This paper deals with approximation numbers of the compact trace operator of an anisotropic Besov space into some Lp-space,trΓ: Bpps,a (Rn) → Lp(Γ), s > 0, 1 < p < ∞,where Γ is an anisotropic d-set, 0 < d < n. We also prove homogeneity estimates, a homogeneous equivalent norm and the localization property in Bpps,a.

The aim of the paper is twofold. First we give a survey of some recent results concerning the asymptotic behavior of the entropy and approximation numbers of compact Sobolev embeddings. Second we prove new estimates of approximation numbers of embeddings of weighted Besov spaces in the so called limiting case.

Soient $W=L^{\text{'}}\left(\mu \right)$ et $V=C\left(X\right)$. Il existe une application (non linéaire) normiquement continue $T\mapsto P\left(T\right)$ de l’espace des opérateurs bornés de $W$ dans $V$ sur l’espace des opérateurs compacts (resp. faiblement compacts) de $W$ dans $V$ telle que $\parallel T-P\left(T\right)\parallel$ coïncide avec la distance de $T$ au sous-espace formé des opérateurs compacts (resp. faiblement compacts). Pour un opérateur donné $T$ de $W$ dans $V$ on étudie les propriétés de l’ensemble $\mathbf{K}\left(T\right)$ (resp. $\mathbf{F}\left(T\right)$) des opérateurs compacts (resp. faiblement compacts) tel que pour tout $\mathbf{R}$ de $\mathbf{K}\left(T\right)$ (resp. $\mathbf{K}\left(T\right)$) la quantité...

We show that the classes ${}_p^{bern}:=T:\left(bₙ\left(T\right)\right)\in l_p$ associated with the Bernstein numbers bₙ fail to be operator ideals. Moreover, ${}_p^{bern}{\circ }_q^{bern}{⊈}_r^{bern}$ for 1/r = 1/p + 1/q.

We determine the set of all triples 1 ≤ p,q,r ≤ ∞ for which the so-called Marcinkiewicz-Zygmund inequality is satisfied: There exists a constant c≥ 0 such that for each bounded linear operator $T:L_q\left(\mu \right)\to L_p\left(\nu \right)$, each n ∈ ℕ and functions $f_1,...,f_n\in L_q\left(\mu \right)$, $\left(ʃ\right({\sum }_{k=1}^n|T{f}_k{|}^r{)}^{p/r}{d\nu )}^{1/p}\le c\parallel T\parallel \left(ʃ\right({\sum }_{k=1}^n|f_k{|}^r{{)}^{q/r}d\mu )}^{1/q}$. This type of inequality includes as special cases well-known inequalities of Paley, Marcinkiewicz, Zygmund, Grothendieck, and Kwapień. If such a Marcinkiewicz-Zygmund inequality holds for a given triple (p,q,r), then we calculate the best constant c ≥ 0 (with the only exception:...

In this paper, we consider the boundary stabilization of a sandwich beam which consists of two outer stiff layers and a compliant middle layer. Using Riesz basis approach, we show that there is a sequence of generalized eigenfunctions, which forms a Riesz basis in the state space. As a consequence, the spectrum-determined growth condition as well as the exponential stability of the closed-loop system are concluded. Finally, the well-posedness and regularity in the sense of Salamon-Weiss class as...

In this paper, we consider the boundary stabilization of a sandwich beam which consists of two outer stiff layers and a compliant middle layer. Using Riesz basis approach, we show that there is a sequence of generalized eigenfunctions, which forms a Riesz basis in the state space. As a consequence, the spectrum-determined growth condition as well as the exponential stability of the closed-loop system are concluded. Finally, the well-posedness and regularity in the sense of Salamon-Weiss class as...

The dual of a Banach space X is of weak type p if and only if the entropy numbers of an r-nuclear operator with values in a Banach space of weak type q belong to the Lorentz sequence space ${\ell }_{s,r}$ with 1/s + 1/p + 1/q = 1 + 1/r (0 < r < 1, 1 ≤ p, q ≤ 2). It is enough to test this for Y = X*. This extends results of Carl, König and Kühn.