Centered Hardy–Littlewood maximal constant in dimension 2

Description of constant

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

$M_{d} f (x) := r > 0 sup \frac{1}{∣ Q ( x , r )∣} \int_{Q (x, r)} ∣ f (y)∣ d y,$

where $Q (x, r)$ is a closed $ℓ_{\infty}$ ball of radius $r$ and center $x$ in $R^{d}$ , that is, a closed cube centered at $x$ , with sides parallel to the coordinate axes, and sidelength $2 r$ , and $∣ \cdot ∣$ denotes Lebesgue measure. <a href="#Ald2011-def-Md">[Ald2011-def-Md]</a> <a href="#Ald2011-def-Qxr">[Ald2011-def-Qxr]</a>

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

$α {x \in R^{d} : M_{d} f (x) \geq α} \leq c_{d} ∥ f ∥_{1} .$

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 $R^{2}$ .

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 $c_{2} \leq 3^{2} = 9$ . [Tao2006-weak-3d] [Tao2006-cubes]
$4$	[Tao2010]	The Vitali-covering proof can be sharpened from $3^{d}$ to $2^{d}$ by covering centers with $(2 + ε)$ -dilates of a disjoint subcollection and letting $ε ↓ 0$ . The same argument applies to $ℓ_{\infty}$ balls, i.e. axis-parallel cubes; hence $c_{2} \leq 2^{2} = 4$ . [Tao2010-ex42]
$3.879$	<a href="#Lin2026">[Lin2026]</a>	Comparison with an explicit kernel for the Cauchy generator $A = - ∣ D_{u} ∣ - ∣ D_{v} ∣$ . In the diamond radius $r = ∣ u ∣ + ∣ v ∣$ , the kernel $K = (R / r - 1)_{+} / (R - 1) - ε (1 - r)_{+}^{2}$ with $R = 3797/2000$ and $ε = 159411/200000$ satisfies $K \geq 1$ on the unit diamond and $A K \geq 0$ away from the origin, which gives $c_{2} \leq 3.879$ . Formalized in Lean 4 as `CenteredMaximal.weakTypeConstant_two_le_upper`.

Known lower bounds

Bound	Reference	Comments
$\frac{3}{4} - \frac{2}{4} + \frac{6}{2} \approx 1.6211915$	[Ald2000]	Aldaz's Proposition 1.4 gives a lower bound in every dimension $n \geq 2$ . Specializing the formula to $n = 2$ gives the displayed value. [Ald2000-prop1.4]
$\frac{11 + 61}{12} \approx 1.5675208$	<a href="#Mel2003">[Mel2003]</a>, <a href="#Ald2011">[Ald2011]</a>	Melas proved $c_{1} = \frac{11 + 61}{12}$ . Since $c_{d + 1} \geq c_{d}$ , we get $c_{2} \geq c_{1}$ . <a href="#Mel2003-c1-formula">[Mel2003-c1-formula]</a> <a href="#Ald2011-monotone">[Ald2011-monotone]</a>
$Φ \approx 1.6855099933$	<a href="#Lin2026">[Lin2026]</a>	Periodic measure: masses $1$ and $w = (17 + 422) /9$ on alternating columns $x = ih$ , rows $y = j V$ , with $h = (5 + 22) /6$ and $V = h + 1$ . Its maximal function is at least $1$ on a period cell except four open $a \times b$ slots, so $Φ = (2 hV - 4 ab) / (1 + w)$ , where $u = (2 + 22) /3$ , $a = 2 h - 2 (2 + w) /2 - u /2$ and $b = V - 2 u /2 - w /2$ . Formalized in Lean 4 and registered on Palomar. <a href="#Lin2026-construction">[Lin2026-construction]</a>

Additional comments and links

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 $\Phipprox 1.6855099933$ lower bound and the $3.879$ upper bound, both [Lin2026]; among the earlier lower bounds, [Ald2000]'s $pprox 1.6211915$ 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 $d > 1$ . <a href="#Ald2011-open-d-gt-1">[Ald2011-open-d-gt-1]</a>

Dimension growth for cubes. For cube averages, the optimal weak-type constants satisfy $c_{d} \to \infty$ as $d \to \infty$ . <a href="#Ald2011-cd-infty">[Ald2011-cd-infty]</a>

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. <a href="#Ald2011-discretization">[Ald2011-discretization]</a>

Wikipedia page on the Hardy–Littlewood maximal function

References

[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 $\frac{3}{4} - \frac{2}{4} + \frac{6}{2}$ .

<a id="Ald2011"></a>[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+bounds+for+the+maximal+function+associated+to+cubes+grow+to+infinity+with+the+dimension+Aldaz). arXiv PDF
<a id="Ald2011-cd-infty"></a>[Ald2011-cd-infty] loc: arXiv PDF p.1, Abstract. quote: “We show that $c_{d} \to \infty$ as $d \to \infty$ .”
<a id="Ald2011-def-Md"></a>[Ald2011-def-Md] loc: arXiv PDF p.1, Introduction (definition of $M_{d}$ ). quote: “(1) $M_{d} f (x) := sup_{r > 0} \frac{1}{∣ Q ( x , r ) ∣} \int_{Q (x, r)} ∣ f (y) ∣ d y$ .”
<a id="Ald2011-def-Qxr"></a>[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 $ℓ_{\infty}$ ball of radius $r$ and center $x$ in $R^{d}$ , that is, a closed cube centered at $x$ , with sides parallel to the coordinate axes, and sidelength $2 r$ .”
<a id="Ald2011-def-cd"></a>[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 $R^{d}$ .”
<a id="Ald2011-monotone"></a>[Ald2011-monotone] loc: arXiv PDF p.1, Introduction. quote: “In fact, these constants approach $\infty$ in a monotone manner, since $c_{d + 1} \geq c_{d}$ by [AV, Theorem 2].”
<a id="Ald2011-discretization"></a>[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 $c_{d}$ .”
<a id="Ald2011-open-d-gt-1"></a>[Ald2011-open-d-gt-1] loc: arXiv PDF p.2, Introduction. quote: “No best constants are known for dimensions larger than one.”

<a id="Lin2026"></a>[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).
<a id="Lin2026-construction"></a>[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 $M f > α$ , 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.

<a id="Mel2003"></a>[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
<a id="Mel2003-c1-formula"></a>[Mel2003-c1-formula] loc: arXiv PDF p.3, Introduction (equation (1.8) and the following sentence). quote: “Hence (1.8) $C = \frac{11 + 61}{12} = 1.5675208 \dots$ is the largest solution of the quadratic equation (1.9) $12 C^{2} - 22 C + 5 = 0$ .”

[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 $2^{d}$ .”

<a id="Tao2006"></a>[Tao2006] Tao, Terence. 247A Notes 3: Maximal theorem of Hardy-Littlewood. Lecture notes (Fall 2006). Google Scholar. Author PDF
<a id="Tao2006-weak-3d"></a>[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 $3^{d}$ ).”
<a id="Tao2006-cubes"></a>[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.”

What counts as progress

A better upper or lower bound, with a proof or a construction whose value is re-computed by published code (reproducible), ideally with a certificate a deterministic checker can validate.
A formal proof (Lean) of a known bound, or a precise error in a claimed one.
New references for the tables above (literature claims).

Source and licence

Imported from the crowdsourced repository of optimization constants (Terence Tao and contributors), commit 2c1968cd520b, Apache License 2.0; reformatted for this page. New records should also be reported there.