On dilation operators and sampling numbers

We consider the dilation operators Tk:f→f(2k.) in the frame of Besov spaces Bpqs(ℝd) with 1 ≤p,q≤∞. If s > 0, Tk is a bounded linear operator from Bpqs(ℝd) into itself and there are optimal bounds for its norm, see [4, 2.3.1]. We study the situation in the case s = 0, an open problem mentioned also in [4]. It turns out, that new effects based on Littlewood-Paley theory appear. In the second part of the paper, we apply these results to the study of the so-called sampling numbers of the embedding id:Bpq1s1(Ω)→Bpq20(Ω), where Ω=(0,1)d. It was observed already in [13] that the estimates from above for the norm of the dilation operator have their immediate counterpart in the estimates from above for the sampling numbers. In this paper we show that even in the limiting case s2=0 (left open so far), this general method supplies optimal results