Pointwise differentiability of higher order for distributions

TL;DR

This paper develops a comprehensive theory of higher order pointwise differentiability for distributions, extending classical notions and establishing new regularity, rectifiability, and approximation results, along with a Poincaré inequality.

Contribution

It introduces a novel framework for higher order differentiability of distributions, including new results for zeroth order and characterizations via distributional derivatives.

Findings

Borel regularity of differentials

Higher order rectifiability of jets

Poincaré inequality involving negative order norms

Abstract

For distributions, we build a theory of higher order pointwise differentiability comprising, for order zero, {\L}ojasiewicz'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 Poincar\'e 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.