Some new function spaces of variable smoothness

The present paper is concerned with new Besov-type space of variable smoothness. Nonlinear spline-approximation approach is used to give atomic decomposition of such space. Characterization of the trace space on hyperplane is also obtained.Comment: to appear in Sbornik Mathematics 201

Almost sharp descriptions of traces of Sobolev Wp1(Rn)W_{p}^{1}(\mathbb{R}^{n})-spaces to arbitrary compact subsets of Rn\mathbb{R}^{n}. The case p∈(1,n]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.Comment: The final version accepted in Ann.~Sc.~Norm.~Super.~Pisa Cl.~Sci.~(5) 26, issue 1 (2025

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

Let $(\operatorname{X},\operatorname{d},\mu)$ be a metric measure space with uniformly locally doubling measure $\mu$. Given $p \in (1,\infty)$, assume that $(\operatorname{X},\operatorname{d},\mu)$ supports a weak local $(1,p)$-Poincar\'e 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

Traces of Sobolev spaces to irregular subsets of metric measure spaces

Given $p \in (1,\infty)$, let $(\operatorname{X},\operatorname{d},\mu)$ be a metric measure space with uniformly locally doubling measure $\mu$ supporting a weak local $(1,p)$-Poincar\'e inequality. For each $\theta \in [0,p)$, we characterize the trace space of the Sobolev $W^{1}_{p}(\operatorname{X})$-space to lower codimension $\theta$ content regular closed subsets $S \subset \operatorname{X}$. This gives a far-reaching generalization of the previously known results obtained for Ahlfors--David codimension $\theta$ regular sets $S \subset \operatorname{X}$. In particular, if $\operatorname{X}$ is Ahlfors $Q$-regular for some $Q \geq 1$ and $p \in (Q,\infty)$ we get an intrinsic description of the trace-space of the Sobolev $W^{1}_{p}(\operatorname{X})$-space to arbitrary closed nonempty set $S \subset \operatorname{X}$.Comment: The proof of Lemma 10.1 was slightly improve

In a wilderness of mirrors: the ethics of translation in Cold-War espionage

The article looks at translation in the context of (counter)intelligence. Here translation finds itself in a situation in which scarcity of information turns originals into fragments rather than proper texts. This changes the way the translator works: s/he has to constantly justify his/her decisions, acting in several capacities or with different experts and foregrounding his/her presence in the text. This presence also points to the most difficult of ethical choices to be made. Translation mediates between conflicting or rivalling parties. To mediate, the translator must learn about and identify with an/the other, yet this other is an enemy.The translator must remember that s/he works not against but for the party to which s/he has pledged allegiance. The allegiance may be towards the party for which the translator works, or for the country in which s/he lives, and where s/he acts as a patriot. But the allegiance may be towards the ‘enemy’ – the absolute other – and here the translator’s loyalty is perceived simultaneously as treachery. The ethical ‘wilderness of mirrors’ to be navigated by translators within the (counter)espionage setting reveals a complex layering of multiple acts of fidelity and betrayal, agency and double agency