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Katrin Fässler

Publications and source records attributed to Katrin Fässler.

29 records · Page 2Linked to original sources
arXiv2019

Riesz transform and vertical oscillation in the Heisenberg group

We study the $L^{2}$-boundedness of the $3$-dimensional (Heisenberg) Riesz transform on intrinsic Lipschitz graphs in the first Heisenberg group $\mathbb{H}$. Inspired by the notion of vertical perimeter, recently defined and studied by Lafforgue, Naor, and Young, we first introduce new scale and translation invariant coefficients $\operatorname{osc}_Ω(B(q,r))$. These coefficients quantify the vertical oscillation of a domain $Ω\subset \mathbb{H}$ around a point $q \in \partial Ω$, at scale $r > 0$. We then proceed to show that if $Ω$ is a domain bounded by an intrinsic Lipschitz graph $Γ$, and $$\int_{0}^{\infty} \operatorname{osc}_Ω(B(q,r)) \, \frac{dr}{r} \leq C < \infty, \qquad q \in Γ,$$ then the Riesz transform is $L^{2}$-bounded on $Γ$. As an application, we deduce the boundedness of the Riesz transform whenever the intrinsic Lipschitz parametrisation of $Γ$ is an $ε$ better than $\tfrac{1}{2}$-Hölder continuous in the vertical direction. We also study the connections between the vertical oscillation coefficients, the vertical perimeter, and the natural Heisenberg analogues of the $β$-numbers of Jones, David, and Semmes. Notably, we show that the $L^{p}$-vertical perimeter of an intrinsic Lipschitz domain $Ω$ is controlled from above by the $p^{th}$ powers of the $L^{1}$-based $β$-numbers of $\partial Ω$.

math.CA↗
arXiv2019

Dorronsoro's theorem in Heisenberg groups

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 variable. As an application, we deduce new proofs for certain vertical vs. horizontal Poincaré inequalities for real-valued functions on the Heisenberg group, originally due to Austin-Naor-Tessera and Lafforgue-Naor.

math.CA↗
arXiv2018

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

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 other hand, we study the problem whether two quasi-isometrically equivalent Lie groups may be made isometric if equipped with suitable left-invariant Riemannian metrics. We show that this is the case for three-dimensional simply connected groups, but it is not true in general for multiply connected groups. The counterexample also demonstrates that `may be made isometric' is not a transitive relation.

math.GR↗
arXiv2018

Metric currents and the Poincaré inequality

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 $s,t \in X$. 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 generalised pencil of curves (GPC). A GPC joining $s$ and $t$ is a normal $1$-current $T$, in the sense of Ambrosio and Kirchheim, with boundary $\partial T = δ_{t} - δ_{s}$, support contained in a ball of radius $\sim d(s,t)$ around $\{s,t\}$, and satisfying $\|T\| \ll μ$, with $$\frac{d\|T\|}{dμ}(y) \lesssim \frac{d(s,y)}{μ(B(s,d(s,y)))} + \frac{d(t,y)}{μ(B(y,d(t,y)))}.$$ We show that the $1$-Poincaré inequality implies the existence of GPCs joining any pair of points in $X$. Then, we deduce the existence of PCs from a recent decomposition result for normal $1$-currents due to Paolini and Stepanov.

math.MG↗
arXiv2018

Intrinsic Lipschitz graphs and vertical $β$-numbers in the Heisenberg group

The purpose of this paper is to introduce and study some basic concepts of quantitative rectifiability in the first Heisenberg group $\mathbb{H}$. In particular, we aim to demonstrate that new phenomena arise compared to the Euclidean theory, founded by G. David and S. Semmes in the 90's. The theory in $\mathbb{H}$ has an apparent connection to certain nonlinear PDEs, which do not play a role with similar questions in $\mathbb{R}^{3}$. Our main object of study are the intrinsic Lipschitz graphs in $\mathbb{H}$, introduced by B. Franchi, R. Serapioni and F. Serra Cassano in 2006. We claim that these $3$-dimensional sets in $\mathbb{H}$, if any, deserve to be called quantitatively $3$-rectifiable. Our main result is that the intrinsic Lipschitz graphs satisfy a weak geometric lemma with respect to vertical $β$-numbers. Conversely, extending a result of David and Semmes from $\mathbb{R}^{n}$, we prove that a $3$-Ahlfors-David regular subset in $\mathbb{H}$, which satisfies the weak geometric lemma and has big vertical projections, necessarily has big pieces of intrinsic Lipschitz graphs.

math.CA↗
arXiv2017

Isometric Embeddings into Heisenberg Groups

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. We discuss a necessary and certain sufficient conditions for the target space to have this `geodesic linearity property', and we provide various examples.

math.MG↗
arXiv2016

Heisenberg quasiregular ellipticity

Following the Euclidean results of Varopoulos and Pankka--Rajala, we provide a necessary topological condition for a sub-Riemannian 3-manifold $M$ to admit a nonconstant quasiregular mapping from the sub-Riemannian Heisenberg group $\mathbb{H}$. As an application, we show that a link complement $S^3\backslash L$ has a sub-Riemannian metric admitting such a mapping only if $L$ is empty, the unknot or Hopf link. In the converse direction, if $L$ is empty, a specific unknot or Hopf link, we construct a quasiregular mapping from $\mathbb{H}$ to $S^3\backslash L$. The main result is obtained by translating a growth condition on $π_1(M)$ into the existence of a supersolution to the $4$-harmonic equation, and relies on recent advances in the study of analysis and potential theory on metric spaces.

math.GT↗
arXiv2016

Curve packing and modulus estimates

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 has always area at least $c$ for some small absolute constant $c > 0$. We strengthen Marstrand's result by showing that for $p > 3$, the $p$-modulus of a Moser family of curves is at least $c_{p} > 0$.

math.CA↗
arXiv2014

On restricted families of projections in R^3

We study projections onto non-degenerate one-dimensional families of lines and planes in $\mathbb{R}^{3}$. Using the classical potential theoretic approach of R. Kaufman, one can show that the Hausdorff dimension of at most $1/2$-dimensional sets $B \subset \mathbb{R}^{3}$ is typically preserved under one-dimensional families of projections onto lines. We improve the result by an $\varepsilon$, proving that if $\dim_{\mathrm{H}} B = s > 1/2$, then the packing dimension of the projections is almost surely at least $σ(s) > 1/2$. For projections onto planes, we obtain a similar bound, with the threshold $1/2$ replaced by $1$. In the special case of self-similar sets $K \subset \mathbb{R}^{3}$ without rotations, we obtain a full Marstrand type projection theorem for one-parameter families of projections onto lines. The $\dim_{\mathrm{H}} K \leq 1$ case of the result follows from recent work of M. Hochman, but the $\dim_{\mathrm{H}} K > 1$ part is new: with this assumption, we prove that the projections have positive length almost surely.

math.CA↗
arXiv2013

Constancy results for special families of projections

Let {\mathbb{V} = V x R^l : V \in G(n-l,m-l)} be the family of m-dimensional subspaces of R^n containing {0} x R^l, and let π_{\mathbb{V}} : R^n --> \mathbb{V} be the orthogonal projection onto \mathbb{V}. We prove that the mapping V \mapsto Dim π_{\mathbb{V}}(B) is almost surely constant for any analytic set B \subset R^n, where Dim denotes either Hausdorff or packing dimension.

math.CA↗