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Roman D. Oleinik

Publications and source records attributed to Roman D. Oleinik.

4 recordsLinked to original sources
arXiv2026

On one relaxation of the bounded-length-distortion condition in the context of metric measure spaces

We reformulate the bounded-length-distortion condition for maps between metric spaces in a certain relaxed form that requires the presence of a reference measure on the source space, which makes the new approach more natural from the perspective of maps from metric measure spaces to metric spaces. In terms of the introduced notion, we establish some mapping results in an entirely singular setting of the following general structure: a metric measure space of finite Hausdorff dimension admits a map with the relaxed bounded-length-distortion condition into a finite-dimensional normed space.

math.FA↗
arXiv2024

Characterization of AC and Sobolev curves via Lipschitz post-compositions

Let $\operatorname{X}:=(\operatorname{X},\operatorname{d})$ be an arbitrary metric space. For each $p \in [1,\infty]$, we prove that a map $γ:[a,b] \to \operatorname{X}$ is $p$-absolutely continuous if and only if, for every Lipschitz function $h:\operatorname{X} \to \mathbb{R}$, the post-composition $h \circ γ$ is a $p$-absolutely continuous function. Furthermore, if $\operatorname{X}$ is complete and separable, then, for each $p \in (1,\infty)$, we show that the equivalence class (up to $\mathcal{L}^{1}$-a.e. equality) of a Borel map $γ:[a,b] \to \operatorname{X}$ belongs to the Sobolev $W_{p}^{1}([a,b],\operatorname{X})$-space if and only if, for every Lipschitz function $h:\operatorname{X} \to \mathbb{R}$, the equivalence class (up to $\mathcal{L}^{1}$-a.e. equality) of the post-composition $h \circ γ$ belongs to the Sobolev $W_{p}^{1}([a,b],\mathbb{R})$-space.

math.FA↗
arXiv2023

Asymptotic relations of the Bourgain-Brezis-Mironescu type for mappings between singular spaces

We explore the asymptotic behavior of families of Bourgain-Brezis-Mironescu type nonlocal functionals for mappings from metric measure spaces to arbitrary metric spaces. As the first outcome, we obtain a characterization of Sobolev maps and of maps of bounded variation via such functionals. As the second outcome, we establish precise expressions of the limits of such functionals for Sobolev maps. All this provides an extension of several Bourgain-Brezis-Mironescu type results to the entirely singular setting.

math.FA↗