On fine differentiability properties of Sobolev functions

Abstract

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.Comment: 24 page