Entropy and approximation numbers of weighted Sobolev spaces via bracketing

TL;DR

This paper studies the asymptotic behavior of entropy and approximation numbers of weighted Sobolev space embeddings with singular weights, using a bracketing technique inspired by Evans and Harris.

Contribution

It introduces a bracketing method to analyze the asymptotics of these embeddings, extending techniques used for unweighted cases to weighted Sobolev spaces with singularities.

Findings

Derived asymptotic formulas for entropy numbers.

Established bounds for approximation numbers.

Extended bracketing techniques to weighted Sobolev spaces.

Abstract

We investigate the asymptotic behaviour of entropy and approximation numbers of the compact embedding $E^m_{p,\sigma}(B)\hookrightarrow L_p(B)$, $1\leq p<\infty,$ defined on the unit ball $B$ in $\mathbb{R}^n$. Here $E^m_{p,\sigma}(B)$ denotes a Sobolev space with a power weight perturbed by a logarithmic function. The weight contains a singularity at the origin. Inspired by Evans and Harris, we apply a bracketing technique which is an analogue to that of Dirichlet-Neumann-bracketing used by Triebel if $p=2$.