arXiv subjects

Hans-Gerd Leopold

Publications and source records attributed to Hans-Gerd Leopold.

3 recordsLinked to original sources

arXiv2022

Nuclear and compact embeddings in function spaces of generalised smoothness

We study nuclear embeddings for function spaces of generalised smoothness defined on a bounded Lipschitz domain $Ω\subset\mathbb{R}^d$. This covers, in particular, the well-known situation for spaces of Besov and Triebel-Lizorkin spaces defined on bounded domains as well as some first results for function spaces of logarithmic smoothness. In addition, we provide some new, more general approach to compact embeddings for such function spaces, which also unifies earlier results in different settings, including also the study of their entropy numbers. Again we rely on suitable wavelet decomposition techniques and the famous Tong result (1969) about nuclear diagonal operators acting in $\ell_r$ spaces, which we could recently extend to the vector-valued setting needed here.

math.FA↗

arXiv2021

Nuclear embeddings in general vector-valued sequence spaces with an application to Sobolev embeddings of function spaces on quasi-bounded domains

We study nuclear embeddings for spaces of Besov and Triebel-Lizorkin type defined on quasi-bounded domains $Ω\subset {\mathbb R}^d$. The counterpart for such function spaces defined on bounded domains has been considered for a long time and the complete answer was obtained only recently. Compact embeddings for function spaces defined on quasi-bounded domains have been studied in detail already, also concerning their entropy and $s$-numbers. We now prove the first and complete nuclearity result in this context. The concept of nuclearity has been introduce by Grothendieck in 1955 already. Our second main contribution is the generalisation of the famous Tong result (1969) which characterises nuclear diagonal operators acting between sequence spaces of $\ell_r$ type, $1\leq r\leq\infty$. We can now extend this to the setting of general vector-valued sequence spaces of type $\ell_q(β_j \ell_p^{M_j})$ with $1\leq p,q\leq\infty$, $M_j\in {\mathbb N}_0$ and weight sequences with $β_j>0$. In particular, we prove a criterion for the embedding $id_β: \ell_{q_1}(β_j \ell_{p_1}^{M_j}) \hookrightarrow \ell_{q_2}(\ell_{p_2}^{M_j})$ to be nuclear.

math.FA↗

arXiv2012

On generalized Besov and Triebel-Lizorkin spaces of regular distributions

We establish conditions on the parameters which are both necessary and sufficient in order that Besov and Triebel-Lizorkin spaces of generalized smoothness contain only regular distributions. We also connect this with the possibility of embedding such spaces in some particular Lebesgue spaces.

math.FA↗