Starting Grant 278558 “Analytic-probabilistic methods for borderline singular integrals”. They are also part of Finnish Centre of Excellence in Analysis and Dynamics Research.
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Anderson, T.C., Hytönen, T. & Tapiola, O. Weak \(A_\infty \) Weights and Weak Reverse Hölder Property in a Space of Homogeneous Type. J Geom Anal 27, 95–119 (2017). https://doi.org/10.1007/s12220-015-9675-6
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DOI: https://doi.org/10.1007/s12220-015-9675-6