In this paper, we consider generic corank 2 sub-Riemannian structures, and we show that the Spherical Hausdorf measure is always a C-smooth volume, which is in fact generically Csmooth out of a stratified subset of codimension 7. In particular, for rank 4, it is generically C 2 . This is the continuation of a previous work by the auhors. subjclass: 53C17, 49J15, 58C35

Let E be a set in R with finite n-dimensional Hausdorff measure H such that lim infr→0 r −n H (B(x, r)∩E) > 0 for H-a.e. x ∈ E. In this paper it is shown that E is n-rectifiable if and only if

In the present paper we prove that for any open connected set Ω ⊂ R, n ≥ 1, and any E ⊂ ∂Ω with 0 < H(E) < ∞ absolute continuity of the harmonic measure ω with respect to the Hausdorff measure on E implies that ω|E is rectifiable. CONTENTS

In this paper we study the nilpotent 2-step, corank 2 sub-Riemannian metrics that are nilpotent approximations of general sub-Riemannian metrics. We exhibit optimal syntheses for these problems. It turns out that in general the cut time is not equal to the first conjugate time but has a simple explicit expression. As a byproduct of this study we get some smoothness properties of the spherical H...

Recently, the theory of quasiregular mappings on Carnot groups has been developed intensively. Let ν stand for the homogeneous dimension of a Carnot group and let m be the index of the last vector space of the corresponding Lie algebra. We prove that the (ν −m− 1)-dimensional Hausdorff measure of the image of the branch set of a quasiregular mapping on the Carnot group is positive. Some estimat...

Given a symplectic space, equipped with a line bundle and a Hamiltonian group action satisfying certain compatibility conditions, it is a basic question to understand the decomposition of the quantization space in irreducible representations of the group. We derive weight multiplicity formulas for the quantization space in terms of data at the fixed points on the symplectic space, which apply t...

We prove various generalizations of classical Sard’s theorem to mappings f : M → N between manifolds in Hölder and Sobolev classes. It turns out that if f ∈ C(M,N), then—for arbitrary k and λ—one can obtain estimates of the Hausdorff measure of the set of critical points in a typical level set f−1(y). The classical theorem of Sard holds true for f ∈ C with sufficiently large k, i.e., k > max(m−...

A theorem of Balogh, Koskela, and Rogovin states that in Ahlfors Q-regular metric spaces which support a p-Poincaré inequality, 1 ≤ p ≤ Q, an exceptional set of σ-finite (Q−p)-dimensional Hausdorff measure can be taken in the definition of a quasiconformal mapping while retaining Sobolev regularity analogous to that of the Euclidean setting. Through examples, we show that the assumption of a Po...

A new model of coherent upper conditional prevision is proposed in a metric space. It is defined by the Choquet integral with respect to the s-dimensional Hausdorff outer measure if the conditioning event has positive and finite Hausdorff outer measure in its dimension s. Otherwise if the conditioning event has Hausdorff outer measure in its dimension equal to zero or infinity it is defined by ...