Classical and approximate Taylor expansions of weakly differentiable functions

Abstract

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.

Norms supporting the Lebesgue differentiation theorem

“…[3,2,11,12]. In particular, in [2] inequality (1.11) [3], where it is proved that (1.11) holds if, and only if, 1 ≤ q ≤ p. In fact, a different notion of maximal operator is considered in [3], which, however, is equivalent to (1.9) when • X(R n ) is a Lorentz norm, as is easily seen from [7,Equation (3.7)]. A simple sufficient condition for the validity of the Riesz-Wiener type inequality for very general maximal operators is proposed in [12].…”

Functions determining locally solid topological Riesz spaces continuously embedded in $L^0$

“…Furthermore, in the case when the measure space ðS; S; mÞ is totally finite, all rearrangement invariant function spaces are continuously embedded into L 1 ðmÞ (see e.g. [4] for relevant definitions and examples, and the monograph [3] for a comprehensive treatment of the topic). Thus, thanks to (1.1) with p ¼ 1, all rearrangement invariant function spaces built upon a totally finite measure space are continuously embedded into ðL 0 ðmÞ; t m Þ.…”

Negative Orlicz–Sobolev norms and strongly nonlinear systems in fluid mechanics

“…for every u ∈ W 1,A (Ω). Inequality (2.12) is established in [15,Lemma 4.1] in the special case when Ω is a ball. Its proof makes use of a rearrangement type inequality for the norm ∇u L A (Ω) which holds, in fact, for Sobolev functions u on any Lipschitz domain Ω [19, Lemma 4.1 and inequality (3.5)].…”