arXiv subjects

Oleksandr V. Maslyuchenko

Publications and source records attributed to Oleksandr V. Maslyuchenko.

4 recordsLinked to original sources

arXiv2026

Operators arising from invariant measures under some class of multidimensional transformations

We investigate a linear operator associated with a functional equation that arises from studying some class of invariant measures under multidimensional transformations. By examining its iterates, we derive an explicit solution formula for the functional equation in some class of functions and establish a result on the existence of an absolutely continuous invariant measure under a multidimensional transformation that can be viewed as a generalization of classical $p$-adic maps to higher dimensions.

math.FA↗

arXiv2025

Classification of Lipschitz derivatives in terms of semicontinuity and the Baire limit functions

We introduce the generalized notion of semicontinuity of a function defined on a topological space and derive the useful classification of the so-called Lipschitz derivatives of functions defined on a metric space. Secondly, we investigate some connections of the Lipschitz derivatives defined on normed spaces to the Fréchet derivative and relations between little, big and local Lipschitz derivatives (denoted by $\lip f$, $\Lip f$ and $\LLip f$ respectively) in terms of Baire limit functions. In particular, we prove that $\lip f$ is $\mathcal{F}_σ$-lower, $\Lip f$ is $\mathcal{F}_σ$-upper, $\LLip f$ is upper semicontinuous. Moreover, for a function $f$ defined on an open or convex subset of a normed space, the upper Baire limit function of functions $\lip f$ and $\Lip f$ are equal to $\LLip f$.

math.FA↗

arXiv2025

Invariant Probability Measures under $p$-adic Transformations

It is well-known that the Lebesgue measure is the unique absolutely continuous invariant probability measure under the $p$-adic transformation. The purpose of this paper is to characterize the family of all invariant probability measures under the $p$-adic transformation and to provide some description of them. In particular, we describe the subfamily of all atomic invariant measures under the $p$-adic transformation as well as the subfamily of all continuous and singular invariant probability measures under the $p$-adic transformation. Iterative functional equations play the base role in our considerations.

math.CA↗

arXiv2025

Centered Takagi--van der Waerden functions and their Lipschitz derivatives

Using a modification of a generalized Takagi-van der Waerden function on a metric space we prove that for any closed subset of a metric space without isolated points there exists a continuous function such that its big and local Lipschitz derivatives are equal to infinity exactly on this set. Moreover, if given space is hermetic (for example, if it is normed) then the little Lipschitz derivative has the same property.

math.FA↗