The Lip-lip equality is stable under blow-up

Schioppa, Andrea

doi:10.1007/s00526-016-0957-z

Cited by 8 publications

(8 citation statements)

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“…A more recent application of this theory is the proof that at generic points blowups / tangents of differentiability spaces are still differentiability spaces, see [Sch15a]. Interestingly, that proof goes by contrapositive, and this theory comes to rescue as it works without the assumption of a differentiable structure.…”

Section: Introductionmentioning

confidence: 99%

Derivations and Alberti representations

Schioppa¹

2016

Advances in Mathematics

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110

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Abstract. We relate generalized Lebesgue decompositions of measures in terms of curve fragments ("Alberti representations") and Weaver derivations. This correspondence leads to a geometric characterization of the local norm on the Weaver cotangent bundle of a metric measure space (X, µ): the local norm of a form df "sees" how fast f grows on curve fragments "seen" by µ. This implies a new characterization of differentiability spaces in terms of the µ-a.e. equality of the local norm of df and the local Lipschitz constant of f . As a consequence, the "Lip-lip" inequality of Keith must be an equality. We also provide dimensional bounds for the module of derivations in terms of the Assouad dimension of X.MSC: 53C23, 46J15, 58C20

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Section: Introductionmentioning

confidence: 99%

Derivations and Alberti representations

Schioppa¹

2016

Advances in Mathematics

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110

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“…Because of (F6) the function F i can be glued together to get a (4/L + √ 2)-Lipschitz function f as in [Sch16b,Thm. 4.8].…”

Section: Preliminary Resultsmentioning

confidence: 99%

Lipschitz functions with prescribed blowups at many points

Marchese

Schioppa

2019

Calc. Var.

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In this paper we prove generalizations of Lusin-type theorems for gradients due to Giovanni Alberti, where we replace the Lebesgue measure with any Radon measure µ. We apply this to go beyond the known result on the existence of Lipschitz functions which are non-differentiable at µ-almost every point x in any direction which is not contained in the decomposability bundle V (µ, x), recently introduced by Alberti and the first author. More precisely, we prove that it is possible to construct a Lipschitz function which attains any prescribed admissible blowup at every point except for a closed set of points of arbitrarily small measure. Here a function is an admissible blowup at a point x if it is null at the origin and it is the sum of a linear function on V (µ, x) and a Lipschitz function on V (µ, x) ⊥ .

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“…if one has a (1, P )-Poincaré inequality but not a (1, P − ∆)-Poincaré, the minimal P -weak upper gradient and the minimal (P − ∆)-weak upper gradient can be different). One may check that this is not the case for our examples; this is unavoidable in the context of taking blow-ups as discussed in [Sch15a].…”

Section: Introductionmentioning

confidence: 91%