CLASSICAL AND APPROXIMATE TAYLOR EXPANSIONS OF WEAKLY DIFFERENTIABLE FUNCTIONS

The pointwise behavior of Sobolev-type functions, whose weak derivatives up to a given order belong to some rearrangement-invariant Banach function space, is investigated. We introduce a notion of approximate Taylor expansion in norm for these functions, which extends the usual definition of Taylor expansion in $L^p$-sense for standard Sobolev functions. An approximate Taylor expansion for functions in arbitrary-order Sobolev-type spaces, with sharp norm, is established. As a consequence, a characterization of those Sobolev-type spaces in which all functions admit a classical Taylor expansion is derived. In particular, this provides a high-order version of a well-known result of Stein on the differentiability of weakly differentiable functions. Applications of our results to customary classes of Sobolev-type spaces are also presented.