Centered Hardy–Littlewood maximal constant in dimension $2$

Upper bound, last-listed row

| $3.879$ | [Lin2026] | Comparison with an explicit kernel for the Cauchy generator $A=-\lvert D\_u\rvert-\lvert D\_v\rvert$. In the diamond radius $r=\lvert u\rvert+\lvert v\rvert$, the kernel $K=(R/r-1)\_+/(R-1)-\varepsilon(1-r)\_+^2$ with $R=3797/2000$ and $\varepsilon=159411/200000$ satisfies $K\ge 1$ on the unit diamond and $AK\ge 0$ away from the origin, which gives $c\_2\le 3.879$. Formalized in Lean 4 as `CenteredMaximal.weakTypeConstant_two_le_upper`. |

exact pinned row

| $3.879$ | <a href="#Lin2026">[Lin2026]</a> | Comparison with an explicit kernel for the Cauchy generator $A=-\lvert D\_u\rvert-\lvert D\_v\rvert$. In the diamond radius $r=\lvert u\rvert+\lvert v\rvert$, the kernel $K=(R/r-1)\_+/(R-1)-\varepsilon(1-r)\_+^2$ with $R=3797/2000$ and $\varepsilon=159411/200000$ satisfies $K\ge 1$ on the unit diamond and $AK\ge 0$ away from the origin, which gives $c\_2\le 3.879$. Formalized in Lean 4 as `CenteredMaximal.weakTypeConstant_two_le_upper`. |

value changed 2026-09-26

Lower bound, last-listed row

| $\Phi\approx 1.6855099933$ | [Lin2026] | Periodic measure: masses $1$ and $w=(17+4\sqrt{22})/9$ on alternating columns $x=ih$, rows $y=jV$, with $h=(5+\sqrt{22})/6$ and $V=h+1$. Its maximal function is at least $1$ on a period cell except four open $a\times b$ slots, so $\Phi=(2hV-4ab)/(1+w)$, where $u=(2+\sqrt{22})/3$, $a=2h-\sqrt{2(2+w)}/2-u/2$ and $b=V-\sqrt{2}u/2-\sqrt{w}/2$. Formalized in Lean 4 and registered on Palomar. [Lin2026-construction] |

exact pinned row

| $\Phi\approx 1.6855099933$ | <a href="#Lin2026">[Lin2026]</a> | Periodic measure: masses $1$ and $w=(17+4\sqrt{22})/9$ on alternating columns $x=ih$, rows $y=jV$, with $h=(5+\sqrt{22})/6$ and $V=h+1$. Its maximal function is at least $1$ on a period cell except four open $a\times b$ slots, so $\Phi=(2hV-4ab)/(1+w)$, where $u=(2+\sqrt{22})/3$, $a=2h-\sqrt{2(2+w)}/2-u/2$ and $b=V-\sqrt{2}u/2-\sqrt{w}/2$. Formalized in Lean 4 and registered on Palomar. <a href="#Lin2026-construction">[Lin2026-construction]</a> |

value changed 2026-09-26