Non-local Gehring Lemmas in Spaces of Homogeneous Type and Applications
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Non-local Gehring Lemmas in Spaces of Homogeneous Type and Applications Pascal Auscher1,2 · Simon Bortz3 · Moritz Egert1 · Olli Saari4,5 Received: 5 July 2017 © Mathematica Josephina, Inc. 2019
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∞ weights and fractional elliptic divergence form equations. We write our results in spaces of homogeneous type. Keywords Gehring’s lemma · (non-local) Reverse Hölder inequalities · Spaces of homogeneous type · (very weak) A∞ weights · C p weights · Fractional elliptic equations · Self-improvement properties Mathematics Subject Classification Primary: 30L99; Secondary: 34A08 · 42B25
1 Introduction Gehring’s lemma [9] establishes the open-ended property of reverse Hölder classes. If 1/q ˆ ˆ 1 1 q u dx u dx (1.1) |B| B |B| B
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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Olli Saari [email protected]
Extended author information available on the last page of the article
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with q > 1 and all Euclidean balls B ⊂ Rn , then
1 |B|
1/(q+)
ˆ
q
u q+ dx B
1 |B|
ˆ u dx B
for a certain > 0 and all Euclidean balls. This self-improving property has proved to be an important tool when studying elliptic [7,10] and parabolic [11] partial differential equations as well as quasiconformal mappings [16]. In this case, one has to enlarge the ball in the right-hand side. We come back to this. In this work, we are concerned with reverse Hölder inequalities when the right-hand side is non-local. Understanding an analogue of Gehring’s lemma in this generality turned out to be crucial in [4], where we prove Hölder continuity in time for solutions of parabolic systems. The non-local nature arises from the use of half-order time derivatives. The ambient space being quasi-metric instead of Euclidean is also an assumption natural from the point of view of parabolic partial differential equations. Hence, we shall explore these non-local Gehring lemmas in spaces of homogeneous type. It is well known that Gehring’s lemma holds for the so-called weak reverse Hölder inequality where the right-hand side of (1.1) is an average over a dilated ball 2B. We replace the single dilate by a significantly weaker non-local tail such as ∞ k=0
2−k
1 k |2 B|
ˆ u dx 2k B
and certain averages over additional functions f and h that have a special meaning in applications. The main result of this pap
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