A microlocal characterization of Lipschitz continuity

TL;DR

This paper provides a microlocal perspective on Lipschitz continuity and strict differentiability of maps between manifolds, using microsupports of sheaves to characterize these properties.

Contribution

It introduces a novel microlocal characterization of Lipschitz continuity and strict differentiability for maps between manifolds.

Findings

Lipschitz continuity characterized via microsupport of the graph's sheaf

Bounds on microsupport of the graph of a continuous map

Microlocal criteria for strict differentiability

Abstract

We study continuous maps between differential manifolds from a microlocal point of view. In particular, we characterize the Lipschitz continuity of these maps in terms of the microsupport of the constant sheaf on their graph. Furthermore, we give lower and upper bounds on the microsupport of the graph of a continuous map and use these bounds to characterize strict differentiability in microlocal terms.