Non-local distance functions and geometric regularity

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Abstract

We establish the equivalence between the regularity (rectifiability) of sets and suitable estimates on the oscillation of the gradient for smooth non-local distance functions. A prototypical example of such a distance was introduced, as part of a larger PDE theory, by David, Feneuil, and Mayboroda in [5]. The results apply to all dimensions and co-dimensions, require no underlying topological assumptions, and provide a surprisingly rich class of analytic characterizations of rectifiability.

Original languageEnglish (US)
Article number109649
JournalAdvances in Mathematics
Volume445
DOIs
StatePublished - May 2024

Bibliographical note

Publisher Copyright:
© 2024 Elsevier Inc.

Keywords

  • Regularized distance
  • Square function estimates
  • Uniform rectifiability

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