Universal differentiability sets in Carnot groups of arbitrarily high step
- PDF / 549,323 Bytes
- 58 Pages / 429.408 x 636.768 pts Page_size
- 75 Downloads / 162 Views
UNIVERSAL DIFFERENTIABILITY SETS IN CARNOT GROUPS OF ARBITRARILY HIGH STEP
BY
Andrea Pinamonti Department of Mathematics, University of Trento Via Sommarive 14, 38050 Povo (Trento), Italy e-mail: [email protected]
AND
Gareth Speight Department of Mathematical Sciences, University of Cincinnati 2815 Commons Way, Cincinnati, OH 45221, USA e-mail: [email protected]
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.
Received October 21, 2019 and in revised form November 13, 2019
1
2
A. PINAMONTI AND G. SPEIGHT
Isr. J. Math.
1. Introduction Rademacher’s theorem asserts that every Lipschitz map f : Rn → Rm is differentiable almost everywhere with respect to the Lebesgue measure. This important result has been extended to many other spaces and measures [2, 6, 18, 27]. It is also interesting to consider whether Rademacher’s theorem admits a converse: given a Lebesgue null set N ⊂ Rn , does there exist a Lipschitz map f : Rn → Rm which is differentiable at no point of N ? The answer to this question is yes if and only if n ≤ m and combines the work of several authors [38, 30, 32, 3, 8]. In the case where n > m = 1, the results in [9, 11, 12] provide a stronger result: there is a compact set of Hausdorff dimension one in Rn which contains some point of differentiability of any Lipschitz map f : Rn → R. Such a set may even be chosen with upper Minkowski dimension one [12]. Sets containing a point of differentiability for any real-valued Lipschitz map are called universal differentiability sets. We refer the reader to [31] and the references therein for more discussion of such sets. The present paper continues the investigation of universal differentiability sets in Carnot groups which was started in [29, 17]; see also the survey [28]. We recall that a Carnot group (Definition 2.1) is a simply connected Lie group whose Lie algebra g admits a stratification, i.e., it admits a decomposition g = V1 ⊕· · ·⊕Vs where Vi+1 = [V1 , Vi ] for i = 1, . . . , s − 1. The subspace V1 is called the horizontal layer while s is the step of the Carnot group and to some extent indicates its complexity (Carnot groups of step one are simply Euclidean spaces). Carnot groups have a rich geometric structure adapted to the horizontal layer, including translations, dilations, Carnot–Carat
Data Loading...
