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Regularity of convex functions on Heisenberg groups

Zoltán M. BaloghMatthieu Rickly — 2003

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

We discuss differentiability properties of convex functions on Heisenberg groups. We show that the notions of horizontal convexity (h-convexity) and viscosity convexity (v-convexity) are equivalent and that h-convex functions are locally Lipschitz continuous. Finally we exhibit Weierstrass-type h-convex functions which are nowhere differentiable in the vertical direction on a dense set or on a Cantor set of vertical lines.

Dimension Distortion by Sobolev Mappings in Foliated Metric Spaces

Zoltán M. BaloghJeremy T. TysonKevin Wildrick — 2013

Analysis and Geometry in Metric Spaces

We quantify the extent to which a supercritical Sobolev mapping can increase the dimension of subsets of its domain, in the setting of metric measure spaces supporting a Poincaré inequality. We show that the set of mappings that distort the dimensions of sets by the maximum possible amount is a prevalent subset of the relevant function space. For foliations of a metric space X defined by a David–Semmes regular mapping Π : X → W, we quantitatively estimate, in terms of Hausdorff dimension in W, the...

Comparison of Hausdorff measures with respect to the Euclidean and the Heisenberg metric.

Zoltán M. BaloghMatthieu RicklyFrancesco Serra Cassano — 2003

Publicacions Matemàtiques

We compare the Hausdorff measures and dimensions with respect to the Euclidean and Heisenberg metrics on the first Heisenberg group. The result is a dimension jump described by two inequalities. The sharpness of our estimates is shown by examples. Moreover a comparison between Euclidean and H-rectifiability is given.

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