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Higher order pointwise differential for distribution

Yu-Tong Liu

TL;DR

This work extends the theory of pointwise differentials to distributions valued in a Banach space via the $\nu_{i,p}$-scale, establishing Borel regularity, Lusin-type approximation, rectifiability of the jets, and a higher-order Rademacher-type theorem. The analysis combines deformation arguments, a detailed PDE toolkit for poly-\(\mathrm{Laplacian}\) operators, and topological methods for measurability in spaces of distributions. The results unify and generalize previous work (notably by Menne) to Sobolev-type norms, enabling robust local regularity descriptions of distributional objects. The findings have implications for the geometric-measure-theoretic understanding of distributions and their higher-order jets, with potential applications to partial differential equations and variational problems.

Abstract

The notion of pointwise differentials for distributions is a way to extract local information of distributions by rescaling the distribution at a point. In this paper, we study the pointwise differentials for distributions corresponding to a negative order Sobolev functions. Our main results prove Borel regularity, Lusin approximation, rectifiability, and a Rademacher theorem for these pointwise differentials.

Higher order pointwise differential for distribution

TL;DR

This work extends the theory of pointwise differentials to distributions valued in a Banach space via the -scale, establishing Borel regularity, Lusin-type approximation, rectifiability of the jets, and a higher-order Rademacher-type theorem. The analysis combines deformation arguments, a detailed PDE toolkit for poly- operators, and topological methods for measurability in spaces of distributions. The results unify and generalize previous work (notably by Menne) to Sobolev-type norms, enabling robust local regularity descriptions of distributional objects. The findings have implications for the geometric-measure-theoretic understanding of distributions and their higher-order jets, with potential applications to partial differential equations and variational problems.

Abstract

The notion of pointwise differentials for distributions is a way to extract local information of distributions by rescaling the distribution at a point. In this paper, we study the pointwise differentials for distributions corresponding to a negative order Sobolev functions. Our main results prove Borel regularity, Lusin approximation, rectifiability, and a Rademacher theorem for these pointwise differentials.

Paper Structure

This paper contains 19 sections, 30 theorems, 120 equations.

Table of Contents

  1. Introduction
  2. A Criteria of differentiability
  3. Borel regularity of the differential
  4. Rectifiability of the family of $k$ jets
  5. Lusin approximation by functions of class $k$
  6. A Rademacher-Stepanov type theorem
  7. Organization of the paper
  8. Acknowledgement
  9. Notation
  10. Symmetric algebra
  11. Properties of weakly differentiable functions
  12. A criterion for pointwise differentiability
  13. The graph of a pointwise differential
  14. Rectifiability of the family of k jets and a Lusin type theorem
  15. Auxiliary PDE estimates
  16. ...and 4 more sections

Key Result

Theorem A

Suppose $i,m$ are nonnegative integers, $k$ is an integer, $0<\alpha \leq 1$, $Y$ is a Banach space, $1\leq p \leq \infty$, $T \in \mathscr D'(\mathbf R^n,Y)$, and $a\in \mathbf R^n$.

Theorems & Definitions (74)

  • Definition 1.1
  • Theorem A: see \ref{['ThmB']}
  • Theorem B
  • Corollary 1.2
  • Theorem C
  • Theorem D
  • Theorem E
  • Lemma 4.1
  • proof
  • Theorem 4.2
  • ...and 64 more