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Refined localization spaces, Kondratiev spaces with fractional smoothness and extension operators

Markus Hansen, Cornelia Schneider

TL;DR

This work extends the Kondratiev scale to fractional smoothness by exploiting its close relationship with refined localization spaces and employing complex interpolation. It establishes a precise bridge between ${\\mathcal{K}}^m_{a,p}$ and refined localization spaces, enabling fractionally smooth spaces ${\\mathfrak{K}}^s_{a,p}$ to be constructed via interpolation and transferred from the whole space to polyhedral domains using Stein extension. The authors prove both wavelet-based and interpolation-based characterizations, provide a universal extension operator for polyhedral cones, and derive Sobolev-type embeddings for fractional Kondratiev spaces on nonsmooth domains. These results yield a robust toolkit for elliptic problems on domains with corners and edges, with concrete implications for regularity theory and numerical approximation. Overall, the paper advances a unified framework for fractional Kondratiev spaces, their interpolation, and their embedding properties on complex geometric domains.

Abstract

In this paper, we introduce Kondratiev spaces of fractional smoothness based on their close relation to refined localization spaces. Moreover, we investigate relations to other approaches leading to extensions of the scale of Kondratiev spaces with integer order of smoothness, based on complex interpolation, and give further results for complex interpolation of those function spaces. As it turns out to be one of the main tools in studying these spaces on domains of polyhedral type, certain aspects of the analysis of Stein's extension operator are revisited. Finally, as an application, we study Sobolev-type embeddings.

Refined localization spaces, Kondratiev spaces with fractional smoothness and extension operators

TL;DR

This work extends the Kondratiev scale to fractional smoothness by exploiting its close relationship with refined localization spaces and employing complex interpolation. It establishes a precise bridge between and refined localization spaces, enabling fractionally smooth spaces to be constructed via interpolation and transferred from the whole space to polyhedral domains using Stein extension. The authors prove both wavelet-based and interpolation-based characterizations, provide a universal extension operator for polyhedral cones, and derive Sobolev-type embeddings for fractional Kondratiev spaces on nonsmooth domains. These results yield a robust toolkit for elliptic problems on domains with corners and edges, with concrete implications for regularity theory and numerical approximation. Overall, the paper advances a unified framework for fractional Kondratiev spaces, their interpolation, and their embedding properties on complex geometric domains.

Abstract

In this paper, we introduce Kondratiev spaces of fractional smoothness based on their close relation to refined localization spaces. Moreover, we investigate relations to other approaches leading to extensions of the scale of Kondratiev spaces with integer order of smoothness, based on complex interpolation, and give further results for complex interpolation of those function spaces. As it turns out to be one of the main tools in studying these spaces on domains of polyhedral type, certain aspects of the analysis of Stein's extension operator are revisited. Finally, as an application, we study Sobolev-type embeddings.
Paper Structure (18 sections, 21 theorems, 105 equations)

This paper contains 18 sections, 21 theorems, 105 equations.

Table of Contents

  1. Introduction
  2. Kondratiev spaces on domains of polyhedral type
  3. Domains of polyhedral type
  4. Kondratiev spaces
  5. Localization principles
  6. Extension operators
  7. Refined localization spaces
  8. Definition and basic properties
  9. Wavelet characterization
  10. Complex interpolation of Kondratiev spaces
  11. A relation between Kondratiev and refined localization spaces
  12. Complex interpolation of refined localization spaces
  13. Complex interpolation of Kondratiev spaces
  14. An alternative approach to Kondratiev spaces with fractional smoothness
  15. A general interpolation result for Kondratiev spaces
  16. ...and 3 more sections

Key Result

Proposition 1

For $b\in{\mathbb R}$ define the mapping This operator is an isomorphism from ${\mathcal{K}}^m_{a,p}(\Omega,M)$ onto ${\mathcal{K}}^m_{a+b,p}(\Omega,M)$.

Theorems & Definitions (49)

  • Remark 1
  • Definition 1
  • Definition 2
  • Remark 2
  • Remark 3
  • Remark 4
  • Definition 3
  • Remark 5
  • Proposition 1
  • Lemma 1
  • ...and 39 more