!Entropy Numbers of Embeddings of Weighted Besov Spaces. Ii

Abstract

We investigate the asymptotic behaviour of the entropy numbers of the compact embed-denotes a weighted Besov space. We present a general approach which allows us to work with a large class of weights.

Entropy numbers of embeddings of weighted Besov spaces III. Weights of logarithmic type

“…This is a continuation of our previous papers [13,14] on entropy numbers of compact embeddings B s 1 p 1 ,q 1 (R d , w 1 ) → B s 2 p 2 ,q 2 (R d , w 2 ) of weighted Besov spaces. In these articles we considered the case where the ratio of the weights w(x) := w 1 (x)/w 2 (x) is of polynomial type, or a small perturbation thereof, and determined the asymptotic behaviour of the entropy numbers of the corresponding embeddings.…”

“…For the compactness of embeddings of weighted function spaces (with smooth weights) only the behaviour of the weight at infinity is relevant. In [13,14] we considered polynomial weights resp. perturbations of polynomial weights.…”

“…(c) In [14] we expressed our results in terms of the so-called upper and lower indices (usually denoted by α ϕ and β ϕ ) of functions ϕ : [1, ∞) → (0, ∞). Our present conditions on ϕ imply α ϕ = β ϕ = 0; this fact explains the subscript 0 in our notation 0 and W 0 .…”

Local Growth Envelopes of Triebel-Lizorkin Spaces of Generalized Smoothness

“…However, the related functions ϕ are usually defined either on R or on (0, ∞). The above definition was given first in connection with weighted Besov spaces in [21]-see also there for details and further references.…”

Entropy and approximation numbers of embeddings of function spaces with Muckenhoupt weights, II. General weights

“…V, §3, Thm. 9]), and were continued and extended by Kühn, Leopold, Sickel and the second author in the series of papers [21,22,23,33]. As an application one obtains spectral estimates of certain pseudo-differential operators in the way of the program proposed by Edmunds and Triebel [10].…”