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Function spaces and trace theorems for maximally subelliptic boundary value problems

Brian Street

TL;DR

This work extends the elliptic trace theory to maximally subelliptic boundary value problems by building Besov and Triebel--Lizorkin spaces intrinsically on manifolds with boundary, adapted to Hörmander vector fields. It develops a boundary-sensitive Littlewood--Paley framework using pre- and elementary operators in place of Fourier multipliers, and proves that trace maps are continuous retractions with explicit image characterizations. The theory carefully handles non-characteristic boundary points, introduces filtrations of sheaves to encode Carnot--Carathéodory geometry, and establishes extension/restriction mechanisms that are robust under ambient choices. Collectively, these results lay the function-space foundation for a general maximally subelliptic boundary-value problem theory and generalize classical trace results to this degenerate, non-elliptic setting.

Abstract

We introduce Besov and Triebel--Lizorkin spaces on a manifold with boundary adapted to Hörmander vector fields, near a so-called non-characteristic point of the boundary. We prove sharp results in these spaces for the corresponding restriction and trace operators, show these operators are retractions, and other related results. This is the second paper in a forthcoming series devoted to a general theory of maximally subelliptic boundary value problems, and lays the function space foundation for this general theory.

Function spaces and trace theorems for maximally subelliptic boundary value problems

TL;DR

This work extends the elliptic trace theory to maximally subelliptic boundary value problems by building Besov and Triebel--Lizorkin spaces intrinsically on manifolds with boundary, adapted to Hörmander vector fields. It develops a boundary-sensitive Littlewood--Paley framework using pre- and elementary operators in place of Fourier multipliers, and proves that trace maps are continuous retractions with explicit image characterizations. The theory carefully handles non-characteristic boundary points, introduces filtrations of sheaves to encode Carnot--Carathéodory geometry, and establishes extension/restriction mechanisms that are robust under ambient choices. Collectively, these results lay the function-space foundation for a general maximally subelliptic boundary-value problem theory and generalize classical trace results to this degenerate, non-elliptic setting.

Abstract

We introduce Besov and Triebel--Lizorkin spaces on a manifold with boundary adapted to Hörmander vector fields, near a so-called non-characteristic point of the boundary. We prove sharp results in these spaces for the corresponding restriction and trace operators, show these operators are retractions, and other related results. This is the second paper in a forthcoming series devoted to a general theory of maximally subelliptic boundary value problems, and lays the function space foundation for this general theory.
Paper Structure (46 sections, 162 theorems, 552 equations)

This paper contains 46 sections, 162 theorems, 552 equations.

Table of Contents

  1. Introduction
  2. Background: Classical trace theorems for elliptic function spaces
  3. Basic definitions and global corollaries
  4. Informal description of the norms
  5. Vector fields and sheaves
  6. Filtrations of sheaves
  7. Filtrations of sheaves on submanifolds and non-characteristic points
  8. Metrics and Volumes
  9. Differential operators
  10. Besov and Triebel--Lizorkin spaces
  11. The classical setting
  12. Littlewood--Paley theory
  13. Main definitions
  14. Scaling
  15. Elementary and pre-elementary operators
  16. ...and 31 more sections

Key Result

Theorem 2.2

The map $f\mapsto f|_{\mathbb{R}^n_{> 0}}$, $\mathscr{D}'\IfNoValueTF{\mathbb{R}^n}{}{(\mathbb{R}^n)}\rightarrow \mathscr{D}'\IfNoValueTF{\mathbb{R}^n_{> 0}}{}{(\mathbb{R}^n_{> 0})}$, is a retraction $\mathscr{B}^{s}_{p,q}(\mathbb{R}^n)\rightarrow \mathscr{B}^{s}_{p,q}(\mathbb{R}^n_{\geq 0})$ and $\

Theorems & Definitions (386)

  • Definition 1.1
  • Remark 2.1
  • Theorem 2.2: TriebelTheoryOfFunctionSpaces
  • Theorem 2.3: TriebelTheoryOfFunctionSpaces and RunstSickelSobolevSpacesOfFractionalOrder
  • Remark 2.4
  • Remark 2.5
  • Theorem 2.6: RunstSickelSobolevSpacesOfFractionalOrder
  • Remark 2.7
  • Definition 3.2
  • Example 3.3
  • ...and 376 more