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Let $M_1$ and $M_2$ be functions on $[0,1]$ such that $M_1(t^{1/p})$ and $M_2(t^{1/p})$ are Orlicz functions for some $p \in (0,1].$ Assume that $M_2^{-1} (1/t)/M_1^{-1} (1/t)$ is non-decreasing for $t \geq 1.$ Let $(\alpha_i)_{i=1}^\infty$ be a non-increasing sequence of non-negative real numbers. Under some conditions on $(\alpha_i)_{i=1}^\infty,$ sharp two-sided estimates for entropy numbers of diagonal operators $T_\alpha :\ell_{M_1} \rightarrow \ell_{M_2}$ generated by $(\alpha_i)_{i=1}^\infty,$ where $\ell_{M_1}$ and $\ell_{M_2}$ are Orlicz sequence spaces, are proved. The results generalise some works of Edmunds and Netrusov and hence a result of Cobos, K\"{u}hn and Schonbek.

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.

This paper settles a long-standing question by showing that in certain circumstances the entropy numbers of a map do not behave well under real interpolation. To do this, lemmas of combinatorial type are established and used to obtain lower bounds for the entropy numbers of a particular diagonal map acting between Lorentz sequence spaces. These lower bounds contradict the estimates from above that would be obtained if the behaviour of entropy numbers under real interpolation was as good as conjectured. The paper also provides sharp two-sided estimates of the entropy number en(T) of diagonal operators \({T:l_{p}\rightarrow l_{q}, T(\left( a_{k}\right)_{k=1}^{\infty}) = (( \lambda_{k}a_{k}) _{k=1}^{\infty}) ,}\) where 0 < p < q ≤ ∞ and \({\{\lambda _{i}\}_{i=1}^{\infty}}\) is a non-increasing sequence of non-negative numbers with λi = λn for all i ≤ n.

We obtain estimates for the counting function of the Neumann Laplacian on a planar domain bounded by the graph of a lower semicontinuous L1-function. These estimates imply necessary and sufficient conditions for the validity of the classical one-term Weyl formula for the counting function and, under certain restrictions, give an order sharp remainder estimate in this formula.

We study asymptotic distribution of eigenvalues of the Laplacian on a bounded domain in ℝ n . Our main results include an explicit remainder estimate in the Weyl formula for the Dirichlet Laplacian on an arbitrary bounded domain, sufficient conditions for the validity of the Weyl formula for the Neumann Laplacian on a domain with continuous boundary in terms of smoothness of the boundary and a remainder estimate in this formula. In particular, we show that the Weyl formula holds true for the Neumann Laplacian on a Lip α -domain whenever (d−1)/α

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.

We present explicit estimates for the remainder in the Weyl formula for the Laplace operator on a domain Ω, which involve only the most basic characteristics of Ω and hold under minimal assumptions about the boundary \(\partial \Omega\).

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 .