Abstract
We prove a self-improving property for reverse Hölder inequalities with non-local right-hand side. We attempt to cover all the most important situations that one encounters when studying elliptic and parabolic partial differential equations. We present applications to non-local extensions of \(A_{\infty }\) weights and fractional elliptic divergence form equations. We write our results in spaces of homogeneous type.
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Acknowledgements
We thank Tuomas Hytönen for an enlightening discussion on the topics of this work that led to the results extending the \(A_{\infty }\) class and Carlos Pérez for pointing out the connection to the \(C_p\) class. We also thank an anonymous referee for suggesting that our results should apply to the fractional divergence form equation of Shieh–Spector [25] rather than the toy model investigated in an earlier version of our manuscript.
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The first and third authors were partially supported by the ANR project “Harmonic Analysis at its Boundaries,” ANR-12-BS01-0013. This material is based upon work supported by National Science Foundation under Grant No. DMS-1440140 while the authors were in residence at the MSRI in Berkeley, California, during the Spring 2017 semester. The second author was supported by the NSF INSPIRE Award DMS 1344235. The third author was supported by a public grant as part of the FMJH.
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Auscher, P., Bortz, S., Egert, M. et al. Non-local Gehring Lemmas in Spaces of Homogeneous Type and Applications. J Geom Anal 30, 3760–3805 (2020). https://doi.org/10.1007/s12220-019-00217-z
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DOI: https://doi.org/10.1007/s12220-019-00217-z
Keywords
- Gehring’s lemma
- (non-local) Reverse Hölder inequalities
- Spaces of homogeneous type
- (very weak) \(A_\infty \) weights
- \(C_p\) weights
- Fractional elliptic equations
- Self-improvement properties