2018
DOI: 10.1016/j.aim.2018.06.002
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Differentiability and Poincaré-type inequalities in metric measure spaces

Abstract: We demonstrate the necessity of a Poincaré type inequality for those metric measure spaces that satisfy Cheeger's generalization of Rademacher's theorem for all Lipschitz functions taking values in a Banach space with the Radon-Nikodym property. This is done by showing the existence of a rich structure of curve fragments that connect nearby points, similar in nature to Semmes's pencil of curves for the standard Poincaré inequality. Using techniques similar to Cheeger-Kleiner [CK15], we show that our conditions… Show more

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Cited by 14 publications
(15 citation statements)
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“…This enlarged curve has a well‐defined metric derivative and integral, and the ones for curve fragments are obtained by restriction. For a similar discussion, see [3, 14].…”
Section: General Poincaré Resultsmentioning
confidence: 99%
See 1 more Smart Citation
“…This enlarged curve has a well‐defined metric derivative and integral, and the ones for curve fragments are obtained by restriction. For a similar discussion, see [3, 14].…”
Section: General Poincaré Resultsmentioning
confidence: 99%
“…If instead rw>r023 for some w, then the claim follows easily from doubling and using a single ball. By applying Maz'ya's trick, that is, applying the above argument with the truncated function uk±false(xfalse)=±false(min(maxfalse(±f,2k1false),2k)2k1false)in place of f and at level 2k1 in place of 2k, and since Lipuk±=1Ek1±Ek±Lipfalmost everywhere (see, for example, [3, Lemma 2.6]), then analogously as (5.7) we obtain μfalse(Ek±false)2p+1D3pC2prp2kp2C2Bfalse(Ek1±Ek±false)Lipfalse[ffalse](x)pdμ,which when multiplied by 2kp and summed over k gives B|f|pdμ2p+<...>…”
Section: General Poincaré Resultsmentioning
confidence: 99%
“…This enlarged curve has a metric derivative and integral, and the ones for curve fragments are obtained by restriction. For a similar discussion see [14,4]. We will employ the proof of the characterization of (global) Poincaré inequalities from [23, Lemma 5.1], in order to prove new characterizations.…”
Section: General Poincaré Resultsmentioning
confidence: 99%
“…All the parameters can be made quantitative. 7 Proof works for geodesic metric spaces, although it is only stated for manifolds. See also [19,Proposition 6.12].…”
Section: Corollaries and Applicationsmentioning
confidence: 99%