Sharp weak-(1,1) constant of the planar cube maximal operator
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For the centered Hardy–Littlewood maximal operator over axis-parallel squares in R², what is the smallest constant c₂ in its weak-type (1,1) inequality?
Why it matters and what counts as progress
Why it matters: Lower and upper bounds quantify the sharp size of maximal-function level sets in the planar cube model.
What counts as progress: A rigorously improved bound on c₂, with the centered square convention retained; an explicit construction must justify its level-set ratio.
How the problem developed
- 2000 · Lower bound [1]
J. M. Aldaz Established a lower bound of approximately 1.6212 for the planar centered cube maximal weak-(1,1) constant.
Yongxi Lin, with Claude Code assistance Introduced a weighted lattice construction and raised the lower bound for the planar centered cube constant above 1.685.
Yongxi Lin, with Claude Code assistance Published a Lean-verified algebraic lower bound c₂≥1.6855099933… for axis-parallel cubes in the plane.
AI contribution
AI-found construction improves the planar cube maximal constant
For the centered Hardy–Littlewood maximal operator over axis-parallel cubes in R², the optimal weak-type (1,1) constant c₂ is at least Φ=1.6855099933…, so c₂>1.685.
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AI-found construction improves the planar cube maximal constant