A median approach to differentiation bases

TL;DR

This paper extends the Lebesgue differentiation theorem by replacing averages with medians over differentiation bases, providing new characterizations and applications to function spaces like Besov and Triebel--Lizorkin spaces.

Contribution

It introduces a median-based differentiation framework and characterizes its properties, broadening the theorem's applicability to non-integrable functions and metric measure spaces.

Findings

Median maximal function characterizations of differentiation

Application to non-integrable functions in Besov and Triebel--Lizorkin spaces

Results applicable to metric measure spaces

Abstract

We study a version of the Lebesgue differentiation theorem in which the integral averages are replaced with medians over Busemann--Feller differentiation bases. Our main result gives several characterizations for the differentiation property in terms of the corresponding median maximal function. As an application, we study pointwise behaviour in Besov and Triebel--Lizorkin spaces, where functions are not necessarily locally integrable. Most of our results apply also for functions defined on metric measure spaces.