Higher order pointwise differential for distribution

TL;DR

This paper investigates the properties of pointwise differentials for distributions associated with negative order Sobolev functions, establishing regularity, approximation, rectifiability, and differentiability results.

Contribution

It provides new theoretical insights into the structure and regularity of pointwise differentials for negative Sobolev distributions, extending classical differential concepts.

Findings

Proves Borel regularity of these differentials

Establishes Lusin approximation results

Demonstrates rectifiability and Rademacher-type theorems

Abstract

The notion of pointwise differentials for distributions is a way to extract local information of distributions by rescaling the distribution at a point. In this paper, we study the pointwise differentials for distributions corresponding to a negative order Sobolev functions. Our main results prove Borel regularity, Lusin approximation, rectifiability, and a Rademacher theorem for these pointwise differentials.