Zoltán M. Balogh

Lifts of Lipschitz maps and horizontal fractals in the Heisenberg group

We consider horizontal iterated function systems in the Heisenberg group

Fundamental solution for the Q-Laplacian and sharp Moser–Trudinger inequality in Carnot groups

\mathbb{H}^1

The Stepanov differentiability theorem in metric measure spaces

, i.e. collections of Lipschitz contractions of

Hausdorff dimensions of self-similar and self-affine fractals in the Heisenberg group

\mathbb{H}^1

Frequency of Sobolev dimension distortion of horizontal subgroups in Heisenberg groups

with respect to the Heisenberg metric. The invariant sets for such systems are so-called horizontal fractals . We study questions related to connectivity of horizontal fractals and regularity of functions whose graph lies within a horizontal fractal. Our construction yields examples of horizontal BV (bounded variation) surfaces in

Dimension Distortion by Sobolev Mappings in Foliated Metric Spaces

\mathbb{H}^1

Scaled-oscillation and regularity

that are in contrast with the non-existence of horizontal Lipschitz surfaces which was recently proved by Ambrosio and Kirchheim (Rectifiable sets in metric and Banach spaces. Math. Ann. 318 (3) (2000), 527–555).

Sphericalization and flattening

Abstract For a general Carnot group G with homogeneous dimension Q we prove the existence of a fundamental solution of the Q -Laplacian u Q and a constant a Q >0 such that exp(− a Q u Q ) is a homogeneous norm on G . This implies a representation formula for smooth functions on G which is used to prove the sharp Carnot group version of the celebrated Moser–Trudinger inequality.

Accessible domains in the Heisenberg group

We extend Cheeger’s theorem on differentiability of Lipschitz functions in metric measure spaces to the class of functions satisfying Stepanov’s condition. As a consequence, we obtain the analogue of Calderon’s differentiability theorem of Sobolev functions in metric measure spaces satisfying a Poincaré inequality.

Geometric inequalities on Heisenberg groups

We study the Hausdorff dimensions of invariant sets for self-similar and self-affine iterated function systems in the Heisenberg group. In our principal result we obtain almost sure formulae for the dimensions of self-affine invariant sets, extending to the Heisenberg setting some results of Falconer and Solomyak in Euclidean space. As an application, we complete the proof of the comparison theorem for Euclidean and Heisenberg Hausdorff dimension initiated by Balogh, Rickly and Serra-Cassano.