Anatoly Golberg

Directional dilatations in space

Two dilatations in ℝ n connected with a given direction are considered. They are spatial counterparts of the angular and radial dilatations in the plane. We establish new geometric estimates of module of ring domains under quasiconformal mappings. These estimates are written in terms of integrals depending on the directional dilatations. The sharpness of the bounds is illustrated by certain examples.

Extension of the Schwarz Lemma to homeomorphisms with controlled p-module

Abstract We extend the classical Schwarz Lemma to homeomorphisms with integrally bounded p-module in ℝn.

Logarithmic Hölder continuity of ring homeomorphisms with controlled -module

We study homeomorphisms preserving integrally quasiinvariant the weighted p-module of the ring domains and provide a condition ensuring the local Hölder continuity of such mappings with respect to logarithms of the distances. The inequality defining the continuity is sharp with respect to the order.

Regularity of Mappings with Integrally Restricted Moduli

We consider certain classes of homeomorphisms of domains in \(\mathbb R^n\) with integrally bounded p-moduli of the families of curves and surfaces, which essentially extend the well-known classes of mappings such as quasiconformal, quasiisometric, Lipschitzian, etc. In the paper we survey the known results in this field regarded to the differential properties of such homeomorphisms, but mainly present a wide range of open related problems.

Extremal Bounds of Teichmüller-Wittich-Belinskiı̆ Type for Planar Quasiregular Mappings

The theorems of TWB (Teichmuller-Wittich-Belinskii) type imply the local conformality (or weaker properties) of quasiconformal mappings at a prescribed point under assumptions of the finiteness of appropriate integral averages of the quantity K μ (z) − 1, where K μ (z) stands for the real dilatation coefficient. We establish the extremal bounds for distortions of the moduli of annuli in terms of integrals in TWB theorems under quasiconformal and quasiregular mappings and illustrate their sharpness by several examples. Some local conditions weaker than the conformality are also discussed.

Absolute continuity on paths of spatial open discrete mappings

We prove that open discrete mappings of Sobolev classes

On lipschitz continuity of quasiconformalmappings in space

W_\mathrm{loc}^{1, p},

Poletskiĭ Type Inequality for Mappings from the Orlicz-Sobolev Classes

Wloc1,p,