On the differentiability of directionally differentiable functions and applications

TL;DR

This paper investigates the structure of non-differentiability sets of Lipschitz functions with directional derivatives and applies these findings to analyze the differentiability of the Hardy-Littlewood maximal function.

Contribution

It establishes optimal estimates for non-differentiability sets using k-tangential sets and applies these to study the differentiability of maximal functions.

Findings

Non-differentiability sets of Lipschitz functions are contained in k-tangential sets.

If a function is differentiable outside a countable union of tangential sets, so is its maximal function.

The Hardy-Littlewood maximal function inherits differentiability properties from the original function under certain conditions.

Abstract

In the first part of this paper we establish, in terms of so called k-tangential sets, a kind of optimal estimate for the size and structure of the set of non-differentiability of Lipshitz functions with one-sided directional derivatives. These results can be applied to many important special functions in analysis, like distance functions or different maximal functions. In the second part, having the results from the first part in our use, we focus more carefully on the differentiability properties of the classical Hardy-Littlewood maximal function. For example, we will show that if f is continuous and differentiable outside a countable union of tangential sets, then the same holds to the maximal function Mf as well (if Mf is not identically infinity). As an another example, our results also imply that if f is differentiable almost everywhere, then Mf is differentiable a.e.