Bounds for the weak type (1,1) constant of the centred maximal operator over squares

Let c₂ be the best constant in the weak type (1,1) inequality for the centred Hardy–Littlewood maximal operator over axis-parallel squares in the plane. We prove 1.7027237335 ≤ c₂ ≤ 3.59123, improving the bounds 1.6855… ≤ c₂ < 3.615749 obtained by Lin. The upper bound comes from a new kernel in Lin's comparison principle, which bounds c₂ by half the integral of a suitable kernel; closed forms for the fractional operator in this principle make its conditions checkable. The lower bound comes from a periodic measure obtained by modifying Lin's lattice of point masses. Both proofs are computer-assisted and formally verified in Lean.

The certificates of both bounds, their verification programs, the Lean proofs of both bounds, the search programs and the logs are archived at doi:10.5281/zenodo.23716078 and available at github.com/tejasvi/square-maximal-weak-type (tag v5, commit 3af09fc359ce).