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Exception Sets of Intrinsic and Piecewise Lipschitz Functions.

Gunther Leobacher1, Alexander Steinicke2,3

  • 1Institute of Mathematics and Scientific Computing, University of Graz, Heinrichstraße 36, 8010 Graz, Austria.

Journal of Geometric Analysis
|February 14, 2022
PubMed
Summary

We introduce "permeability" to define exception sets for Lipschitz continuity on metric spaces. Functions continuous outside these permeable sets are Lipschitz continuous intrinsically.

Keywords:
Intrinsic metricPermeable setsPiecewise Lipschitz continuity

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Area of Science:

  • Real Analysis
  • Metric Geometry
  • Functional Analysis

Background:

  • Piecewise Lipschitz continuous functions are crucial in analysis on Euclidean spaces.
  • Understanding the failure of Lipschitz continuity requires characterizing exception sets.
  • Generalizing these concepts to metric spaces presents significant challenges.

Purpose of the Study:

  • To introduce and define a novel concept of "permeability" for sets in metric spaces.
  • To investigate the properties of functions exhibiting Lipschitz continuity outside these permeable sets.
  • To establish conditions under which such functions possess global Lipschitz continuity with respect to the intrinsic metric.

Main Methods:

  • Generalization of piecewise Lipschitz continuity to arbitrary metric spaces.
  • Introduction of the concept of permeable sets as natural exceptions to Lipschitz continuity.
  • Analysis of intrinsic Lipschitz continuity and its relationship with global Lipschitz continuity.

Main Results:

  • A new class of functions on metric spaces is defined, generalizing piecewise Lipschitz functions.
  • Permeable sets are formally introduced and characterized as exception sets for Lipschitz continuity.
  • A key theorem demonstrates that continuous functions intrinsically Lipschitz continuous off a permeable set are globally Lipschitz continuous with respect to the intrinsic metric.
  • Examples of permeable sets in Euclidean spaces, including Lipschitz submanifolds, are provided.

Conclusions:

  • The notion of permeability offers a precise way to understand and handle exceptions to Lipschitz continuity in metric spaces.
  • The main result provides a powerful tool for analyzing functions with localized regularity.
  • The findings extend the applicability of Lipschitz continuity concepts to more general settings beyond Euclidean domains.