Page 11 - Textos de Matemática Vol. 34

REMARKS ON SOBOLEV EMBEDDINGS
DAVID E. EDMUNDS
Dedicated to Professor J. A. Sampaio Martins on the occasion of his sixtieth birthday
Abstract. A survey is given of recent work concerning the asymptotic be- haviour of the approximation numbers of Sobolev embeddings, and some open problems are discussed.
1. Introduction
Let Ω be a bounded domain in Rn with smooth boundary, let s ∈ N, suppose
that 1 < p2 ≤ p1 < ∞ and let id be the natural embedding of the Sobolev space
Ws (Ω) in L (Ω). Then it is very well known that the approximation numbers p1 p2
ak(id) of id satisfy the inequalities
(1.1) c1k−s/n ≤ ak(id) ≤ c2k−s/n, k ∈ N,
where c1 and c2 are positive constants independent of k. Results of this kind go
back a long way, certainly to 1967 and the famous work of Birman and Solomyak [1]
in which the method of piecewise-polynomial approximations was used. Since then
there has been a flood of papers in which upper and lower estimates were obtained
for the approximation or entropy numbers of embeddings between function spaces
in circumstances of considerable generality and often by means of the technique of
atomic and subatomic decompositions. For example, the spaces concerned could
be of (possibly weighted) Besov (Bs (Ω)) or Lizorkin-Triebel (Fs (Ω)) type, with p,q p,q
s ∈ R and p,q ∈ (0,∞] (p < ∞ for the F spaces); various generalisations of the notion of smoothness can be allowed. Moreover, anisotropy is also permitted, as is shown, for example, by the work [17] of Jorge Martins, whose birthday we are celebrating. In much (but by no means all) of this work the upper and lower estimates are sharp in the sense that they decay at the same rate as k → ∞. Further details will be found in [3], [10], [12], [13], [18], [21] and the references given in these works.
Here I focus attention on the question of whether estimates such as (1.1), sharp though they are, can be refined to show that the approximation numbers have a genuine asymptotic behaviour. In terms of (1.1), this amounts to asking whether
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