Norms supporting the Lebesgue differentiation theorem

TL;DR

This paper generalizes the Lebesgue differentiation theorem by replacing the $L^p$ norm with any rearrangement-invariant norm, establishing conditions under which the theorem still holds.

Contribution

It provides necessary and sufficient conditions for rearrangement-invariant norms to support the Lebesgue differentiation theorem, extending classical results to broader norm classes.

Findings

Lorentz and Orlicz norms support the theorem

Characterization of norms supporting Lebesgue differentiation

Extension of Lebesgue's theorem to rearrangement-invariant norms

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.