On nuclearity of embeddings between Besov spaces

Let Bp,qs,α(Ω) be the Besov space with classical smoothness s and additional logarithmic smoothness of order α on a bounded Lipschitz domain Ω in Rd. For s1,s2∈R,1≤p1,p2,q1,q2≤∞ and s1−s2=d−d(1∕p2−1∕p1)+, we show a sufficient condition on q1,q2 for nuclearity of embedding Bp1,q1s1,α1(Ω)↪Bp2,q2s2,α2(Ω). We also show that the condition is necessary in a wide range of parameters.