## Whitney's Problem on Extendability of Functions and an Intrinsic Metric

Published in Advances in Mathematics

Published in Advances in Mathematics

Published in Mathematical Notes

Published in The Journal of Geometric Analysis

Consider the Sobolev space W∞k(Ω) of functions with bounded kth derivatives defined in a planar domain. We study the problem of extendability of functions from W∞k(Ω) to the whole ℝ2 with preservation of class, i.e., surjectivity of the restriction operator W∞k(ℝ2) → W∞k(Ω).

Published in Potential Analysis

In this paper, we obtain the Varadhan type small time asymptotics for diffusion processes on path groups.

Published in Potential Analysis

For conservative symmetric diffusions on a general state space (X,m), the short time asymptotic behavior of tlog∫X1A⋅Tt1B dm is investigated, where Tt is the associated semigroup and A and B are measurable subsets of X. It is proved that the superior limit is dominated by the inferior limit up to some absolute constant. When Γ2 of the associated Di...

Published in Siberian Mathematical Journal

We study the algebraic conditions for all intrinsic metrics to be Finsler on a homogeneous space. These conditions were firstly found by Berestovskiĭ in terms of Lie algebras and their subalgebras (the corresponding subalgebras will be called strong). We obtain a description of the structure of strong subalgebras in semisimple solvable Lie algebras...

Published in Russian Mathematics

In this article, we generalize partially the theorem of V. N. Berestovskii on characterization of similarity homogeneous (nonhomogeneous) Riemannian manifolds, i.e., Riemannian manifolds admitting transitive group of metric similarities other than motions to the case of locally compact similarity homogeneous (nonhomogeneous) spaces with intrinsic m...

Published in Siberian Advances in Mathematics

We say that a domain U ⊂ ℝn is uniquely determined by the relative metric (which is the extension by continuity of the intrinsic metric of the domain on its boundary) of its Hausdorff boundary if any domain V ⊂ ℝn such that its Hausdorff boundary is isometric in the relative metric to the Hausdorff boundary of U, is isometric to U in the Euclidean ...

Published in Functional Analysis and Its Applications

We suggest a new method for describing invariant measures on Markov compacta and on path spaces of graphs and, thereby, for describing characters of certain groups and traces of AF-algebras. The method relies on properties of filtrations associated with a graph and, in particular, on the notion of a standard filtration. The main tool is an intrinsi...

Published in Siberian Advances in Mathematics

We study locally compact spaces with an intrinsic metric such that the group of metric similarities is transitive and the group of isometries is not transitive. We confirm Berestovskiĭ’s conjecture on the structure of such spaces.