Centered Hardy–Littlewood maximal constant in dimension $2$
Description of constant
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 $\alpha>0$,
\[\alpha\,\bigl\lvert\{x\in\mathbb{R}^d:\ M_d f(x)\ge \alpha\}\bigr\rvert\ \le\ c_d\,\lVert f\rVert_1.\]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 $\mathbb{R}^2$.
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] |
Known lower bounds
| Bound | Reference | Comments |
|---|---|---|
| $\dfrac{11+\sqrt{61}}{12}\approx 1.5675208$ | [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] |
Additional comments and links
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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]
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Dimension growth for cubes. For cube averages, the optimal weak-type constants satisfy $c_d\to\infty$ as $d\to\infty$. [Ald2011-cd-infty]
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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]
References
- [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.”
- [Ald2011-cd-infty]
- [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
- [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.”
- [Tao2006-weak-3d]
Contribution notes
Prepared with assistance from ChatGPT 5.2 Pro.