Pointwise differentiability of higher-order for distributions

Ulrich Menne*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

For distributions, we build a theory of higher-order pointwise differentiability comprising, for order zero, Łojasiewicz’s notion of point value. Results include Borel regularity of differentials, higher-order rectifiability of the associated jets, a Rademacher–Stepanov-type differentiability theorem, and a Lusin-type approximation. A substantial part of this development is new also for zeroth order. Moreover, we establish a Poincaré inequality involving the natural norms of negative order of differentiability. As a corollary, we characterise pointwise differentiability in terms of point values of distributional partial derivatives.

Original languageEnglish
Pages (from-to)323-354
Number of pages32
JournalAnalysis and PDE
Volume14
Issue number2
DOIs
Publication statusPublished - 2021

Keywords

  • asymptotic expansion
  • distribution
  • higher-order pointwise differentiability
  • higher-order rectifiability
  • inequality
  • Lusin-type approximation
  • Poincaré
  • Rademacher–Stepanov-type theorem
  • Łojasiewicz point value

ASJC Scopus subject areas

  • Analysis
  • Numerical Analysis
  • Applied Mathematics

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