A note on quasiconformal maps with Holder-continuous dilatation

Here we give an alternate proof of a sufficient condition due to J. Mateu, J. Orobitg, and J. Verdera for a quasiconformal map of the plane with dilatation supported in a smooth domain to be bi-Lipschitz. We also extend this theorem to cover boundaries with certain types of corners.Comment: 13 pages, 2 figure

Discrete Approximations of Metric Measure Spaces of Controlled Geometry

We find a necessary and sufficient condition for a doubling metric space to carry a (1,p)-Poincare inequality. The condition involves discretizations of the metric space and Poincare inequalities on graphs.Comment: 23 Page