Compact embeddings of Sobolev spaces with power weights perturbed by slowly varying functions

Abstract

Key words Weighted Sobolev spaces, entropy numbers, approximation numbers, degenerate elliptic operators, continuous and compact embeddings, slowly varying functions, quadratic forms MSC (2010) 46E35We study compact embeddings of weighted Sobolev spaces into Lebesgue spaces on the unit ball in R n . The weight is of slowly varyingly disturbed polynomial growth with a singularity at the origin. It extends [21], [27] to a wider class of weights. Special attention is paid to the influence of the growth rate of the weight on the quality of compactness, measured in terms of entropy and approximation numbers. In case of Hilbert spaces, the results are related to the distribution of eigenvalues of some degenerate elliptic operators.

Fractional Hardy-Sobolev inequalities for canceling elliptic differential operators

“…holds for some C > 0 and all u ∈ C ∞ c (R N ; E) such that u(0) = 0. Estimate (1.6) is somewhat similar to an estimate given by [12, Theorem 3.3] where the weight |x| ν is present on the right-hand side (this phenomenon was also observed in [8,Remark 4.20]).…”

Existence and regularity of positive solutions of a degenerate elliptic problem

“…Such kind of embeddings of weighted Sobolev spaces with different weights have been studied by some authors, see . Especially, some results of embeddings with power weights in or can be found in .…”

“…We know from the Caffarelli-Kohn-Nirenberg inequality and the related results in [ ) and ≥ 3. Such kind of embeddings of weighted Sobolev spaces with different weights have been studied by some authors, see [2,3,20,27,36,37]. Especially, some results of embeddings with power weights in or ∖{0} can be found in [33,34].…”

Entropy and approximation numbers of weighted Sobolev spaces via bracketing

“…where the maximum is taken over all finite-dimensional linear subspaces S of E m p,σ (Ω). This approach attempts to mimic the Dirichlet-Neumann bracketing described in [9,8,14] if p = 2. This is emphasised by Proposition 3.4.…”

“…where E m p,σ (B) is the closure of C m 0 (B) with respect to the norm and B = {x ∈ R n : |x| < 1} is the unit ball. Problems of this type have been considered so far in [14,9,8]. In case of Hilbert spaces Triebel obtained in [14] sharp results for the corresponding entropy and approximation numbers.…”