COMPACT EMBEDDINGS OF BESOV SPACES IN EXPONENTIAL ORLICZ SPACES

Abstract

Let $1 < p <\infty, 0 < v < p^\prime$ , let $\Omega$ be a bounded domain in ${\bb R}^n$ , and denote by ${\rm id}_{\omega}$ the limiting compact embedding of the Besov space $B^{n/p}_{pp}({\bb R}^n)$ into the exponential Orlicz space $L_{\exp (t^v)}(\Omega)$ , mapping a function $f$ onto its restriction $f\vert_{\Omega}$ . In 1993 Triebel established, among others, two-sided estimates for the entropy numbers of ${\rm id}_{\omega}$ , which are even asymptotically optimal for ‘small’ $\nu$ . The aim of the paper is to improve the upper bounds in the case of ‘large’ $\nu$ , where Triebel's estimates are not yet sharp, thus making a further step towards the conjectured correct asymptotic behaviour.