Kai Rajala

Lipschitz continuity of Cheeger-harmonic functions in metric measure spaces

Abstract 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)-Poincare inequality and in addition supporting a corresponding Sobolev–Poincare-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.

Mappings of finite distortion : removable singularities

We show that certain small sets are removable for bounded mappings of finite distortion for which the distortion function satisfies a suitable subexponential integrability condition. We also give an example demonstrating the sharpness of this condition.

Mappings of finite distortion: Removable singularities for locally homeomorphic mappings

Let f be a locally homeomorphic mapping of finite distortion in dimension larger than two. We show that when the distortion of f satisfies a certain subexponential integrability condition, small sets are removable. The smallness is measured by a weighted modulus.

MAPPINGS OF FINITE DISTORTION: THE RICKMAN-PICARD THEOREM FOR MAPPINGS OF FINITE LOWER ORDER

We show that an entire mappingf of finite distortion with finite lower order can omit at most finitely many points when the distortion function off is suitably controlled. The proof uses the recently established modulus inequalities for mappings of finite distortion [15] and comparison inequalities for the averages of the counting function. A similar technique also gives growth estimates for mappings having asymptotic values.

Uniformization of two-dimensional metric surfaces

We establish uniformization results for metric spaces that are homeomorphic to the Euclidean plane or sphere and have locally finite Hausdorff 2-measure. Applying the geometric definition of quasiconformality, we give a necessary and sufficient condition for such spaces to be QC equivalent to the Euclidean plane, disk, or sphere. Moreover, we show that if such a QC parametrization exists, then the dilatation can be bounded by 2. As an application, we show that the Euclidean upper bound for measures of balls is a sufficient condition for the existence of a 2-QC parametrization. This result gives a new approach to the Bonk–Kleiner theorem on parametrizations of Ahlfors 2-regular spheres by quasisymmetric maps.

A lower bound for the Bloch radius of K-quasiregular mappings

We give a quantitative proof to Eremenkos theorem (2000), which extends Blochs classical theorem to the class of n-dimensional K-quasiregular mappings.

Sharp exponential integrability for traces of monotone Sobolev functions

We answer a question posed in (12) on exponential integrability of functions of restricted n-energy. We use geomet- ric methods to obtain a sharp exponential integrability result for boundary traces of monotone Sobolev functions defined on the unit ball.