Abstract
We regard the compact Sobolev embeddings between Besov and Sobolev spaces of radial functions on noncompact symmetric spaces of rank one. The asymptotic formula for the behaviour of approximation numbers of these embeddings is described.
Highlights
Approximation numbers measure the closeness by which a bounded operator may be approximated by linear maps of finite range, whereas entropy numbers measure compactness of the operator by means of finite coverings of an image of the unit ball
Approximation and entropy numbers, of compact Sobolev embeddings between function spaces of Sobolev and Besov type on the Euclidean case have been investigated by several authors: D
The necessary and sufficient conditions for compactness of Sobolev embeddings for Besov and Sobolev function spaces defined on the Euclidean space Rd with d ≥ 2, are the same as for the functions on the noncompact Riemannian symmetric space of rank one cf. [21], [22]
Summary
Approximation numbers measure the closeness by which a bounded operator may be approximated by linear maps of finite range, whereas entropy numbers measure compactness of the operator by means of finite coverings of an image of the unit ball. Approximation and entropy numbers, of compact Sobolev embeddings between function spaces of Sobolev and Besov type on the Euclidean case have been investigated by several authors: D. W. Sickel and the first named author found the necessary and sufficient conditions for compactness of the embeddings of radial Besov and Triebel-Lizorkin spaces, cf [21]. The necessary and sufficient conditions for compactness of Sobolev embeddings for Besov and Sobolev function spaces defined on the Euclidean space Rd with d ≥ 2 , are the same as for the functions on the noncompact Riemannian symmetric space of rank one cf [21], [22]. Even though the necessary and sufficient conditions for compactness of Sobolev embedding are the same for Rd and X the asymptotic behaviour of corresponding approximation (as well as entropy) numbers is quite different.
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