The paper deals with weighted function spaces of type B s p;q (R n ; w(x)) and F s p;q (R n ; w(x)), where w(x) is a weight function of at most polynomial growth. Of special interest are weight functions of type w(x) = (1 + jxj 2 ) ff=2 (log(2 + jxj)) with ff 0 and 2 R. Our main result deals with estimates for the entropy numbers of compact embeddings between spaces of this type; more precisely, we may extend and tighten some of our previous results in [12]. AMS Subject Classification: 46E 35 Key Words: weighted function spaces, compact embeddings, entropy numbers Introduction 1 Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 Weighted embeddings -- the non-limiting case 3 2 Limiting embeddings, entropy numbers 7 2.1 Estimates from above, an approach via duality arguments . . . . . . . . . . . . . . 8 2.2 Estimates from above, an approach via approximation numbers . . . . . . . . . . . 15 2.3 Estimates...

In this survey we summarise recent results concerning compact embeddings of some weighted function spaces of type B s p;q and F s p;q on R n . We study the asymptotic behaviour of their entropy and approximation numbers. We give some motivation how to apply results of this type in spectral theory. This survey provides a unified approach of recent outcomes (which have mostly been published before) and marks some present `state of the art' in this field of research. AMS Subject Classification: 46 E 35, 41 A 46, 47 B 06 Key words: (weighted) function spaces, compact embeddings, entropy numbers, approximation numbers Introduction This survey reflects some recent developments concerning the obviously symbiotic relationship between weighted function spaces on R n , entropy and approximation numbers of compact embeddings and applications to Received December 6, 1997 Analele Universitat¸ii din Craiova, Seria Matematica - Informatica, XXIV, 1997 2 dorothee d. haroske spectral theo...

. In [14] and [11] we have studied compact embeddings of weighted function spaces on R n , idH : H s 1 p 1 (w(\Delta); R n ) \Gamma! H s 2 p 2 (R n ), s1 ? s2 , 1 ! p1 p2 ! 1, s1 \Gamma n=p1 ? s2 \Gamma n=p2 , and w(x) of the type w(x) = (1 + jxj) ff (log(2 + jxj)) fi , ff 0, fi 2 R. We have determined the asymptotic behaviour of the corresponding entropy numbers ek (idH ). Now we are interested in the limiting case s1 \Gamma n=p1 = s2 \Gamma n=p2 . Let w(x) = log fi hxi, fi ? 0. Then idH cannot be compact (for any fi ? 0), but replacing the Sobolev spaces H s i p i , i = 1; 2, by their logarithmic counterparts, H s i p i (log H)a i , a i 2 R, i = 1; 2, one can prove compactness of the so modified 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 ek (idH;a ). Finally we apply our result to estimate ...

limiting embeddings in weighted function spaces and related entropy numbers