On differentiability of Sobolev functions with respect to the Sobolev norm

TL;DR

This paper investigates the relationship between different notions of differentiability in Sobolev spaces, establishing that $W^1_p$-differentiability implies $L_p$-differentiability, but not vice versa, and discusses approximate differentiability.

Contribution

It clarifies the implications between $W^1_p$- and $L_p$-differentiability for Sobolev functions and explores the concept of approximate differentiability.

Findings

$W^1_p$-differentiability implies $L_p$-differentiability

The converse implication does not hold

Discussion of approximate differentiability in Sobolev functions

Abstract

We study connections between the $W^1_p$-differentiability and the $L_p$-differentiability of Sobolev functions. We prove that, $W^1_p$-differentiability implies the $L_p$-differentiability, but the opposite implication is not valid. The notion of approximate differentiability is discussed as well. In addition, we consider the $W^1_p$-differentiability of Sobolev functions $\cp_p$-almost everywhere.