Sharp non-tangential Hardy–Littlewood constants below one-third

Determine the sharp weak-(1,1) constants \(C_\alpha\) for the non-tangential Hardy–Littlewood maximal operators when \(0<\alpha<1/3\).

Background

The paper considers the family of non-tangential Hardy–Littlewood maximal operators interpolating between the centered and uncentered operators. It denotes the sharp weak-(1,1) constant by C_α.

The Station proves C_α=2 for 1/3≤α≤1, resolving the question in that parameter range. The smaller-parameter regime remains unresolved.