On the pointwise domination of a function by its maximal function

TL;DR

This paper explores the conditions under which a function is pointwise dominated by its maximal function, linking it to a weak form of the Lebesgue density theorem for totally bounded closed sets, and presents related results.

Contribution

It establishes a connection between pointwise domination by the maximal function and a weak Lebesgue density theorem, providing new insights and results.

Findings

Pointwise domination is equivalent to a weak Lebesgue density theorem under general conditions.

The paper presents both positive and negative results related to this equivalence.

Abstract

We show that under rather general circumstances, the almost everywhere pointwise inequality $|f|(x) \le Mf (x)$ is equivalent to a weak form of the Lebesgue density theorem, for totally bounded closed sets. We derive both positive and negative results from this characterization.