Jeremy T. Tyson

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

Quasiconformal dimensions of self-similar fractals

, i.e. collections of Lipschitz contractions of

On the lack of density of Lipschitz mappings in Sobolev spaces with Heisenberg target

\mathbb{H}^1

Convexity and horizontal second fundamental forms for hypersurfaces in Carnot 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

Global conformal Assouad dimension in the Heisenberg group

\mathbb{H}^1

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

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).

Modulus and Poincaré Inequalities on Non-Self-Similar Sierpiński Carpets

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.

Riesz potentials and p-superharmonic functions in Lie groups of Heisenberg type

The Sierpinski gasket and other self-similar fractal subsets of Rd, d = 2, can be mapped by quasiconformal self-maps of Rd onto sets of Hausdorff dimension arbitrarily close to one. In R2 we construct explicit mappings. In Rd, d = 3, the results follow from general theorems on the equivalence of invariant sets for iterated function systems under quasisymmetric maps and global quasiconformal maps. More specifically, we present geometric conditions ensuring that (i) isomorphic systems have quasisymmetrically equivalent invariant sets, and (ii) one-parameter isotopies of systems have invariant sets which are equivalent under global quasiconformal maps.

Frequency of Sobolev dimension distortion of horizontal subgroups in Heisenberg groups

We study the question: when are Lipschitz mappings dense in the Sobolev space W (M, H)? Here M denotes a compact Riemannian manifold with or without boundary, while H denotes the nth Heisenberg group equipped with a sub-Riemannian metric. We show that Lipschitz maps are dense in W (M, H) for all 1 ≤ p < ∞ if dim M ≤ n, but that Lipschitz maps are not dense in W (M, H) if dim M ≥ n + 1 and n ≤ p < n + 1. The proofs rely on the construction of smooth horizontal embeddings of the sphere S into H. We provide two such constructions, one arising from complex hyperbolic geometry and the other arising from symplectic geometry. The nondensity assertion can be interpreted as nontriviality of the nth Lipschitz homotopy group of H. We initiate a study of Lipschitz homotopy groups for sub-Riemannian spaces.