Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 39, 2014, 527-544

CLASSICAL AND APPROXIMATE TAYLOR EXPANSIONS OF WEAKLY DIFFERENTIABLE FUNCTIONS

Paola Cavaliere and Andrea Cianchi

Università di Salerno, Dipartimento di Matematica
Via Giovanni Paolo II, 84084 Fisciano (SA), Italy; pcavaliere 'at' unisa.it

Università di Firenze, Dipartimento di Matematica e Informatica "U. Dini"
Piazza Ghiberti 27, 50122 Firenze, Italy; andrea.cianchi 'at' unifi.it

Abstract. 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 higher-order version of a well-known result of Stein [27] on the differentiability of weakly differentiable functions. Applications of our results to customary classes of Sobolev-type spaces are also presented.

2010 Mathematics Subject Classification: Primary 46E35, 46E30.

Key words: Sobolev spaces, approximate differentiability, approximate Taylor expansion, rearrangement invariant spaces, Lorentz spaces, Orlicz spaces.

Reference to this article: P. Cavaliere and A. Cianchi: Classical and approximate Taylor expansions of weakly differentiable functions. Ann. Acad. Sci. Fenn. Math. 39 (2014), 527-544.

Full document as PDF file

doi:10.5186/aasfm.2014.3933