Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Relations between Kondratiev spaces and refined localization Triebel-Lizorkin spaces

Markus Hansen, Benjamin Scharf, Cornelia Schneider

TL;DR

This work develops a sharp bridge between Kondratiev weighted Sobolev spaces and refined localization Triebel–Lizorkin spaces to translate boundary singularity regularity into Besov/Triebel–Lizorkin adaptivity scales. It proves that for suitable domains ${\mathcal{K}}^m_{m,p}(D)$ coincides with ${F}^{m,\text{rloc}}_{p,2}(D)$ and, more generally, establishes embeddings ${\mathcal{K}}^m_{a,p}(D)\hookrightarrow F^{m,\text{rloc}}_{\tau,2}(D)$ (and into $F^m_{\tau,2}(D)$) under precise conditions on $(m,a,p,\tau)$ and the domain’s singular set. Two complementary proofs are offered: a Hölder-inequality approach and a localization-based method, both aided by diffeomorphism invariance to extend results to polyhedral domains. The reverse embeddings are also analyzed, clarifying when refined localization spaces sit inside Kondratiev spaces. Collectively, these embeddings enable sharper predictions for convergence rates of adaptive wavelet and finite element schemes by aligning PDE regularity with appropriate adaptivity scales.

Abstract

We investigate the close relation between certain weighted Sobolev spaces (Kondratiev spaces) and refined localization spaces from introduced by Triebel [39,40]. In particular, using a characterization for refined localization spaces from Scharf [32], we considerably improve an embedding from Hansen [17]. This embedding is of special interest in connection with convergence rates for adaptive approximation schemes.

Relations between Kondratiev spaces and refined localization Triebel-Lizorkin spaces

TL;DR

This work develops a sharp bridge between Kondratiev weighted Sobolev spaces and refined localization Triebel–Lizorkin spaces to translate boundary singularity regularity into Besov/Triebel–Lizorkin adaptivity scales. It proves that for suitable domains coincides with and, more generally, establishes embeddings (and into ) under precise conditions on and the domain’s singular set. Two complementary proofs are offered: a Hölder-inequality approach and a localization-based method, both aided by diffeomorphism invariance to extend results to polyhedral domains. The reverse embeddings are also analyzed, clarifying when refined localization spaces sit inside Kondratiev spaces. Collectively, these embeddings enable sharper predictions for convergence rates of adaptive wavelet and finite element schemes by aligning PDE regularity with appropriate adaptivity scales.

Abstract

We investigate the close relation between certain weighted Sobolev spaces (Kondratiev spaces) and refined localization spaces from introduced by Triebel [39,40]. In particular, using a characterization for refined localization spaces from Scharf [32], we considerably improve an embedding from Hansen [17]. This embedding is of special interest in connection with convergence rates for adaptive approximation schemes.
Paper Structure (14 sections, 21 theorems, 95 equations)

This paper contains 14 sections, 21 theorems, 95 equations.

Table of Contents

  1. Introduction
  2. Function spaces
  3. Triebel-Lizorkin spaces
  4. Kondratiev spaces
  5. Kondratiev spaces on $\mathbb{R}^d$
  6. Refined localization spaces
  7. A relation between Kondratiev and refined localization spaces
  8. The spaces ${\mathcal{K}}^m_{m,p}(D)$
  9. Localization of Kondratiev spaces
  10. Invariance under diffeomorphisms
  11. An embedding result
  12. Approach via Hölder inequalities
  13. Second approach via localization
  14. The reverse embedding

Key Result

Proposition 2.1

Let $D$ be some bounded polyhedral domain without cracks in $\mathbb{R}^d$, $d=2,3$. Consider the problem where $A=(a_{i,j})_{i,j=1}^d$ is symmetric and Let the bilinear form satisfy for all $v,w\in H^1_0(D)$ and for some constants $0<r\leq R<\infty$. Then there exists some $\overline a>0$ such that for any $m\in\mathbb{N}_0$, any $|a|<\overline a$, and any $f\in{\mathcal{K}}^{m-1}_{a-1}(D)$ t

Theorems & Definitions (45)

  • Proposition 2.1
  • Definition 2.2
  • Remark 2.3
  • Lemma 2.4
  • proof
  • Lemma 3.1
  • Theorem 3.2
  • proof
  • Remark 3.3
  • Corollary 3.4
  • ...and 35 more