Continuity envelopes of spaces of generalised smoothness: a limiting case; embeddings and approximation numbers

António M. Caetano; Dorothee D. Haroske

doi:10.1155/2005/165785

Continuity envelopes of spaces of generalised smoothness: a limiting case; embeddings and approximation numbers

Journal of Function Spaces and Applications ◽

10.1155/2005/165785 ◽

2005 ◽

Vol 3 (1) ◽

pp. 33-71 ◽

Cited By ~ 10

Author(s):

António M. Caetano ◽

Dorothee D. Haroske

Keyword(s):

Asymptotic Behaviour ◽

Function Spaces ◽

Approximation Numbers ◽

Local Growth ◽

Recent Developments ◽

Limiting Case ◽

Growth Envelopes

Continuity envelopes for the spaces of generalised smoothnessBpq(s,Ψ)(ℝn)andFpq(s,Ψ)(ℝn)are studied in the so-called supercriticals=1+n/p, paralleling recent developments for a corresponding limiting case for local growth envelopes of spaces of such a type. In addition, the power of the concept is used in proving conditions for some embeddings between function spaces to hold, as well as in the study of the asymptotic behaviour of approximation numbers of related embeddings.

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Entropy numbers of embedding maps between Besov spaces with an application to eigenvalue problems

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500015341 ◽

1981 ◽

Vol 90 (1-2) ◽

pp. 63-70 ◽

Cited By ~ 5

Author(s):

Bernd Carl

Keyword(s):

Banach Space ◽

Asymptotic Behaviour ◽

Besov Spaces ◽

Eigenvalue Problems ◽

Function Spaces ◽

Sequence Spaces ◽

Entropy Numbers ◽

Diagonal Operators

SynopsisIn this paper we determine the asymptotic behaviour of entropy numbers of embedding maps between Besov sequence spaces and Besov function spaces. The results extend those of M. Š. Birman, M. Z. Solomjak and H. Triebel originally formulated in the language of ε-entropy. It turns out that the characterization of embedding maps between Besov spaces by entropy numbers can be reduced to the characterization of certain diagonal operators by their entropy numbers.Finally, the entropy numbers are applied to the study of eigenvalues of operators acting on a Banach space which admit a factorization through embedding maps between Besov spaces.The statements of this paper are obtained by results recently proved elsewhere by the author.

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