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This paper is concerned with the regularity of solutions to linear and nonlinear evolution equations extending our findings in [22] to domains of polyhedral type. In particular, we study the smoothness in the specific scale $B^r_{\tau,\tau}$, $\frac{1}{\tau}=\frac rd+\frac 1p$ of Besov spaces. The regularity in these spaces determines the approximation order that can be achieved by adaptive and other nonlinear approximation schemes. We show that for all cases under consideration the Besov regularity is high enough to justify the use of adaptive algorithms., Comment: arXiv admin note: substantial text overlap with arXiv:1811.09428

For a compact subgroup G of the group of isometries acting on a Riemannian manifold M we investigate subspaces of Besov and Triebel–Lizorkin type which are invariant with respect to the group action. Our main aim is to extend the classical Strauss lemma under suitable assumptions on the Riemannian manifold by proving that G-invariance of functions implies certain decay properties and better local smoothness. As an application we obtain inequalities of Caffarelli–Kohn–Nirenberg type for G-invariant functions. Our results generalize those obtained in Skrzypczak (Rev Mat Iberoam 18:267–299, 2002). The main tool in our investigations are atomic decompositions adapted to the G-action in combination with trace theorems.

We study unboundedness of smoothness Morrey spaces on bounded domains Ω ⊂ Rn in terms of growth envelopes. It turns out that in this situation the growth envelope function is finite—in contrast to the results obtained by Haroske et al. (2016) for corresponding spaces defined on Rn. A similar effect was already observed by Haroske et al. (2017), where classical Morrey spaces Mu,p(Ω) were investigated. We deal with all cases where the concept is reasonable and also include the tricky limiting cases. Our results can be reformulated in terms of optimal embeddings into the scale of Lorentz spaces Lp,q(Ω).

We deal with Morrey spaces on bounded domains \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega $$\end{document}Ω obtained by different approaches. In particular, we consider three settings \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {M}_{u,p}(\Omega )$$\end{document}Mu,p(Ω), \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {M}_{u,p}(\Omega )$$\end{document}Mu,p(Ω) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak {M}_{u,p}(\Omega )$$\end{document}Mu,p(Ω), where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0

We study necessary and sufficient conditions for embeddings of Besov and Triebel-Lizorkin spaces of generalized smoothness \(B^{(n/p,\Psi)}_{p,q}(\mathbb{R}^{n})\) and \(F^{(n/p,\Psi)}_{p,q}(\mathbb{R}^{n})\), respectively, into generalized Holder spaces \(\Lambda_{\infty,r}^{\mu(\cdot)}(\ensuremath {\ensuremath {\mathbb {R}}^{n}})\). In particular, we are able to characterize optimal embeddings for this class of spaces provided q>1. These results improve the embedding assertions given by the continuity envelopes of \(B^{(n/p,\Psi)}_{p,q}(\mathbb{R}^{n})\) and \(F^{(n/p,\Psi)}_{p,q}(\mathbb{R}^{n})\), which were obtained recently solving an open problem of D.D. Haroske in the classical setting.

Previous large epidemiological studies reporting on the association between chronic obstructive pulmonary disease (COPD) and cardiovascular diseases mainly focussed on prevalent diseases rather than on the incidence of newly diagnosed cardiovascular outcomes. We used the UK-based General Practice Research Database (GPRD) to assess the prevalence and incidence of cardiovascular diseases in COPD patients aged 40-79 between 1995 and 2005, and we randomly matched COPD-free comparison patients to COPD patients. In nested-case control analyses, we compared the risks of developing an incident diagnosis of cardiac arrhythmias, venous thromboembolism, myocardial infarction, or stroke between patients with and without COPD, stratifying the analyses by COPD-severity, using COPD-treatment as proxy for disease severity. We identified 35,772 patients with COPD and the same number of COPD-free patients. Most cardiovascular diseases were more prevalent among COPD patients than among the comparison group of COPD-free patients. The relative risk estimates of developing an incident diagnosis of cardiac arrhythmia (OR 1.19, 95% CI 0.98-1.43), deep vein thrombosis (OR 1.35, 95% CI 0.97-1.89), pulmonary embolism (OR 2.51, 95% CI 1.62-3.87), myocardial infarction (OR 1.40, 95% CI 1.13-1.73), or stroke (OR 1.13, 95% CI 0.92-1.38), tended to be increased for patients with COPD as compared to COPD-free controls. The findings of this large observational study provide further evidence that patients with COPD are at increased risk for most cardiovascular diseases.