Density results for Sobolev, Besov and Triebel--Lizorkin spaces on rough sets

We investigate two density questions for Sobolev, Besov and Triebel--Lizorkin spaces on rough sets. Our main results, stated in the simplest Sobolev space setting, are that: (i) for an open set $\Omega\subset\mathbb R^n$, $\mathcal{D}(\Omega)$ is dense in $\{u\in H^s(\mathbb R^n):{\rm supp}\, u\subset \overline{\Omega}\}$ whenever $\partial\Omega$ has zero Lebesgue measure and $\Omega$ is "thick" (in the sense of Triebel); and (ii) for a $d$-set $\Gamma\subset\mathbb R^n$ ($0<d<n$), $\{u\in H^{s_1}(\mathbb R^n):{\rm supp}\, u\subset \Gamma\}$ is dense in $\{u\in H^{s_2}(\mathbb R^n):{\rm supp}\, u\subset \Gamma\}$ whenever $-\frac{n-d}{2}-m-1<s_{2}\leq s_{1}<-\frac{n-d}{2}-m$ for some $m\in\mathbb N_0$. For (ii), we provide concrete examples, for any $m\in\mathbb N_0$, where density fails when $s_1$ and $s_2$ are on opposite sides of $-\frac{n-d}{2}-m$. The results (i) and (ii) are related in a number of ways, including via their connection to the question of whether $\{u\in H^s(\mathbb R^n):{\rm supp}\, u\subset \Gamma\}=\{0\}$ for a given closed set $\Gamma\subset\mathbb R^n$ and $s\in \mathbb R$. They also both arise naturally in the study of boundary integral equation formulations of acoustic wave scattering by fractal screens. We additionally provide analogous results in the more general setting of Besov and Triebel--Lizorkin spaces.Comment: 38 pages, 6 figure

Liga promotora dos interesses materiais do país (1846-1849) : o crescimento económico-social adiado

Entre 1846 e 1849 surgiu um movimento cívico que visava “colocar a nação a par das mais adiantadas no regime económico”. Ocorreu, após a Maria da Fonte, saído de uma proclamação de Claudio Adriano da Costa, que não chegou a ser publicada, mas foi difundida por entre personalidades que se movimentavam nos meios económicos e políticos da capital. Rapidamente granjeou aderentes e suscitou grande entusiasmo nas várias reuniões que se realizaram em menos de três semanas, entre 19 de Julho e 6 de Agosto de 1846. A situação política agitada, que desembocou na guerra civil da Patuleia, determinou que os trabalhos da Liga cessassem, mas não impediu - ou, até, estimulou - o seu regresso, exactamente dois anos depois, ainda com maior pujança, envolvendo gente de todos os quadrantes políticos e de norte a sul do país. Após um reinício fulgurante, e quando terminara a discussão dos estatutos, o marechal Saldanha mandou encerrar a Liga. Esta comunicação procura historiar a vida breve da associação e descortinar as motivações que poderão ter determinado o seu fim, para além de, mesmo sem recorrer à técnica do contrafactual, asseverar que o nosso desenvolvimento económico-social terá sido retardado mercê desse golpe. [...

Embeddings of Besov spaces on fractal h-sets

Let $\Gamma$ be a fractal $h$-set and ${\mathbb{B}}^{{\sigma}}_{p,q}(\Gamma)$ be a trace space of Besov type defined on $\Gamma$. While we dealt in [9] with growth envelopes of such spaces mainly and investigated the existence of traces in detail in [12], we now study continuous embeddings between different spaces of that type on $\Gamma$. We obtain necessary and sufficient conditions for such an embedding to hold, and can prove in some cases complete characterisations. It also includes the situation when the target space is of type $L_r(\Gamma)$ and, as a by-product, under mild assumptions on the $h$-set $\Gamma$ we obtain the exact conditions on $\sigma$, $p$ and $q$ for which the trace space ${\mathbb{B}}^{{\sigma}}_{p,q}(\Gamma)$ exists. We can also refine some embedding results for spaces of generalised smoothness on $\mathbb R^n$

Traces for Besov spaces on fractal h-sets and dichotomy results

We study the existence of traces of Besov spaces on fractal h-sets Γ with a special focus on assumptions necessary for this existence; in other words, we present criteria for the non-existence of traces. In that sense our paper can be regarded as an extension of Bricchi (2004) and a continuation of Caetano (2013). Closely connected with the problem of existence of traces is the notion of dichotomy in function spaces: We can prove that—depending on the function space and the set Γ —there occurs an alternative: either the trace on Γ exists, or smooth functions compactly supported outside Γ are dense in the space. This notion was introduced by Triebel (2008) for the special case of d-sets

Decompositions with atoms and molecules for variable exponent Triebel-Lizorkin-Morrey spaces

We continue the study of the variable exponent Morreyfied Triebel-Lizorkin spaces introduced in a previous paper. Here we give characterizations by means of atoms and molecules. We also show that in some cases the number of zero moments needed for molecules, in order that an infinite linear combination of them (with coefficients in a natural sequence space) converges in the space of tempered distributions, is much smaller than what is usually required. We also establish a Sobolev type theorem for related sequence spaces, which might have independent interest.publishe

How good is teleworking? Development and validation of the tele attitude scale

The objective of this study was to develop and validate a measure called the Tele Attitude Scale (TAS). This measure aims to evaluate relevant aspects of the teleworking experience related to its perceived effects regarding, for instance: job characteristics, perceived productivity, quality of work-related interactions, work-non-work balance, and well-being. Four studies were conducted between 2021 and 2022. First, a qualitative study was conducted to develop the scale (N = 80). Afterward, a second study to explore the scale’s factorial structure (N = 602) was developed. A third study served to analyze its internal validity and reliability (N = 232). A fourth study analyzed the criterion validity of the scale by exploring its correlations with measures of health, affect, and performance (N = 837 teleworkers). The findings revealed that the 10-item scale accounted for a unique factor and that it was a reliable measure. Moreover, the results also showed that the scale was significantly related to measures of health, affect, and performance, thus supporting its convergent and criterion validity. This research advances the knowledge about telework by proposing a user-friendly scale to measure teleworking, specifically how workers perceive their experience of it and how it may impact them at several levels. Thus, the TAS can not only fill a gap in the research but also help organizations evaluate and support teleworkers’ needs and subsequent satisfaction while teleworking. © The Author(s) 2024.Fundação para a Ciência e Tecnologia - FCTinfo:eu-repo/semantics/publishedVersio