Sharp Maximal Function

The sharp maximal function of a locally integrable function on measures its mean oscillation over intervals containing a point. It is defined by

where ranges over bounded intervals and is the average on (Lerner 2020). This differs from the Hardy-Littlewood maximal function, which averages rather than the deviation from its average.

For and , the weight class in the Muckenhoupt C_p conjecture, Lerner and Ombrosi (2026) proved the weighted Fefferman-Stein inequality

for every locally integrable such that has finite Lebesgue measure for each . Lerner and Ombrosi (2026) report AI assistance with the mathematical proof.