OP-12059OpenAnalysis

Centered Hardy-Littlewood maximal constant in dimension $2$ ( $C_{47}$ )

Centered Hardy-Littlewood maximal constant in dimension $2$ : determine $C_{47}$ or improve its bounds; the repository records 1.68550999 (lower) and 3.879 (upper).

MSC 42B25 · Maximal functions, Littlewood-Paley theory

Editorial MSC2020 classification. Original article codes and source evidence are recorded in the literature directory.

Licensed source statementAs of Repository table at commit 2c1968c (2026-09-26); open while the recorded lower and upper bounds differSource & evidence ↓

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Problem statement

In $\mathbb{R}^d$ ( $d\ge 1$ ), let $M_d$ denote the centered Hardy–Littlewood maximal operator associated to cubes, defined by

M_d f(x)\ :=\ \sup_{r>0}\ \frac{1}{\lvert Q(x,r)\rvert}\int_{Q(x,r)} \lvert f(y)\rvert\,dy,

where $Q(x,r)$ is a closed $\ell_\infty$ ball of radius $r$ and center $x$ in $\mathbb{R}^d$ , that is, a closed cube centered at $x$ , with sides parallel to the coordinate axes, and sidelength $2r$ , and $\lvert\cdot\rvert$ denotes Lebesgue measure. [Ald2011-def-Md] [Ald2011-def-Qxr]

Let $c_d$ be the smallest constant such that for every $f\in L^1(\mathbb{R}^d)$ and every $\alpha>0$ ,

\alpha\,\bigl\lvert\{x\in\mathbb{R}^d:\ M_d f(x)\ge \alpha\}\bigr\rvert\ \le\ c_d\,\lVert f\rVert_1.

[Ald2011-def-cd]

We define

C_{47}\ :=\ c_2,

the optimal weak-type $(1,1)$ constant of the centered Hardy–Littlewood maximal operator associated to axis-parallel squares in $\mathbb{R}^2$ .

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Background

Reading the tables. Rows are in chronological order, not in order of strength, so the last row is not always the record. The current records are the $\Phi\approx 1.6855099933$ lower bound and the $3.879$ upper bound, both [Lin2026]; among the earlier lower bounds, [Ald2000]'s $\approx 1.6211915$ is stronger than the [Mel2003] row that follows it.
Status. The exact value of $c_2$ is unknown; in fact, no best constants $c_d$ are known for $d>1$ . [Ald2011-open-d-gt-1]
Dimension growth for cubes. For cube averages, the optimal weak-type constants satisfy $c_d\to\infty$ as $d\to\infty$ . [Ald2011-cd-infty]
Discretization (Dirac deltas). For cubes, studying sums of Dirac deltas suffices for upper bounds and growth questions; moreover discretization does not change the best constants. [Ald2011-discretization]
Wikipedia page on the Hardy–Littlewood maximal function

References (from the repository page)

[Ald2000] Aldaz, José M. A remark on the centered $n$ -dimensional Hardy-Littlewood maximal function. Czechoslovak Mathematical Journal 50 (2000), no. 1, 103-112. MR 1745465. DML-CZ. PDF.
[Ald2000-prop1.4] loc: p. 110, Proposition 1.4. note: Specializing Aldaz's formula to $n=2$ gives $\dfrac{3}{4}-\dfrac{\sqrt{2}}{4}+\dfrac{\sqrt{6}}{2}$ .
[Ald2011] Aldaz, José M. The weak type (1,1) bounds for the maximal function associated to cubes grow to infinity with the dimension. Annals of Mathematics (2) 173 (2011), no. 2, 1013–1023. DOI: 10.4007/annals.2011.173.2.10. Google Scholar. arXiv PDF
[Ald2011-cd-infty] loc: arXiv PDF p.1, Abstract. quote: “We show that $c_d \to \infty$ as $d \to \infty$ .”
[Ald2011-def-Md] loc: arXiv PDF p.1, Introduction (definition of $M_d$ ). quote: “(1) $M_d f(x):=\sup_{r>0}\frac{1}{|Q(x,r)|}\int_{Q(x,r)}|f(y)|\,dy$ .”
[Ald2011-def-Qxr] loc: arXiv PDF p.1, Introduction (definition of $Q(x,r)$ ). quote: “By a cube $Q(x,r)$ we mean a closed $\ell_\infty$ ball of radius $r$ and center $x$ in $\mathbb{R}^d$ , that is, a closed cube centered at $x$ , with sides parallel to the coordinate axes, and sidelength $2r$ .”
[Ald2011-def-cd] loc: arXiv PDF p.1, Introduction (definition of $c_d$ ). quote: “Denote by $c_d$ the best (i.e. lowest) constant satisfying (2) in $\mathbb{R}^d$ .”
[Ald2011-monotone] loc: arXiv PDF p.1, Introduction. quote: “In fact, these constants approach $\infty$ in a monotone manner, since $c_{d+1}\ge c_d$ by [AV, Theorem 2].”
[Ald2011-discretization] loc: arXiv PDF p.2, Introduction. quote: “We mention for completeness that considering Dirac deltas also suffices to give upper bounds, as shown by M. de Guzmán, see [Gu, Theorem 4.1.1]. Furthermore, M. Trinidad Menárguez and F. Soria proved that discretizing does not alter constants, cf. [MS, Theorem 1], so it can be used to study the precise values of $c_d$ .”
[Ald2011-open-d-gt-1] loc: arXiv PDF p.2, Introduction. quote: “No best constants are known for dimensions larger than one.”
[Lin2026] Lin, Yongxi. Bounds 1.6855 and 3.879 for the planar centred Hardy-Littlewood maximal constant over squares. Lean 4 formalization, GitHub at commit 8e1ee3d, registered on the Palomar registry as PALOMAR-2026-09-19-000002 (2026).
[Lin2026-construction] loc: README.md (section The construction) and docs/PROOF.md. note: The Lean statement CenteredMaximal.ofReal_phi_le_weakTypeConstant_two uses closed cubes and the strict level set $\{Mf>\alpha\}$ , which gives the same constant as the non-strict level set used above. The compared theorems depend only on the axioms propext, Classical.choice and Quot.sound.
[Mel2003] Melas, Antonios D. The best constant for the centered Hardy–Littlewood maximal inequality. Annals of Mathematics (2) 157 (2003), no. 2, 647–688. DOI: 10.4007/annals.2003.157.647. Google Scholar. arXiv PDF
[Mel2003-c1-formula] loc: arXiv PDF p.3, Introduction (equation (1.8) and the following sentence). quote: “Hence (1.8) $C=\frac{11+\sqrt{61}}{12}=1.5675208\ldots$ is the largest solution of the quadratic equation (1.9) $12C^2-22C+5=0$ .”
[Tao2010] Tao, Terence. 245A, Notes 5: Differentiation theorems. What's new, 16 October 2010. Blog post.
[Tao2010-ex42] loc: Exercise 42. quote: “Improve the constant $3^d$ in the Hardy-Littlewood maximal inequality to $2^d$ .”
[Tao2006] Tao, Terence. 247A Notes 3: Maximal theorem of Hardy-Littlewood. Lecture notes (Fall 2006). Google Scholar. Author PDF
[Tao2006-weak-3d] loc: Author PDF (notes3.pdf) p.3, end of proof of Theorem 1.2. quote: “and then on summing (1) we get (2) (with an explicit constant of $3^d$ ).”
[Tao2006-cubes] loc: Author PDF (notes3.pdf) p.3, Remark 1.4. quote: “One can also replace balls by similar objects, such as cubes; the main property that one needs is that if two such objects overlap, then the smaller one is contained in some dilate of the larger.”

Progress & source assessment

Repository table (commit 2c1968c, 26 September 2026): best known lower bound 1.68550999; best known upper bound 3.879. Values marked with an asterisk have minimal verification.

Known upper bounds

Bound	Reference	Comments
$9$	[Tao2006]	The standard covering-lemma proof gives an explicit constant $3^d$ in the weak-type $(1,1)$ inequality, and the same argument applies to cubes; hence $c_2\le 3^2=9$ . [Tao2006-weak-3d] [Tao2006-cubes]
$4$	[Tao2010]	The Vitali-covering proof can be sharpened from $3^d$ to $2^d$ by covering centers with $(2+\varepsilon)$ -dilates of a disjoint subcollection and letting $\varepsilon\downarrow 0$ . The same argument applies to $\ell_\infty$ balls, i.e. axis-parallel cubes; hence $c_2\le 2^2=4$ . [Tao2010-ex42]
$3.879$	[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`.

Known lower bounds

Bound	Reference	Comments
$\dfrac{3}{4}-\dfrac{\sqrt{2}}{4}+\dfrac{\sqrt{6}}{2}\approx 1.6211915$	[Ald2000]	Aldaz's Proposition 1.4 gives a lower bound in every dimension $n\ge 2$ . Specializing the formula to $n=2$ gives the displayed value. [Ald2000-prop1.4]
$\dfrac{11+\sqrt{61}}{12}\approx 1.5675208$	[Mel2003], [Ald2011]	Melas proved $c_1=\dfrac{11+\sqrt{61}}{12}$ . Since $c_{d+1}\ge c_d$ , we get $c_2\ge c_1$ . [Mel2003-c1-formula] [Ald2011-monotone]
$\Phi\approx 1.6855099933$	[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]

References

[1]Davis, Damek; Ivanisvili, Paata; Tao, Terence (maintainers). Optimization Constants in Mathematics, github.com/teorth/optimizationproblems (commit 2c1968c, 26 September 2026). — Centered Hardy-Littlewood maximal constant in dimension $2$ ($C_{47}$) linkApache-2.0 (repository license); constant pages reproduced with attribution; repository references retained.

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Source & evidence

https://teorth.github.io/optimizationproblems/constants/47a.html

Constant page from github.com/teorth/optimizationproblems at commit 2c1968c, Apache-2.0. The maintainers review submissions for plausibility and replicability but do not certify correctness; asterisked bounds have minimal verification.

Source date: Repository table at commit 2c1968c (2026-09-26); open while the recorded lower and upper bounds differ
Imported: 2026-10-10

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Difficulty: HardResisted sustained effort by specialists.
Posed by: Davis, Damek; Ivanisvili, Paata; Tao, Terence (maintainers) and contributors
Posed: 2026
Source: Curated problem listOptimization Constants in Mathematics — Davis, Ivanisvili & TaoOptimization Constants in Mathematics — Davis, Ivanisvili & TaoOriginal statement ↗
Citation: Davis, Damek; Ivanisvili, Paata; Tao, Terence (maintainers). Optimization Constants in Mathematics, github.com/teorth/optimizationproblems (commit 2c1968c, 26 September 2026). — Centered Hardy-Littlewood maximal constant in dimension $2$ ($C_{47}$)
Added: 5 Nov 2026
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