Norms supporting the Lebesgue differentiation theorem

Abstract

Abstract. A version of the Lebesgue differentiation theorem is offered, where the L p norm is replaced with any rearrangement-invariant norm. Necessary and sufficient conditions for a norm of this kind to support the Lebesgue differentiation theorem are established. In particular, Lorentz, Orlicz and other customary norms for which Lebesgue's theorem holds are characterized.

Diffeomorphic approximation of planar Sobolev homeomorphisms in Orlicz–Sobolev spaces

“…In order to introduce our main result we include the definition of the Lebesgue point property. This property is stronger than the absolute continuity (see (2.6)) of the norm and has been thoroughly characterised in [9]. We refer to the points x for which (1.2) hold as Lebesgue points in X.…”

Diffeomorphic approximation of planar Sobolev homeomorphisms in rearrangement invariant spaces

“…In order to introduce our main result we include the definition of the Lebesgue point property. This property is stronger than the absolute continuity (see (2.8)) of the norm and has been thoroughly characterised in [5]. Definition 1.1 (Lebesgue point property).…”

“…Another interesting case is X = Λ ϕ , the Lorentz endpoint space associated with a (non-identically vanishing) concave function ϕ : [0, ∞) → [0, ∞) satisfying lim s→0+ ϕ(s) = 0. For details, see [5].…”

“…It is not hard to observe that the absolute continuity of the norm of X is a necessary condition for diffeomorphic approximation of homeomorphisms in W 1 X and by [5] it is also necessary for the Lebesgue point property. Our technique relies heavily on estimates derived directly from the Lebesgue point property.…”

“…Remark 2.3. If X has the Lebesgue point property, we have by [5,Proposition 3.1], that the norm is locally absolutely continuous. Therefore one has (2.9) and Lemma 2.2 can be applied for such a space.…”