Non-horizontal submanifolds and coarea formula

Let k be a positive integer and let m be the dimension of the horizontal subspace of a stratified group. Under the condition k ≤ m, we show that all submanifolds of codimension k are generically non-horizontal. For these submanifolds, we prove an area-type formula that allows us to compute their Q − k dimensional spherical Hausdorff measure. Finally, we observe that a.e. level set of a sufficiently regular vector-valued mapping on a stratified group is a non-horizontal submanifold. This allows us to establish a sub-Riemannian coarea formula for vector-valued Riemannian Lipschitz mappings on stratified groups.     

A Blow-up Theorem for regular hypersurfaces on nilpotent groups

 We obtain an intrinsic Blow-up Theorem for regular hypersurfaces on graded nilpotent groups. This procedure allows us to represent explicitly the Riemannian surface measure in terms of the spherical Hausdorff measure with respect to an intrinsic distance of the group, namely homogeneous distance. We apply this result to get a version of the Riemannian coarea forumula on sub-Riemannian groups, that can be expressed in terms of arbitrary homogeneous distances. We introduce the natural class of horizontal isometries in sub-Riemannian groups, giving examples of rotational invariant homogeneous distances and rotational groups, where the coarea formula takes a simpler form. By means of the same Blow-up Theorem we obtain an optimal estimate for the Hausdorff dimension of the characteristic set relative to C ^1,1 hypersurfaces in 2-step groups and we prove that it has finite Q–2 Hausdorff measure, where Q is the homogeneous dimension of the group. Received: 6 February 2002 Mathematics Subject Classification (2000): 28A75 (22E25)     

Blow-up of regular submanifolds in Heisenberg groups and applications

We obtain a blow-up theorem for regular submanifolds in the Heisenberg group, where intrinsic dilations are used. Main consequence of this result is an explicit formula for the density of (p+1)-dimensional spherical Hausdorff measure restricted to a p-dimensional submanifold with respect to the Riemannian surface measure. We explicitly compute this formula in some simple examples and we present a lower semicontinuity result for the spherical Hausdorff measure with respect to the weak convergence of currents. Another application is the proof of an intrinsic coarea formula for vector-valued mappings on the Heisenberg group.     

Lipschitz continuity, Aleksandrov theorem and characterizations for H-convex functions

In the geometries of stratified groups, we show that H-convex functions locally bounded from above are locally Lipschitz continuous and that the class of v-convex functions exactly corresponds to the class of upper semicontinuous H-convex functions. As a consequence, v-convex functions are locally Lipschitz continuous in every stratified group. In the class of step 2 groups we characterize locally Lipschitz H-convex functions as measures whose distributional horizontal Hessian is positive semidefinite. In Euclidean space the same results were obtained by Dudley and Reshetnyak. We prove that a continuous H-convex function is a.e. twice differentiable whenever its second order horizontal derivatives are Radon measures.     

A compact universal differentiability set with Hausdorff dimension one

We give a short proof that any non-zero Euclidean space has a compact subset of Hausdorff dimension one that contains a differentiability point of every real-valued Lipschitz function defined on the space.     

A differential operator and weak topology for Lipschitz maps

We show that the Scott topology induces a topology for real-valued Lipschitz maps on Banach spaces which we call the L-topology. It is the weakest topology with respect to which the L-derivative operator, as a second order functional which maps the space of Lipschitz functions into the function space of non-empty weak^∗ compact and convex valued maps equipped with the Scott topology, is continuous. For finite dimensional Euclidean spaces, where the L-derivative and the Clarke gradient coincide, we provide a simple characterization of the basic open subsets of the L-topology. We use this to verify that the L-topology is strictly coarser than the well-known Lipschitz norm topology. A complete metric on Lipschitz maps is constructed that is induced by the Hausdorff distance, providing a topology that is strictly finer than the L-topology but strictly coarser than the Lipschitz norm topology. We then develop a fundamental theorem of calculus of second order in finite dimensions showing that the continuous integral operator from the continuous Scott domain of non-empty convex and compact valued functions to the continuous Scott domain of ties is inverse to the continuous operator induced by the L-derivative. We finally show that in dimension one the L-derivative operator is a computable functional.     

Porosity,Differentiability and Pansu’s Theorem

We use porosity to study differentiability of Lipschitz maps on Carnot groups. Our first result states that directional derivatives of a Lipschitz function act linearly outside a \(\sigma \)-porous set. The second result states that irregular points of a Lipschitz function form a \(\sigma \)-porous set. We use these observations to give a new proof of Pansu’s theorem for Lipschitz maps from a general Carnot group to a Euclidean space.     

Intrinsic Lipschitz Graphs Within Carnot Groups

A Carnot group is a connected, simply connected, nilpotent Lie group with stratified Lie algebra. We study the notions of intrinsic graphs and of intrinsic Lipschitz graphs within Carnot groups. Intrinsic Lipschitz graphs are the natural local analogue inside Carnot groups of Lipschitz submanifolds in Euclidean spaces, where “natural” emphasizes that the notion depends only on the structure of the algebra. Intrinsic Lipschitz graphs unify different alternative approaches through Lipschitz parameterizations or level sets. We provide both geometric and analytic characterizations and a clarifying relation between these graphs and Rumin’s complex of differential forms.     

A note on the area and coarea formulas for general volume densities and some applications

We present the area and coarea formulas for Lipschitz maps, valid for general volume densities. We apply these formulas to derive an anisotropic tube formula for hypersurfaces in \({{\mathbb{R}}^n}\) and to give a short euclidean proof of the anisotropic Sobolev inequality. A discussion about the first variation of the anisotropic area is also included.     

Rectifiable sets and coarea formula for metric-valued mappings

We study Lipschitz mappings defined on an Hn-rectifiable metric space with values in an arbitrary metric space. We find necessary and sufficient conditions on the image and the preimage of a mapping for the validity of the coarea formula. As a consequence, we prove the coarea formula for some classes of mappings with Hk-σ-finite image. We also obtain a metric analog of the Implicit Function Theorem. All these results are extended to large classes of mappings with values in a metric space, including Sobolev mappings and BV-mappings.     

Best Constants for Moser-Trudinger Inequalities, Fundamental Solutions and One-Parameter Representation Formulas on Groups of Heisenberg Type

We derive the explicit fundamental solutions for a class of degenerate (or singular) one-parameter subelliptic differential operators on groups of Heisenberg (H) type. This extends the results of Kaplan of the sub-Laplacian on H-type groups, which in turn generalizes Folland's result on the Heisenberg group. As an application, we obtain a one-parameter representation formula for Sobolev functions of compact support on H-type groups. By choosing the parameter equal to the homogeneous dimension Q and using the Moser-Trudinger inequality for the convolutional type operator on stratified groups obtained in [18], we get the following theorem which gives the best constant for the Moser-Trudinger inequality for Sobolev functions in H-type groups. Let ${\Bbb G}We derive the explicit fundamental solutions for a class of degenerate (or singular) one-parameter subelliptic differential operators on groups of Heisenberg (H) type. This extends the results of Kaplan of the sub-Laplacian on H-type groups, which in turn generalizes Folland's result on the Heisenberg group. As an application, we obtain a one-parameter representation formula for Sobolev functions of compact support on H-type groups. By choosing the parameter equal to the homogeneous dimension Q and using the Moser-Trudinger inequality for the convolutional type operator on stratified groups obtained in [18], we get the following theorem which gives the best constant for the Moser-Trudinger inequality for Sobolev functions in H-type groups. Let ? be any group of Heisenberg type whose Lie algebra is g enerated by m left invariant vector fields and with a q-dimensional center. Let and Then, with A _Q as the sharp constant, where ∇_? denotes the subellitpic gradient on ? This continues the research originated in our earlier study of the best constants in Moser-Trudinger inequalities and fundamental solutions for one-parameter subelliptic operators on the Heisenberg group [18]. Received March 15, 2001, Accepted September 21, 2001     

Dimension of invariant sets for mappings with the squeezing property

We show an estimate of the fractal and Hausdorff dimension of sets invariant with respect to families of transformations. This estimate is proved under assumption that the transformations satisfy a squeezing property which is more general than the Lipschitz condition. Our results generalize the classical Moran formula [Moran PAP. Additive functions of intervals and Hausdorff measure. Proc Camb Philos Soc 1946;42:15–23].     

On four-colourings of the rational four-space

Summary LetQ ⁴ denote the graph, obtained from the rational points of the 4-space, by connecting two points iff their Euclidean distance is one. It has been known that its chromatic number is 4. We settle a problem of P. Johnson, showing that in every four-colouring of this graph, every colour class is every-where dense.     

Gauges for the self‐similar sets

For a self‐similar set E with the open set condition we completely determine the class of its Hausdorff gauges and the class of its prepacking gauges. Moreover, its Hausdorff measures and its packing premeasures with respect to the corresponding gauges are estimated. Without the open set condition we prove that a doubling gauge function is a packing gauge of E if and only if it is a prepacking gauge of E. Also, we give some extensions and applications of these results. Here a gauge function is called a Hausdorff, a prepacking, and a packing gauge of a set, if with respect to the function the set has positive and finite Hausdorff measure, packing premeasure, and packing measure, respectively. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The Coarea Formula for Vector Functions on Carnot Groups with Sub-Lorentzian Structure

Siberian Mathematical Journal - We establish the coarea formula as an expression for the measure of a subset of a Carnot group in terms of the sub-Lorentzian measure of...     

Curvature-direction measures of self-similar sets

We obtain fractal Lipschitz–Killing curvature-direction measures for a large class of self-similar sets $F$ in $\mathbb{R }^{d}$ . Such measures jointly describe the distribution of normal vectors and localize curvature by analogues of the higher order mean curvatures of differentiable sub-manifolds. They decouple as independent products of the unit Hausdorff measure on $F$ and a self-similar fibre measure on the sphere, which can be computed by an integral formula. The corresponding local density approach uses an ergodic dynamical system formed by extending the code space shift by a subgroup of the orthogonal group. We then give a remarkably simple proof for the resulting measure version under minimal assumptions.     

On the Lipschitz equivalence of self‐affine sets

Recently Lipschitz equivalence of self‐similar sets on has been studied extensively in the literature. However for self‐affine sets the problem is more awkward and there are very few results. In this paper, we introduce a w‐Lipschitz equivalence by repacing the Euclidean norm with a pseudo‐norm w. Under the open set condition, we prove that any two totally disconnected integral self‐affine sets with a common matrix are w‐Lipschitz equivalent if and only if their digit sets have equal cardinality. The main methods used are the technique of pseudo‐norm and Gromov hyperbolic graph theory on iterated function systems.     

A Frequency Function and Singular Set Bounds for Branched Minimal Immersions

We establish a frequency function monotonicity formula for two‐valued C^1,α solutions to the minimal surface system on n‐dimensional domains. We also establish the sharp regularity result that such solutions are of class C^{1, 1/2}, and that their branch sets, if nonempty, have Hausdorff dimension equal to n‐2.© 2016 Wiley Periodicals, Inc.     

The disintegration of the Lebesgue measure on the faces of a convex function

We consider the disintegration of the Lebesgue measure on the graph of a convex function f:Rn→R w.r.t. the partition into its faces, which are convex sets and therefore have a well defined linear dimension, and we prove that each conditional measure is equivalent to the k-dimensional Hausdorff measure on the k-dimensional face on which it is concentrated. The remarkable fact is that a priori the directions of the faces are just Borel and no Lipschitz regularity is known. Notwithstanding that, we also prove that a Green-Gauss formula for these directions holds on special sets.     

Optimal Mutual Location of Compact Sets in Spaces Endowed with Euclidean Invariant Gromov–Hausdorif Metric

Non–empty compact subsets of the Euclidean space located optimally (i.e., the Hausdorff distance between them cannot be decreased) are studied. It is shown that if one of them is a single point, then it is located at the Chebyshev center of the other one. Many other particular cases are considered too. As an application, it is proved that each three–point metric space cari be isometrically embedded into the orbit space of the group of proper motions acting on the compact subsets of the Euclidean space. In addition, it is proved that for each pair of optimally located compact subsets all intermediate compact sets in the sense of Hausdorff metric are also intermediate in the sense of Euclidean Gromov–Hausdorff metric.