Generalized Hausdorff dimension distortion in Euclidean spaces under Sobolev mappings

We investigate how the integrability of the derivatives of Orlicz-Sobolev mappings defined on open subsets of Rn affect the sizes of the images of sets of Hausdorff dimension less than n. We measure the sizes of the image sets in terms of generalized Hausdorff measures.     

Invertibility of Sobolev mappings under minimal hypotheses

We prove a version of the Inverse Function Theorem for continuous weakly differentiable mappings. Namely, a nonconstant W1,n$W^{1,n}$ mapping is a local homeomorphism if it has integrable inner distortion function and satisfies a certain differential inclusion. The integrability assumption is shown to be optimal.     

A lower bound for the Bloch radius of -quasiregular mappings

We give a quantitative proof to Eremenko's theorem (2000), which extends Bloch's classical theorem to the class of -dimensional -quasiregular mappings.

     

A Density Result for Homogeneous Sobolev Spaces on Planar Domains

Potential Analysis - We show that in a bounded simply connected planar domain Ω the smooth Sobolev functions Wk,∞(Ω) ∩ C∞(Ω) are dense in the homogeneous Sobolev...     

Quasiregularly elliptic link complements

We extend the theorem of B. Daniel about the existence and uniqueness of immersions into \mathbbSⁿ × \mathbbR or \mathbbHⁿ × \mathbbR{\mathbb{S}^{n}\,\times\,\mathbb{R}\, {\rm or}\, \mathbb{H}^{n}\,\times\,\mathbb{R}} to the Riemannian product of two space forms. More precisely, we prove the existence and uniqueness of an isometric immersion of a Riemannian manifold into the Riemannian product of two space forms.     

The local homeomorphism property of spatial quasiregular mappings with distortion close to one

We give a quantitative proof for a theorem of Martio, Rickman and V?is?l? [MRV] on the rigidity of the local homeomorphism property of spatial quasiregular mappings with distortion close to one. The proof is based on a distortion theory established by using two main tools. First, we use a conformal invariant between sphere families and components of their preimages under quasiregular mappings. Secondly, we use Hall’s quantitative isoperimetric inequality result [H] to relate two different types of distortion. Received: April 2004 Revision: October 2004 Accepted: December 2004     

Lipschitz continuity of Cheeger-harmonic functions in metric measure spaces

We use the heat equation to establish the Lipschitz continuity of Cheeger-harmonic functions in certain metric spaces. The metric spaces under consideration are those that are endowed with a doubling measure supporting a (1,2)-Poincaré inequality and in addition supporting a corresponding Sobolev-Poincaré-type inequality for the modification of the measure obtained via the heat kernel. Examples are given to illustrate the necessity of our assumptions on these spaces. We also provide an example to show that in the general setting the best possible regularity for the Cheeger-harmonic functions is Lipschitz continuity.     

Optimal Maps and Exponentiation on Finite-Dimensional Spaces with Ricci Curvature Bounded from Below

We prove existence and uniqueness of optimal maps on \(\mathsf{RCD}^*(K,N)\) spaces under the assumption that the starting measure is absolutely continuous. We also discuss how this result naturally leads to the notion of exponentiation and to the local-to-global property of \(\mathsf{RCD}^*(K,N)\) bounds.     

An egg-yolk principle and exponential integrability for quasiregular mappings

Quasiregular mappings f:ⁿⁿ are a natural generalization of analyticfunctions from complex analysis and provide a theory which isrich with new phenomena. In this paper we extend a well-knownresult of Chang and Marshall on exponential integrability ofanalytic functions in the disk, to the case of quasiregularmappings defined in the unit ball of ⁿ. To this end, an ‘egg-yolk’principle is first established for such maps, which extendsa recent result of the first author. Our work leaves open aninteresting problem regarding n-harmonic functions.     

Optimal Assumptions for Discreteness

There exists, in every dimension ${n \geq 3}$ , a Lipschitz mapping of finite distortion such that the (outer) distortion satisfies ${K\in L^{n-1}}$ but it maps a line segment to a point.