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Entropy numbers of operators acting between vector-valued sequence spaces are estimated using information about the coordinate mappings. To do this some new ideas of combinatorial type are used. The results are applied to give sharp two-sided estimates of the entropy numbers of some embeddings of Besov spaces. For instance, our main result allows us to give exact two-sided estimates of the entropy numbers of the natural embedding of in where Q = (0, 1)d; θ1, θ2, p1, p2 ∈ (0, ∞], when the condition 1/θ1 − 1/θ2 ≥ 1/p1 − 1/p2 > 0 is satisfied. This work enables us to construct an example showing that the behaviour under real interpolation of entropy numbers can be even worse than in the example of 7.

The purpose of this talk is to present a certain method of obtaining upper estimates of eigenvalues of Schrodinger type operators on Riemannian manifolds, which was introduced in the paper Grigor’yan A., Netrusov Yu., Yau S.-T. Eigenvalues of elliptic operators and geometric applications, Surveys in Differential Geometry IX (2004), pp.147-218. The core of the method is the construction that allows to choose a given number of disjoint sets on a manifold such that one can control simultaneously their volumes from below and their capacities from above. The main technical tool for that is the following theorem. Theorem 1 Let (X, d) be a metric space satisfying the following covering property: there exists a constant N such that any metric ball of radius r in X can be covered by at most N balls of radii r/2. Let all metric balls in X be precompact sets, and let ν be a non-atomic Radon measure on X. Then, for any positive integer k, there exists a sequence {Ai}ki=1 of k annuli in X such that the annuli {2Ai}ki=1 are disjoint and, for any i = 1, 2, ..., k

It is shown that some well-known properties of the Sobolev spaceL (Ω) do not admit extension to the spaceL (Ω) of the functions withl-th order derivatives inL p (Ω),l>1, without requirements to the domain Ω. Namely, we give examples of Ω such that In the Appendix necessary and sufficient conditions are given for the imbeddingsL (Ω)⊂L q (Ω, μ) andH (R n )⊂L q (R n , μ), wherep≥1,p>q>0, μ is a measure andH (Ω) is the Bessel potential space, 1 0.

Let ϰi(Hl(V)) denote the negative eigenvalues of the operatorHlu≔(−Δ)lu−V≧0,x ℝd onL2(ℝd). We prove the two-sided estimate Open image in new window . We discuss bounds on the Riesz means Open image in new window .