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Complex interpolation of weighted Sobolev spaces with boundary conditions

Floris B. Roodenburg

TL;DR

The paper develops a comprehensive framework for complex interpolation of weighted Sobolev spaces with boundary conditions, using power weights that measure distance to the boundary. The authors determine high-order trace spaces for vector-valued Besov, Triebel–Lizorkin, Bessel potential, and Sobolev spaces, prove trace theorems for boundary operators, and establish density results, all within a unified vector-valued, non-A_p-weight setting. These foundations are then combined to obtain explicit interpolation identities for weighted spaces with and without boundary conditions on the half-space and locally on smooth domains, enabling fractional power domain descriptions and applications to PDE techniques. The results extend known unweighted and $A_p$-weighted theories to a broader class of weights, with potential implications for maximal regularity, $H^cal{∞}$-calculus, and evolution equations in weighted settings.

Abstract

We characterise the complex interpolation spaces of weighted vector-valued Sobolev spaces with and without boundary conditions on the half-space and on smooth bounded domains. The weights we consider are power weights that measure the distance to the boundary and do not necessarily belong to the class of Muckenhoupt $A_p$ weights. First, we determine the higher-order trace spaces for weighted vector-valued Besov, Triebel-Lizorkin, Bessel potential and Sobolev spaces. This allows us to derive a trace theorem for boundary operators and to interpolate spaces with boundary conditions. Furthermore, we derive density results for weighted Sobolev spaces with boundary conditions.

Complex interpolation of weighted Sobolev spaces with boundary conditions

TL;DR

The paper develops a comprehensive framework for complex interpolation of weighted Sobolev spaces with boundary conditions, using power weights that measure distance to the boundary. The authors determine high-order trace spaces for vector-valued Besov, Triebel–Lizorkin, Bessel potential, and Sobolev spaces, prove trace theorems for boundary operators, and establish density results, all within a unified vector-valued, non-A_p-weight setting. These foundations are then combined to obtain explicit interpolation identities for weighted spaces with and without boundary conditions on the half-space and locally on smooth domains, enabling fractional power domain descriptions and applications to PDE techniques. The results extend known unweighted and -weighted theories to a broader class of weights, with potential implications for maximal regularity, -calculus, and evolution equations in weighted settings.

Abstract

We characterise the complex interpolation spaces of weighted vector-valued Sobolev spaces with and without boundary conditions on the half-space and on smooth bounded domains. The weights we consider are power weights that measure the distance to the boundary and do not necessarily belong to the class of Muckenhoupt weights. First, we determine the higher-order trace spaces for weighted vector-valued Besov, Triebel-Lizorkin, Bessel potential and Sobolev spaces. This allows us to derive a trace theorem for boundary operators and to interpolate spaces with boundary conditions. Furthermore, we derive density results for weighted Sobolev spaces with boundary conditions.

Paper Structure

This paper contains 20 sections, 28 theorems, 200 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Weighted function spaces
  5. Properties of Banach spaces
  6. Embeddings
  7. The complex interpolation method
  8. Trace spaces of Besov and Triebel-Lizorkin spaces
  9. The zeroth-order trace operator
  10. The higher-order trace operator
  11. Trace spaces of Bessel potential and Sobolev spaces
  12. Trace spaces of weighted spaces on the half-space
  13. Density results
  14. Bessel potential spaces
  15. Sobolev spaces
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $p\in(1,\infty)$, $k\in\{2,3,\dots\}$ and $\gamma\in (-1,\infty)\setminus\{jp-1:j\in\mathbb{N}_1\}$. Then for $\ell\in \{1,\dots, k-1\}$ we have Moreover, by localisation, the results also hold for smooth bounded domains.

Theorems & Definitions (62)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 2.1
  • Proposition 2.2
  • Theorem 2.3
  • proof
  • Definition 2.4
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • ...and 52 more