arXiv subjects

Alexander Tyulenev

Publications and source records attributed to Alexander Tyulenev.

9 recordsLinked to original sources

arXiv2026

Descriptions of traces of weighted Sobolev spaces to Ahlfors--David regular sets in the case $p=1$

Given $n \in \mathbb{N}$, an Ahlors--David $n$-regular set $S \subset \mathbb{R}^{n+1}$, and a weight $\gamma$ satisfying the local Muckenhoupt $A_{1}$-condition, we present a complete intrinsic description of the trace-space $W_{1}^{1}(\mathbb{R}^{n+1},\gamma)|_{S}$ of the weighted first-order Sobolev space $W_{1}^{1}(\mathbb{R}^{n+1},\gamma)$ to $S$. Furthermore, we construct a new family of nonlinear bounded extension operators acting from $W_{1}^{1}(\mathbb{R}^{n+1},\gamma)|_{S}$ to $W_{1}^{1}(\mathbb{R}^{n+1},\gamma)$. Finally, we find conditions on $\gamma$ that sufficient for the existence of a bounded linear extension operator from $W_{1}^{1}(\mathbb{R}^{n+1},\gamma)|_{S}$ to $W_{1}^{1}(\mathbb{R}^{n+1},\gamma)$.

math.FA↗

arXiv2026

Catching jumps of metric-valued mappings with Lipschitz functions

It follows from recent results of V. Bakhtin, R. Oleinik, and the second named author that, given a metric space $\mathcal{X}$, a continuous map $\gamma\colon [a,b] \to \mathcal{X}$ is a map of bounded variation if and only if $f \circ \gamma$ is a function of bounded variation for every Lipschitz function $f\colon\mathcal{X} \to \mathbb{R}$. In this note, we show that the continuity assumption is of crucial importance: for many interesting examples of metric spaces there are no analogs of that characterization without the continuity assumption on $\gamma$. The interesting examples are: $\ell_2$, infinite metric trees, and Laakso-type spaces. However, for ultrametric spaces the said characterization holds without any continuity assumptions.

math.CA↗

arXiv2023

Traces of Sobolev spaces to piecewise Ahlfors--David regular sets

Let $(\operatorname{X},\operatorname{d},μ)$ be a metric measure space with uniformly locally doubling measure $μ$. Given $p \in (1,\infty)$, assume that $(\operatorname{X},\operatorname{d},μ)$ supports a weak local $(1,p)$-Poincaré inequality. We characterize trace spaces of the first-order Sobolev $W^{1}_{p}(\operatorname{X})$-spaces to subsets $S$ of $\operatorname{X}$ that can be represented as a finite union $\cup_{i=1}^{N}S^{i}$, $N \in \mathbb{N}$, of Ahlfors--David regular subsets $S^{i} \subset \operatorname{X}$, $i \in \{1,...,N\}$, of different codimensions. Furthermore, we explicitly compute the corresponding trace norms up to some universal constants.

math.FA↗

arXiv2023

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

Let $S \subset \mathbb{R}^{n}$ be an arbitrary nonempty compact set such that the $d$-Hausdorff content $\mathcal{H}^{d}_{\infty}(S) > 0$ for some $d \in (0,n]$. For each $p \in (\max\{1,n-d\},n]$, an almost sharp intrinsic description of the trace space $W_{p}^{1}(\mathbb{R}^{n})|_{S}$ of the Sobolev space $W_{p}^{1}(\mathbb{R}^{n})$ to the set $S$ is obtained. Furthermore, for each $p \in (\max\{1,n-d\},n]$ and $\varepsilon \in (0, \min\{p-(n-d),p-1\})$, new bounded linear extension operators from the trace space $W_{p}^{1}(\mathbb{R}^{n})|_{S}$ into the space $W_{p-\varepsilon}^{1}(\mathbb{R}^{n})$ are constructed.

math.FA↗

arXiv2023

Traces of Sobolev spaces to irregular subsets of metric measure spaces

Given $p \in (1,\infty)$, let $(\operatorname{X},\operatorname{d},μ)$ be a metric measure space with uniformly locally doubling measure $μ$ supporting a weak local $(1,p)$-Poincaré inequality. For each $θ\in [0,p)$, we characterize the trace space of the Sobolev $W^{1}_{p}(\operatorname{X})$-space to lower codimension-$θ$ content regular closed subsets $S \subset \operatorname{X}$. In particular, if the space $(\operatorname{X},\operatorname{d},μ)$ is Ahlfors $Q$-regular for some $Q \geq 1$ and $p \in (Q,\infty)$, then we get an intrinsic description of the trace-space of the Sobolev $W^{1}_{p}(\operatorname{X})$-space to arbitrary closed nonempty set $S \subset \operatorname{X}$.

math.FA↗

arXiv2022

Some porosity-type properties of sets related to the $d$-Hausdorff content

Let $S \subset \mathbb{R}^{n}$ be a nonempty set. Given $d \in [0,n)$ and a cube $\overline{Q} \subset \mathbb{R}^{n}$ with $l=l(\overline{Q}) \in (0,1]$, we show that if the $d$-Hausdorff content $\mathcal{H}^{d}_{\infty}(\overline{Q} \cap S) < \overlineλl^{d}$ for some $\overlineλ \in (0,1)$, then the set $\overline{Q} \setminus S$ contains a specific cavity. More precisely, we prove existence of a pseudometric $ρ=ρ_{S,d}$ such that for each sufficiently small $δ> 0$ the $δ$-neighborhood $U^ρ_{δl}(S)$ of $S$ in the pseudometric $ρ$ does not contain the whole $\overline{Q}$. Moreover, we establish the existence of constants $\overlineδ=\overlineδ(n,d,\overlineλ)>0$ and $\underlineγ=\underlineγ(n,d,\overlineλ)>0$ such that $\mathcal{L}^{n}(\overline{Q} \setminus U^ρ_{δl}(S)) \geq \underlineγ l^{n}$ for all $δ\in (0,\overlineδ)$. If, in addition, the set $S$ is $d$-lower content regular, we prove existence of a constant $\underlineτ=\underlineτ(n,d,\overlineλ)>0$ such that the cube $\overline{Q}$ is $\underlineτ$-porous. The sharpness of the results is illustrated by several examples.

math.FA↗

arXiv2021

Korevaar-Schoen's energy on strongly rectifiable spaces

We extend Korevaar-Schoen's theory of metric valued Sobolev maps to cover the case of the source space being an RCD space. In this situation it appears that no version of the `subpartition lemma' holds: to obtain both existence of the limit of the approximated energies and the lower semicontinuity of the limit energy we shall rely on: - the fact that such spaces are `strongly rectifiable' a notion which is first-order in nature (as opposed to measure-contraction-like properties, which are of second order). This fact is particularly useful in combination with Kirchheim's metric differentiability theorem, as it allows to obtain an approximate metric differentiability result which in turn quickly provides a representation for the energy density, - the differential calculus developed by the first author which allows, thanks to a representation formula for the energy that we prove here, to obtain the desired lower semicontinuity from the closure of the abstract differential. When the target space is CAT(0) we can also identify the energy density as the Hilbert-Schmidt norm of the differential, in line with the smooth situation.

math.FA↗

arXiv2021

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

We construct explicit examples of Frostman-type measures concentrated on arbitrary planar rectifiable curves of positive length. Based on such constructions we obtain for each $p \in (1,\infty)$ an exact description of the trace space of the first-order Sobolev space $W^{1}_{p}(\mathbb{R}^{2})$ to an arbitrary planar rectifiable curve $Γ\subset \mathbb{R}^{2}$ of positive length.

math.FA↗

arXiv2020

Korevaar-Schoen's directional energy and Ambrosio's regular Lagrangian flows

We develop Korevaar-Schoen's theory of directional energies for metric-valued Sobolev maps in the case of ${\sf RCD}$ source spaces; to do so we crucially rely on Ambrosio's concept of Regular Lagrangian Flow. Our review of Korevaar-Schoen's spaces brings new (even in the smooth category) insights on some aspects of the theory, in particular concerning the notion of `differential of a map along a vector field' and about the parallelogram identity for ${\sf CAT}(0)$ targets. To achieve these, one of the ingredients we use is a new (even in the Euclidean setting) stability result for Regular Lagrangian Flows.

math.FA↗