We study compact embeddings for weighted spaces of Besov and Triebel-Lizorkin type where the weight belongs to some Muckenhoupt Ap class. For weights of purely polynomial growth, both near some singular point and at infinity, we obtain sharp asymptotic estimates for the entropy numbers and approximation numbers of this embedding. The main tool is a discretization in terms of wavelet bases.

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We investigate asymptotic behaviour of approximation numbers of Sobolev embeddings between weighted function spaces of Sobolev-Hardy-Besov type with polynomials weights. The exact estimates are proved in almost all cases.Approximation numbers of Sobolev embeddings between functions spaces of SobolevBesov-Hardy type have been studied in recent years by several authors. The approximation numbers of Sobolev embeddings of function spaces defined on bounded domains were studied by Edmunds and Triebel; cf. [6][7][8]. Later some of their estimates were improved by Caetano; cf. [3]. For weighted function spaces the counterpart of the theory was studied by Mynbaev and Otel'baev [20] in the case of fractional Sobolev spaces, and then more generally by Haroske, cf. [11], and by Caetano, cf. [3]. Some later results are described in [12]. In contrast to the function spaces on bounded domains, where the exact estimates were proved in almost all cases, the estimates for weighted spaces are less final. The importance of the asymptotic behavior of approximation and entropy numbers of Sobolev embeddings for the spectral theory of operators is discussed in [8]; cf. also [5,15].

We study embeddings of spaces of Besov-Morrey type, idΩ :where Ω ⊂ R d is a bounded domain, and obtain necessary and sufficient conditions for the continuity and compactness of idΩ. This continues our earlier studies relating to the case of R d . Moreover, we also characterise embeddings into the scale of Lp spaces or into the space of bounded continuous functions.

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