We show that any Carnot group G with sufficiently many deformable directions contains a measure zero set N such that every Lipschitz map f: G→ ℝ is differentiable at some point of N. We also prove that model filiform groups satisfy this condition, extending some previous results to a class of Carnot groups of arbitrarily high step. Essential to our work is the question of whether the existence of an (almost) maximal directional derivative Ef(x) in a Carnot group implies the differentiability of a Lipschitz map f at x. We show that such an implication is valid in model Filiform groups for directions that are outside a one-dimensional subspace of horizontal directions. Conversely, we show that this implication fails for every horizontal direction in the free Carnot group of step three and rank two.

Universal differentiability sets in Carnot groups of arbitrarily high step / Pinamonti, A.; Speight, G.. - In: ISRAEL JOURNAL OF MATHEMATICS. - ISSN 0021-2172. - 2020/240:(2020), pp. 445-502. [10.1007/s11856-020-2069-x]

Universal differentiability sets in Carnot groups of arbitrarily high step

Pinamonti A.;
2020-01-01

Abstract

We show that any Carnot group G with sufficiently many deformable directions contains a measure zero set N such that every Lipschitz map f: G→ ℝ is differentiable at some point of N. We also prove that model filiform groups satisfy this condition, extending some previous results to a class of Carnot groups of arbitrarily high step. Essential to our work is the question of whether the existence of an (almost) maximal directional derivative Ef(x) in a Carnot group implies the differentiability of a Lipschitz map f at x. We show that such an implication is valid in model Filiform groups for directions that are outside a one-dimensional subspace of horizontal directions. Conversely, we show that this implication fails for every horizontal direction in the free Carnot group of step three and rank two.
2020
Pinamonti, A.; Speight, G.
Universal differentiability sets in Carnot groups of arbitrarily high step / Pinamonti, A.; Speight, G.. - In: ISRAEL JOURNAL OF MATHEMATICS. - ISSN 0021-2172. - 2020/240:(2020), pp. 445-502. [10.1007/s11856-020-2069-x]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11572/276261
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