On Clifford-type structures
Volume 440 / 2006
Abstract
We study several techniques which are well known in the case of Besov and Triebel–Lizorkin spaces and extend them to spaces with dominating mixed smoothness. We use the ideas of Triebel to prove three important {\it decomposition theorems.} We deal with so-called {\it atomic, subatomic} and {\it wavelet} decompositions. All these theorems have much in common. Roughly speaking, they say that a function $f$ belongs to some function space (say $S^{\overline{r}}_{p,q}A$) if, and only if, it can be decomposed as $$ f(x)=\sum_{\nu}\sum_{m}\lambda_{\nu\, m}a_{\nu\, m}(x),\quad\ {\rm convergence\ in}\ S', $$ with coefficients $\lambda=\{\lambda_{\nu\, m}\}$ in a corresponding sequence space (say $s^{\overline r}_{p,q}a$). These decomposition theorems establish a very useful connection between function and sequence spaces. We use them in the study of the decay of entropy numbers of compact embeddings between two function spaces of dominating mixed smoothness, reducing this problem to the same question on the sequence space level. The scales considered cover many important specific spaces (Sobolev, Zygmund, Besov) and we get generalisations of respective assertions of Belinsky, Dinh Dung and Temlyakov.
