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...

On the interpolation properties of certain generalized spaces of Besov type A.G.Baghdasaryan * We investigate generalized spaces of Besov type defined by some positive infinitely differentiable functions of polynomial growth and determine the K − functional for pairs of this type of H and B − spaces. We prove interpolation theorems for spaces with different anisotropy. Spaces of functions of mixed smoothness are characterized as “B-products’ ” and interpolation theorems for these spaces are proved. Moreover, we establish embedding theorems and a trace theorem. Keywords: Spaces of Besov type, spaces of Sobolev-Liouville type, Spaces of functions of mixed smoothness, interpolation, K − functional, “B-product”, embeddings, traces.

limiting embeddings in weighted function spaces and related entropy numbers