Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Widths and entropy numbers of embeddings of Sobolev classes on a Hölder domain

A. A. Vasil'eva

TL;DR

This work advances the understanding of Sobolev embeddings on Hölder domains by deriving sharper upper bounds for entropy numbers and widths, using a tree-structured partition and piecewise polynomial approximations. It generalizes to domains with Hölder singularities concentrated on $h$-sets, yielding a refined decay rate that can match Lipschitz-domain behavior under suitable parameter relations. The paper also provides lower-bound constructions on specific $h$-sets to establish sharpness and includes a supplementary treatment showing Besov–Holder domains decompose into model subdomains with cone-like geometry. Overall, the results illuminate how geometric irregularities of the domain govern approximation properties of Sobolev embeddings through precise exponent formulas.

Abstract

In the present paper we improve Besov's recent result about upper estimates for the entropy numbers of Sobolev classes on a Hölder domain (in the case when the definition of the Sobolev class involves all partial derivatives of order $r$). We also obtain upper estimates for the Kolmogorov, linear and the Gelfand widths.

Widths and entropy numbers of embeddings of Sobolev classes on a Hölder domain

TL;DR

This work advances the understanding of Sobolev embeddings on Hölder domains by deriving sharper upper bounds for entropy numbers and widths, using a tree-structured partition and piecewise polynomial approximations. It generalizes to domains with Hölder singularities concentrated on -sets, yielding a refined decay rate that can match Lipschitz-domain behavior under suitable parameter relations. The paper also provides lower-bound constructions on specific -sets to establish sharpness and includes a supplementary treatment showing Besov–Holder domains decompose into model subdomains with cone-like geometry. Overall, the results illuminate how geometric irregularities of the domain govern approximation properties of Sobolev embeddings through precise exponent formulas.

Abstract

In the present paper we improve Besov's recent result about upper estimates for the entropy numbers of Sobolev classes on a Hölder domain (in the case when the definition of the Sobolev class involves all partial derivatives of order ). We also obtain upper estimates for the Kolmogorov, linear and the Gelfand widths.

Paper Structure

This paper contains 7 sections, 8 theorems, 226 equations.

Table of Contents

  1. Introduction
  2. Auxilliary assertions
  3. The tree-like structure of the domain $\Omega \in {\cal G}_{\varphi_1, \, \dots, \, \varphi_{d-1}}$
  4. Proof of Theorems \ref{['main_entr']}, \ref{['main_widths']}
  5. Proof of Theorem \ref{['h_set_e_w']} and the upper estimate in Example \ref{['exa1']}
  6. The example for the lower estimate
  7. Supplement

Key Result

Theorem 1

Let $1\leqslant p\leqslant q\leqslant\infty$, $\Omega \in {\cal G}'_{\varphi_1, \, \dots, \, \varphi_{d-1}}$, where $\varphi_i$ satisfy phi_prop. Suppose that where $\sigma \geqslant 1$ and the function $\Lambda$ is locally absolutely continuous and satisfies lam_slow. Let $r\in \mathbb{N}$, We set Let $\alpha_1\ne \alpha_2$, $j_*\in \{1, \, 2\}$, $\alpha_{j_*}= \min \{\alpha_1, \, \alpha_2\}$.

Theorems & Definitions (19)

  • Definition 1
  • Theorem 1
  • Theorem 2
  • Remark 1
  • Definition 2
  • Remark 2
  • Definition 3
  • Remark 3
  • Theorem 3
  • Remark 4
  • ...and 9 more