Université de Fribourg

Dorronsoro’s theorem in Heisenberg groups

Fässler, Katrin ; Orponen, Tuomas

In: Bulletin of the London Mathematical Society, 2020, vol. 52, no. 3, p. 472–488

A theorem of Dorronsoro from the 1980s quantifies the fact that real‐valued Sobolev functions on Euclidean spaces can be approximated by affine functions almost everywhere, and at all sufficiently small scales. We prove a variant of Dorronsoro's theorem in Heisenberg groups: functions in horizontal Sobolev spaces can be approximated by affine functions which are independent of the last...

Université de Fribourg

A Koebe distortion theorem for quasiconformal mappings in the Heisenberg group

Adamowicz, Tomasz ; Fässler, Katrin ; Warhurst, Ben

In: Annali di Matematica Pura ed Applicata (1923 -), 2020, vol. 199, no. 1, p. 147–186

We prove a Koebe distortion theorem for the average derivative of a quasiconformal mapping between domains in the sub-Riemannian Heisenberg group H1. Several auxiliary properties of quasiconformal mappings between subdomains of H1 are proven, including BMO estimates for the logarithm of the Jacobian. Applications of the Koebe theorem include diameter bounds for images of curves, comparison of...

Université de Fribourg

On the quasi-isometric and bi-Lipschitz classification of 3D Riemannian Lie groups

Fässler, Katrin ; Donne, Enrico Le

In: Geometriae Dedicata, 2020, p. -

This note is concerned with the geometric classification of connected Lie groups of dimension three or less, endowed with left-invariant Riemannian metrics. On the one hand, assembling results from the literature, we give a review of the complete classification of such groups up to quasi-isometries and we compare the quasi- isometric classification with the bi-Lipschitz classification. On the...

Université de Fribourg

Boundedness of singular integrals on C1,α intrinsic graphs in the Heisenberg group

Chousionis, Vasileios ; Fässler, Katrin ; Orponen, Tuomas

In: Advances in Mathematics, 2019, vol. 354, p. 106745

We study singular integral operators induced by 3-dimensional Calderón-Zygmund kernels in the Heisenberg group. We show that if such an operator is L2 bounded on vertical planes, with uniform constants, then it is also L2 bounded on all intrinsic graphs of compactly supported C1,α functions over vertical planes. In particular, the result applies to the operator R induced by the...

Université de Fribourg

Metric currents and the Poincaré inequality

Fässler, Katrin ; Orponen, Tuomas

In: Calculus of Variations and Partial Differential Equations, 2019, vol. 58, no. 2, p. 69

We show that a complete doubling metric space (X,d,μ) supports a weak 1-Poincaré inequality if and only if it admits a pencil of curves (PC) joining any pair of points s,t∈X . This notion was introduced by S. Semmes in the 90’s, and has been previously known to be a sufficient condition for the weak 1-Poincaré inequality. Our argument passes through the intermediate notion of a...

Université de Fribourg

Isometric embeddings into Heisenberg groups

Balogh, Zoltán M. ; Fässler, Katrin ; Sobrino, Hernando

In: Geometriae Dedicata, 2018, vol. 195, no. 1, p. 163–192

We study isometric embeddings of a Euclidean space or a Heisenberg group into a higher dimensional Heisenberg group, where both the source and target space are equipped with an arbitrary left-invariant homogeneous distance that is not necessarily sub-Riemannian. We show that if all infinite geodesics in the target are straight lines, then such an embedding must be a homogeneous homomorphism....

Consortium of Swiss Academic Libraries

Modulus of curve families and extremality of spiral-stretch maps

Balogh, Zoltán ; Fässler, Katrin ; Platis, Ioannis

In: Journal d'Analyse Mathématique, 2011, vol. 113, no. 1, p. 265-291

Consortium of Swiss Academic Libraries

Rectifiability and Lipschitz extensions into the Heisenberg group

Balogh, Zoltán ; Fässler, Katrin

In: Mathematische Zeitschrift, 2009, vol. 263, no. 3, p. 673-683

Université de Fribourg

Curve packing and modulus estimates

Fässler, Katrin ; Orponen, Tuomas

In: Transactions of the American Mathematical Society, 2018, vol. 370, no. 7, p. 4909–4926

A family of planar curves is called a Moser family if it contains an isometric copy of every rectifiable curve in $ \mathbb{R}^{2}$ of length one. The classical ``worm problem'' of L. Moser from 1966 asks for the least area covered by the curves in any Moser family. In 1979, J. M. Marstrand proved that the answer is not zero: the union of curves in a Moser family always has area at least $ c$...