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Nikita Evseev

Publications and source records attributed to Nikita Evseev.

11 recordsLinked to original sources
arXiv2025

Weak derivatives and metric differentiability almost everywhere

It is known that a Lipschitz continuous map from the Euclidean domain to a metric space is metrically differentiable almost everywhere. When the metric space is a Banach space dual to separable, the metric differential has its linear counterpart -- weak* differential. However, for an arbitrary metric or Banach space, a Lipschitz map is not necessarily weak* differentiable. This paper introduces an approach based on a concept of weak weak* derivatives. This framework yields a linear representation for the metric differential, allowing for its calculation as the norm of an associated linear operator.

math.FA↗
arXiv2024

Zero-extension convergence and Sobolev spaces on changing domains

We extend the definition of weak and strong convergence to sequences of Sobolev-functions whose underlying domains themselves are converging. In contrast to previous works, we do so without ever assuming any sort of reference configuration. We then develop the respective theory and counterparts to classical compactness theorems from the fixed domain case. Finally, we illustrate the usefulness of these definitions with some examples from applications and compare them to other approaches.

math.AP↗
arXiv2023

Weak differentiability of metric space valued Sobolev maps

We show that Sobolev maps with values in a dual Banach space can be characterized in terms of weak derivatives in a weak* sense. Since every metric space embeds isometrically into a dual Banach space, this implies a characterization of metric space valued Sobolev maps in terms of such derivatives. Furthermore, we investigate for which target spaces Sobolev maps are weak* differentiable almost everywhere.

math.FA↗
arXiv2022

Decomposable operators acting between distinct $L^p$-direct integrals of Banach spaces

The notion of decomposable operators acting between distinct $L^p$-direct integrals of Banach spaces is introduced. We show that these operators generalize the composition operator, in sense that a mapping is replaced by a binary relation. The necessary and sufficient conditions for the boundedness of those operators are the main results of the paper.

math.FA↗
arXiv2020

Vector-valued Sobolev spaces based on Banach function spaces

It is known that for Banach valued functions there are several approaches to define a Sobolev class. We compare the usual definition via weak derivatives with the Reshetnyak-Sobolev space and with the Newtonian space; in particular, we provide sufficient conditions when all three agree. As well we revise the difference quotient criterion and the property of Lipschitz mapping to preserve Sobolev space when it acting as a superposition operator.

math.FA↗
arXiv2020

Sobolev space of functions valued in a monotone Banach family

We apply the metrical approach to Sobolev spaces, which arise in various evolution PDEs. Functions from those spaces are defined on an interval and take values in a family of Banach spaces. In this case we adapt the definition of Newtonian spaces. For a monotone family, we show the existence of weak derivative, obtain an isomorphism to the standard Sobolev space, and provide some scalar characteristics.

math.FA↗
arXiv2017

Bounded operators on mixed norm Lebesgue spaces

We study two classes of bounded operators on mixed norm Lebesgue spaces, namely composition operators and product operators. A complete description of bounded composition operators on mixed norm Lebesgue spaces are given. For a certain class of integral operators, we provide sufficient conditions for boundedness. We conclude by applying the developed technique to the investigation of Hardy-Steklov type operators.

math.FA↗
arXiv2017

Bounded composition operator on Lorentz spaces

We study a composition operator on Lorentz spaces. In particular we provide necessary and sufficient conditions under which a measurable mapping induces a bounded composition operator.

math.FA↗
arXiv2015

On measurability of Banach indicatrix

Given two metric measure spaces $X$ and $Y$. Let $f:X\to Y$ be a measurable mapping and $A\subset X$. The Banach indicatrix (multiplicity function) is defined as $N(y,f,A) = \#\{x\in A \mid f(x) = y\}$. We prove measurability of this multiplicity function. The question was also discussed on http://mathoverflow.net/q/206780/15946.

math.CA↗