Logarithmic Sobolev Spaces on R N ; Entropy Numbers, and Some Application 4 Applications 37 Logarithmic Sobolev Spaces on R N ; Entropy Numbers, and Some Application

In 14] and 11] we have studied compact embeddings of weighted function spaces on R n , p 2 (R n), s1 > s2, 1 < p1 p2 < 1, s1 ? n=p1 > s2 ? n=p2, and w(x) of the type w(x) = (1 + jxj) (log(2 + jxj)) , 0, 2 R. We have determined the asymptotic behaviour of the corresponding entropy numbers e k (idH). Now we are interested in the limiting case s1 ?n=p1 = s2 ?n=p2. of the so modiied embedding idH;a in some cases. In 13] we have followed this idea, introducing logarithmic Lebesgue spaces Lp(log L)a(R n) for this purpose. We continue and extend these results now, and study the entropy numbers e k (idH;a). Finally we apply our result to estimate eigenvalues of the compact operator B = b2 b(; D) b1 acting in some Lp space, where b(; D) belongs to some HH ormander class ?{ 1;; , { > 0, 0 1, and b1; b2 are in (weighted) logarithmic Lebesgue spaces on R n. This gives some counterpart of 15] in the limiting situation.