6533b7d9fe1ef96bd126cda8
RESEARCH PRODUCT
Universal differentiability sets and maximal directional derivatives in Carnot groups
Gareth SpeightEnrico Le DonneAndrea Pinamontisubject
Pure mathematicsCarnot groupGeneral MathematicsDirectional derivative01 natural sciencesdifferentiaaligeometriasymbols.namesake0103 physical sciencesFOS: MathematicsCarnot group; Directional derivative; Lipschitz map; Pansu differentiable; Universal differentiability set; Mathematics (all); Applied MathematicsMathematics (all)Point (geometry)Differentiable function0101 mathematicsUniversal differentiability setEngel groupMathematics43A80 46G05 46T20 49J52 49Q15 53C17Directional derivativeuniversal differentiability setApplied Mathematicsta111010102 general mathematicsCarnot group16. Peace & justiceLipschitz continuityPansu differentiableFunctional Analysis (math.FA)Mathematics - Functional AnalysisHausdorff dimensionsymbols010307 mathematical physicsLipschitz mapfunktionaalianalyysiCarnot cycledirectional derivativedescription
We show that every Carnot group G of step 2 admits a Hausdorff dimension one `universal differentiability set' N such that every real-valued Lipschitz map on G is Pansu differentiable at some point of N. This relies on the fact that existence of a maximal directional derivative of f at a point x implies Pansu differentiability at the same point x. We show that such an implication holds in Carnot groups of step 2 but fails in the Engel group which has step 3.
| year | journal | country | edition | language |
|---|---|---|---|---|
| 2019-01-01 |