Stretching and Rotation of Planar Quasiconformal Mappings on a Line

In this article, we examine stretching and rotation of planar quasiconformal mappings on a line. We show that for almost every point on the line, the set of complex stretching exponents (describing stretching and rotation jointly) is contained in the disk $ \overline {B}(1/(1-k^{4}),k^{2}/(1-k^{4}))$ B ¯ ( 1 / ( 1 − k 4 ) , k 2 / ( 1 − k 4 ) ) . This yields a quadratic improvement over the known optimal estimate for general sets of Hausdorff dimension 1. Our proof is based on holomorphic motions and estimates for dimensions of quasicircles. We also give a lower bound for the dimension of the image of a 1-dimensional subset of a line under a quasiconformal mapping.

Quasi-symmetric mappings and their generalizations on Riemannian manifolds

A relation between $\eta$-quasi-symmetric homomorphisms and $K$-quasiconformal mappings on $n$-dimensional smooth connected Riemannian manifolds has been studied. The main results of the research are presented in Theorems 2.6 and 2.7. Several conditions for the boundary behavior of $\eta$-quasi-symmetric homomorphisms between two arbitrary domains with weakly flat boundaries and compact closures, QED and uniform domains on the Riemannian mani\-folds, which satisfy the obtained results, were also formulated. In addition, quasiballs, $c$-locally connected domains, and the corresponding results were also considered.

Composition operators on Hardy-Sobolev spaces and BMO-quasiconformal mappings

In this paper, we consider composition operators on Hardy-Sobolev spaces in connections with $\BMO$-quasiconformal mappings. Using the duality of Hardy spaces and $\BMO$-spaces, we prove that $\BMO$-quasiconformal mappings generate bounded composition operators from Hardy--Sobolev spaces to Sobolev spaces.

Hardy Spaces for Quasiregular Mappings and Composition Operators

We define Hardy spaces $${\mathcal {H}}^p$$ H p for quasiregular mappings in the plane, and show that for a particular class of these mappings many of the classical properties that hold in the classical setting of analytic mappings still hold. This particular class of quasiregular mappings can be characterised in terms of composition operators when the symbol is quasiconformal. Relations between Carleson measures and Hardy spaces play an important role in the discussion. This program was initiated and developed for Hardy spaces of quasiconformal mappings by Astala and Koskela in 2011 in their paper $${\mathcal {H}}^p$$ H p -theory for Quasiconformal Mappings (Pure Appl Math Q 7(1):19–50, 2011).