Almost sharp descriptions of traces of Sobolev $W_{p}^{1}(\mathbb{R}^{n})$-spaces to arbitrary compact subsets of $\mathbb{R}^{n}$. The case $p \in (1,n]$.

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Almost sharp descriptions of traces of Sobolev $W_{p}^{1}(\mathbb{R}^{n})$-spaces to arbitrary compact subsets of $\mathbb{R}^{n}$. The case $p \in (1,n]$.

Annali della Scuola Normale Superiore di Pisa, Classe di Scienze. 2025. Vol. 26. No. 1. P. 397–467.

Tyulenev A.

Language: English

Keywords: Sobolev spaces

On the (p,q)-Eigenvalues of the No-Flux p-Laplacian

Menovshchikov A., Journal of Mathematical Sciences 2026 Vol. 298 P. 608–618

We study the set of (p, q)-eigenvalues of the p-Laplace operator with no-flux boundary conditions. We show that this set is closed and that its smallest positive element (the first nontrivial eigenvalue) admits a variational characterization. Moreover, we establish lower bounds for this eigenvalue in cuspidal domains. ...

Added: July 27, 2026

The Sobolev space W_2^{1/2}: Simultaneous improvement of functions by a homeomorphism of the circle

Lebedev V., Journal of Mathematical Analysis and Applications 2026 Vol. 563 No. 2 Article 130787

It is known that for every continuous real-valued function $f$ on the circle $\mathbb T=\mathbb R/2\pi\mathbb Z$ there exists a change of variable, i.e., a self-homeomorphism $h$ of $\mathbb T$, such that the superposition $f\circ h$ is in the Sobolev space $W_2^{1/2}(\mathbb T)$. We obtain new results on simultaneous improvement of functions by a single change of variable in relation ...

Added: May 14, 2026

Restrictions of Sobolev $W^{1}_{p}(\mathbb{R}^{2})$-spaces to planar rectifiable curves

Tyulenev A., Annales Academiae Scientiarum Fennicae Mathematica 2022 Vol. 47 No. 1 P. 507–531

Added: December 25, 2025

The Sobolev space W_2^{1/2} : Simultaneous improvement of functions by a homeomorphism of the circle

Vladimir Lebedev, / Series arXiv "math". 2025. No. arXiv:2511.07840v3.

Added: December 2, 2025

Injectivity in second-gradient nonlinear elasticity

Campbell D., Hencl S., Menovshchikov A. et al., Calculus of Variations and Partial Differential Equations 2025 Vol. 64 No. 8 Article 256

We study injectivity for models of Nonlinear Elasticity that involve the second gradient. We assume that $\Omega \subset \mathbb{R}^n$ is a domain, $f \in W^{2,q}(\Omega, \mathbb{R}^n)$ satisfies $|J_f^{-a}| \in L^1$ and that $f$ equals a given homeomorphism on $\partial\Omega$. Under suitable conditions on $q$ and $a$ we show that $f$ must be a homeomorphism. As a main new tool we find an ...

Added: October 8, 2025

On mappings generating embedding operators in Sobolev classes on metric measure spaces

Menovshchikov A., Ukhlov A., Journal of Mathematical Analysis and Applications 2025 Vol. 551 No. 2 Article 129716

In this article, we study homeomorphisms $\varphi: \Omega \to \widetilde\Omega$ that generate embedding operators in Sobolev classes on metric measure spaces $X$ by the composition rule $\varphi^*(f)=f \circ \varphi$. In turn, this leads to Sobolev type embedding theorems for a wide class of domains $\widetilde\Omega \subset X$. ...

Added: October 8, 2025

Regularity of Distributions of Sobolev Mappings in Abstract Settings

Kosov E., Mathematical notes 2023 Vol. 114 No. 5 P. 862–874

The paper studies the regularity properties of images of Fomin differentiable measures under mappings in Sobolev classes. The obtained results generalize recently obtained bounds in Gaussian case to more abstract settings. ...

Added: February 7, 2025

Composition operators on Hardy-Sobolev spaces and BMO-quasiconformal mappings

Menovshchikov A., Ukhlov A., Journal of Mathematical Sciences 2021 Vol. 258 No. 3 P. 313–325

In this paper, we consider composition operators on Hardy-Sobolev spaces in connections with BMO-quasiconformal mappings. Using the duality of Hardy spaces and BMO-spaces, we prove that BMO-quasiconformal mappings generate bounded composition operators from Hardy–Sobolev spaces to Sobolev spaces. ...

Added: November 20, 2024

Composition Operators on Sobolev Spaces and Q-Homeomorphisms

Alexander Menovschikov, Ukhlov A., Computational Methods and Function Theory 2023 Vol. 24 No. 1 P. 149–162

In this paper, we give connections between mappings which generate bounded composition operators on Sobolev spaces and Q-mappings. Based on this, we obtain measure distortion properties of Q-homeomorphisms. ...

Added: November 20, 2024

Mappings Generating Embedding Operators in Orlicz-Sobolev Spaces

Menovshchikov A., Ukhlov A., Journal of Mathematical Sciences 2023 Vol. 276 No. 1 P. 117–136

We study embedding operators on Orlicz–Sobolev spaces generated by the composition rule. Using the composition operators we consider embeddings of the Orlicz–Sobolev spaces into weighted Orlicz spaces. ...

Added: November 20, 2024

Capacity of rings and mappings generate embeddings of Sobolev spaces

Menovshchikov A., Ukhlov A., Journal of Mathematical Analysis and Applications 2024 Vol. 531 No. 1 Article 127826

In this paper we give characterizations of mappings generate embeddings of Sobolev spaces in the terms of ring capacity inequalities. In addition we prove that such mappings are Lipschitz mappings in the sub-hyperbolic type capacitory metrics. ...

Added: November 20, 2024

Eigenvalues of the Neumann Laplacian with Density and Sharp Sobolev–Orlicz Embeddings

Alexander Menovschikov, Journal of Mathematical Sciences 2024 Vol. 281 No. 5 P. 782–804

We provide estimates for the constant in the weighted Sobolev–Poincaré inequality for a special class of planar domains and weights. We obtain lower bounds for the first nonzero eigenvalue $\mu_\rho$ of the Neumann Laplacian with density $\rho$. These estimates depend on the density function and geometry of the domain. We show that $\mu_\rho$ can be ...

Added: November 20, 2024

The Bohr--Pal Theorem and the Sobolev Space W_2^{1/2}

Vladimir Lebedev, Studia Mathematica 2015 Vol. 231 No. 1 P. 73–81

The well-known Bohr--Pal theorem asserts that for every continuous real-valued function f on the circle T there exists a change of variable, i.e., a homeomorphism h of T onto itself, such that the Fourier series of the superposition foh converges uniformly. Subsequent improvements of this result imply that actually there exists a homeomorphism that brings f into the Sobolev space W_2^{1/2}(T). This ...

Added: February 16, 2016

The Bohr--Pal Theorem and the Sobolev Space W_2^{1/2}

Vladimir Lebedev, / Series math "arxiv.org". 2015. No. 1508.07167.

The well-known Bohr--Pal theorem asserts that for every continuous real-valued function f on the circle T there exists a change of variable, i.e., a homeomorphism h of T onto itself, such that the Fourier series of the superposition f o h converges uniformly. Subsequent improvements of this result imply that actually there exists a homeomorphism ...

Added: September 3, 2015

Hessian metrics, CD(K,N)-spaces, and optimal transportation of log-concave measures

Kolesnikov A., Discrete and Continuous Dynamical Systems 2014 Vol. 34 No. 4 P. 1511–1532

We study the optimal transportation mapping VΦ: ℝd → ℝd pushing forward a probability measure μ = e -V dx onto another probability measure ν = e-W dx. Following a classical approach of E. Calabi we introduce the Riemannian metric g = D2 Φ on ℝd and study spectral properties of the metric-measure space M ...

Added: November 12, 2013

On continuity equations in infinite dimensions with non-Gaussian reference measure

Kolesnikov A., Roeckner M., / Series math "arxiv.org". 2012.

Let $\gamma$ be a Gaussian measure on a locally convex space and $H$ be the corresponding Cameron-Martin space. It has been recently shown by L. Ambrosio and A. Figalli that the linear first-order PDE $$ \dot{\rho} + \mbox{div}_{\gamma} (\rho \cdot {b}) =0, \ \ \rho|_{t=0} = \rho_0, $$ where $\rho_0 \cdot \gamma $ is a ...

Added: May 13, 2013

Hessian manifolds, CD(K,N) -spaces, and optimal transportation of log-concave measures

Kolesnikov A., / Series math "arxiv.org". 2012. No. 1201.2342.

We study the optimal transportation mapping $\nabla \Phi : \mathbb{R}^d \mapsto \mathbb{R}^d$ pushing forward a probability measure $\mu = e^{-V} \ dx$ onto another probability measure $\nu = e^{-W} \ dx$. Following a classical approach of E. Calabi we introduce the Riemannian metric $g = D^2 \Phi$ on $\mathbb{R}^d$ and study spectral properties of the ...

Added: March 28, 2013

Соболевская регулярность для бесконечномерного уравнения Монжа-Ампера

Kolesnikov A., Богачев В. И., Доклады Академии наук 2012 Т. 44 № 2 С. 131–136

Работа связана с изучением соболевской регулярности отображений оптимальной транспортировки в бесконечномерных пространствах, наделенных гауссовской мерой. Найдены условия принадлежности соболевскому классу для таких отображений. Доказана формула замены переменных. ...

Added: February 19, 2013