On fine differentiability properties of Sobolev functions

Paz Hashash, Vladimir Gol'dshtein, Alexander Ukhlov

Published 2022-07-18Version 1

We study fine differentiability properties of functions in Sobolev spaces. We prove that the difference quotient of $f\in W^{1}_{p}(\mathbb R^n)$ converges to the formal differential of this function in the $W^{1}_{p,\loc}$-topology $\cp_p$-a.~e. under an additional assumption of existence of a refined weak gradient. This result is extended to convergence of remainders in the corresponding Taylor formula for functions in $W^{k}_{p}$ spaces. In addition we prove $L_p-$approximately differentiability $\cp_p-$a.e for functions $f\in W^1_p(\mathbb{R}^n)$ with a refined weak gradient.