Gelfand and Kolmogorov numbers of Sobolev embeddings of weighted function spaces

In this paper we study the Gelfand and Kolmogorov numbers of Sobolev embeddings between weighted function spaces of Besov and Triebel-Lizorkin type with polynomial weights. The sharp asymptotic estimates are determined in the so-called non-limiting case.Comment: 18 pages, 4 sections, publishe

Limiting embeddings, entropy numbers and envelopes in function spaces

Zumindest seit Sobolev's berühmtem Resultat (1938) sind sogenannte "Limes-Einbettungen" von besonderem Interesse und deshalb intensiv untersucht worden. Gerade in diesen Situationen versagen aber oft auch Standardmethoden oder bringen nur unzureichende Ergebnisse. In der Arbeit wird das Problem über zwei verschiedene Ansätze betrachtet: einerseits kann man die zugrundeliegenden Funktionenräume, zwischen denen ein "gerade nicht mehr stetiger/kompakter" Einbettungsoperator agiert, entsprechend modifizieren. Wir charakterisieren anschließend - in Abhängigkeit von der Art der Modifikation - den Grad der Kompaktheit mittels Entropiezahlen. Alternativ dazu bietet es sich an, die beteiligten Funktionenräume hinsichtlich ihrer Eigenschaften (Wachstum, Lipschitz-Stetigkeit) separat zu studieren. Dazu haben wir die Methode der "Envelopes" entwickelt, die in dieser Arbeit vorgestellt wird. Sie überzeugt nicht nur durch die klassische Schlichtheit der Definition, sondern vor allem durch die erreichten präzisen Aussagen und vielfältigen Anwendungen

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

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 $\mathbb R^n$

Entropy and Approximation Numbers of Embeddings of Function Spaces with Muckenhoupt Weights, I

We study compact embeddings for weighted spaces of Besov and Triebel-Lizorkin type where the weight belongs to some Muckenhoupt Ap class. For weights of purely polynomial growth, both near some singular point and at infinity, we obtain sharp asymptotic estimates for the entropy numbers and approximation numbers of this embedding. The main tool is a discretization in terms of wavelet bases.We study compact embeddings for weighted spaces of Besov and Triebel-Lizorkin type where the weight belongs to some Muckenhoupt Ap class. For weights of purely polynomial growth, both near some singular point and at infinity, we obtain sharp asymptotic estimates for the entropy numbers and approximation numbers of this embedding. The main tool is a discretization in terms of wavelet bases

Nuclear embeddings of Morrey sequence spaces and smoothness Morrey spaces

We study nuclear embeddings for spaces of Morrey type, both in its sequence space version and as smoothness spaces of functions defined on a bounded domain $\Omega \subset {\mathbb R}^d$. This covers, in particular, the meanwhile well-known and completely answered situation for spaces of Besov and Triebel-Lizorkin type defined on bounded domains which has been considered for a long time. The complete result was obtained only recently. Compact embeddings for function spaces of Morrey type have already been studied in detail, also concerning their entropy and approximation numbers. We now prove the first and complete nuclearity result in this context. The concept of nuclearity has already been introduced by Grothendieck in 1955. Again we rely on suitable wavelet decomposition techniques and the famous Tong result (1969) which characterises nuclear diagonal operators acting between sequence spaces of $\ell_r$ type, $1 \leq r \leq\infty$

Widths of embeddings in weighted function spaces

We study the asymptotic behaviour of the approximation, Gelfand and Kolmogorov numbers of the compact embeddings of weighted function spaces of Besov and Triebel-Lizorkin type in the case where the weights belong to a large class. We obtain the exact estimates in almost all nonlimiting situations where the quasi-Banach setting is included. At the end we present complete results on related widths for polynomial weights with small perturbations, in particular the sharp estimates in the case $\alpha=d(\frac 1{p_2}-\frac 1{p_1})>0$ therein.Comment: 20 pages, 4 section