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Sobolev sheaves on the plane

M'hammed Oudrane

TL;DR

The paper constructs a Sobolev sheaf $\mathcal{F}^k$ on the definable site of $\mathbb{R}^2$ such that $\mathcal{F}^k(U)=W^{k,2}(U)$ on small open definable $L$-regular cells, and provides a complete cohomology calculation using Valette's Good direction. It shows higher cohomology vanishes for suitable domains and characterizes obstructions from punctured disks, establishing when $\mathcal{F}^k$ recovers classical Sobolev spaces. The construction relies on a detailed local analysis of boundary cusps (via L-regular decomposition) and a global gluing that yields a sheaf with acyclicity properties. The work sets a foundation for extending Sobolev-sheafifications to higher dimensions and fractional regularity, linking analytic extension, interpolation, and categorical approaches to Sobolev spaces on definable sites.

Abstract

In this paper, we show that for any integer $k \in \mathbb{N}$ there exists a Sobolev sheaf (in the sense of Lebeau) on any definable site of $\mathbb{R}^2$ that agrees with Sobolev spaces on cuspidal domains. We also provide a complete computation of the cohomology of these sheaves using the notion of 'Good direction' introduced by Valette. This paper serves as an introduction to a more general project on the sheafification of Sobolev spaces in higher dimensions.

Sobolev sheaves on the plane

TL;DR

The paper constructs a Sobolev sheaf on the definable site of such that on small open definable -regular cells, and provides a complete cohomology calculation using Valette's Good direction. It shows higher cohomology vanishes for suitable domains and characterizes obstructions from punctured disks, establishing when recovers classical Sobolev spaces. The construction relies on a detailed local analysis of boundary cusps (via L-regular decomposition) and a global gluing that yields a sheaf with acyclicity properties. The work sets a foundation for extending Sobolev-sheafifications to higher dimensions and fractional regularity, linking analytic extension, interpolation, and categorical approaches to Sobolev spaces on definable sites.

Abstract

In this paper, we show that for any integer there exists a Sobolev sheaf (in the sense of Lebeau) on any definable site of that agrees with Sobolev spaces on cuspidal domains. We also provide a complete computation of the cohomology of these sheaves using the notion of 'Good direction' introduced by Valette. This paper serves as an introduction to a more general project on the sheafification of Sobolev spaces in higher dimensions.
Paper Structure (15 sections, 14 theorems, 60 equations, 4 figures)

This paper contains 15 sections, 14 theorems, 60 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Definitions and Preliminaries
  3. Notations:
  4. O-minimal structures
  5. L-regular decomposition
  6. $L^2$-based Sobolov spaces revisited.
  7. The definable site and the main problem.
  8. The spaces $W^{s,2}$ for $s\in ]-\frac{1}{2}, \frac{1}{2}[$.
  9. Construction of the sheaf $\mathcal{F}^k$ on $\mathbb{R}^2$ for $k\in \mathbb{N}$.
  10. The local nature of open definable sets in $\mathbb{R}^2$.
  11. Local definition of the sheaf $\mathcal{F}^k$.
  12. The global definition of $\mathcal{F}^k$ on the site $X_{\mathcal{A}}(\mathbb{R}^2)$ .
  13. Cohomology of the sheaf $\mathcal{F}^k$.
  14. $(W^{1,2} , W^{0,2})$-double extension is a sufficient condition for the sheafification of $W^{s,2}$.
  15. Further discussion

Key Result

Theorem 2.1

Let $p\in \mathbb{N}$ and $\{ X_{1},...,X_{n} \}$ be a finite family of definable sets of $\mathbb{R}^{n}$. Then there is a $C^{p}$-cell decomposition of $\mathbb{R}^{n}$ compatible with this family, i.e. each $X_{i}$ is a union of some cells.

Figures (4)

  • Figure 1: Example of building L-regular cells by induction.
  • Figure 2: The domain $R(r,\gamma _1, \gamma _2)$.
  • Figure 9: Example of a bi-Lipschitz transformation to get a good direction for a closed hypersurface in $\mathbb{R}^n$
  • Figure 10: The cover $\mathcal{V}$.

Theorems & Definitions (43)

  • Theorem 2.1
  • proof
  • Definition 2.2
  • Theorem 2.3
  • proof
  • Definition 3.1
  • Theorem 3.2
  • Proposition 3.3
  • proof
  • Definition 4.1
  • ...and 33 more