Dimension-free $L^p$ bounds for the centered Hardy–Littlewood maximal operator over arbitrary centrally symmetric convex bodies

For a centrally symmetric convex body $B\subset\mathbb{R}^n$, the centered Hardy–Littlewood maximal operator $M_B$ averages a function over all homothetic copies of $B$ centered at the point and takes the supremum over the scale; $C_p(B)$ denotes its operator norm on $L^p(\mathbb{R}^n)$. A bound is dimension-free if it is independent of $n$, and since the constants are invariant under linear maps the question is uniform in the body as well. The problem, left open by Bourgain's 2014 paper on the cube and explicitly conjectured in the 2021 lecture notes of Bourgain–Mirek–Stein–Wróbel, is whether a single constant $C_p$ bounds $C_p(B)$ over all dimensions and all centrally symmetric convex bodies for every $p>1$. This is known for all such bodies when $p>3/2$ (Bourgain; Carbery) and for all $p>1$ for special families — Euclidean balls (Stein), $\ell^q$-balls with $1\le q<\infty$ (Müller), products of Euclidean balls (Sommer), and the cube (Bourgain 2014) — and no counterexample body is known in the continuous setting. By interpolation, the open content is exactly the range $1<p\le\frac32$. Bourgain's Fourier-analytic method is tied to the product structure of the cube's multiplier; the general case remains a major open problem [17].

Background

The Hardy–Littlewood maximal operator is the basic tool behind the differentiation of integrals and behind covering arguments throughout harmonic analysis, ergodic theory and PDE. For a fixed averaging set it averages a locally integrable function over all scaled copies of the set and takes the supremum over the scale. The classical Vitali-covering proof of its boundedness produces constants that grow exponentially with the ambient dimension $n$, because the covering geometry degenerates as $n$ grows; a bound is called dimension-free if it is independent of $n$. Whether dimension-free bounds hold is a sharp test of whether the constants in a maximal inequality reflect genuine structure or an artifact of the covering argument.

Let $B\subset\mathbb{R}^n$ be a centrally symmetric convex body, i.e. a compact convex set with non-empty interior such that $B=-B$ (for example a Euclidean ball, an $\ell^q$-ball, or a cube). The associated centered maximal operator is

$$M_B f(x)=\sup_{t>0}\frac{1}{|B|}\int_B |f(x+ty)|\,dy,$$

where $|B|$ is the volume of $B$ and the supremum runs over all homothetic copies of $B$ centered at $x$. Let $C_p(B)$ be the best constant in $\|M_B f\|_{L^p(\mathbb{R}^n)}\le C_p(B)\|f\|_{L^p(\mathbb{R}^n)}$; the weak-type constant $C_1(B)$ is defined by the corresponding inequality against the weak-$L^1$ quasinorm $\|g\|_{L^{1,\infty}}=\sup_{\lambda>0}\lambda\,|\{x:|g(x)|>\lambda\}|$. The constants are invariant under invertible linear maps, $C_p(uB)=C_p(B)$, so the question below is really a question about the geometry of finite-dimensional normed spaces rather than about any particular body.

Stein initiated the dimension-free program by proving that $C_p(B_2)$ for the Euclidean ball is bounded independently of $n$ for every $p>1$; the detailed arguments, and the general bound $C_1(B)\lesssim n\log n$ valid for every centrally symmetric convex body, are due to Stein and Strömberg. Bourgain then proved that $C_2(B)$ is bounded by an absolute constant for every such body in every dimension, and he and Carbery independently extended the uniform bound to every $p>3/2$. Müller identified a geometric sufficient condition: placing $B$ in isotropic position — normalized to volume one so that $\int_B\langle x,\xi\rangle^2dx=L(B)^2$ for every unit vector $\xi$, the constant $L(B)$ being the isotropic constant — one lets $Q(B)$ be the largest $(n-1)$-dimensional volume of a central hyperplane projection of this isotropic image; Müller proved that $C_p(B)$ is controlled by $L(B)$ and $Q(B)$, which yields dimension-free bounds for all $p>1$ whenever $Q(B)$ stays bounded. This covers the $\ell^q$-balls for $1\le q<\infty$, but fails for the cube, where $Q(B_\infty)=\sqrt n$.

Bourgain's 2014 paper settled the cube: $C_p(B_\infty)\le C_p$ for all $p>1$. The proof works in Fourier space, where the averaging operator has multiplier $m(\xi)=\prod_{j=1}^n\frac{\sin(\pi\xi_j)}{\pi\xi_j}$, the Fourier transform of the cube's indicator function. The only decay available uniformly is the slow worst-case bound $|m(\xi)|\lesssim|\xi|^{-1}$, attained only on narrow cones around the coordinate axes; Bourgain localizes in frequency space away from these cones and proves a dimension-free vector-valued estimate for the associated square function. As Bourgain states, the argument 'is based on a very explicit analysis which does not immediately carry over to other convex symmetric bodies'.

The open problem is whether the dimension-free bound nevertheless holds for every centrally symmetric convex body: for each $p>1$, is there a single constant $C_p$ with $C_p(B)\le C_p$ simultaneously for every dimension $n$ and every centrally symmetric convex body $B\subset\mathbb{R}^n$? Bourgain calls the general statement 'reasonable to believe'; it is stated as a conjecture in the 2021 lecture notes of Bourgain, Mirek, Stein and Wróbel and was still described as 'a major open problem in this field' in work from 2026. Since uniform bounds are known for the whole range $p\in(3/2,\infty]$ — including the trivial endpoint $p=\infty$, where $C_\infty(B)=1$ because $M_B$ averages — and since $\|M_B\|_{L^p\to L^p}$ is log-convex in $1/p$ (interpolation applied to one and the same operator), the open content of the conjecture is exactly the range $1<p\le\frac32$.

Three pieces of context delimit the problem. First, the endpoint $p=1$ genuinely fails: Aldaz proved that the weak-type $(1,1)$ constants for cubes grow to infinity with the dimension, with quantitative lower bounds $(\log n)^{1-\varepsilon}$ due to Aubrun and $n^{1/4}$ due to Iakovlev and Strömberg, against Stein–Strömberg's $O(n\log n)$ upper bound; the conjecture concerns strong $L^p$ only. Second, the isotropic constant $L(B)$ appearing in Müller's criterion is the subject of Bourgain's slicing problem, which Bourgain recalls; that problem was resolved affirmatively in 2024–2025 by Guan and Klartag–Lehec, but this does not decide the maximal-function conjecture — Bourgain notes the isotropic issue 'did not impact the proofs of the dimension free bounds', and Müller's criterion also involves $Q(B)$, which is unbounded in general. Third, in the discrete setting of $\ell^p(\mathbb{Z}^d)$, dimension-free estimates for maximal functions over lattice points of dilated bodies fail even for ellipsoids with bounded axis ratios, so no transfer from discrete analogs is available and the continuous conjecture must be attacked directly.

Problem Statement

Let $B\subset\mathbb{R}^n$ be a centrally symmetric convex body (compact, convex, with non-empty interior and $B=-B$), and let

$$M_B f(x)=\sup_{t>0}\frac{1}{|B|}\int_B |f(x+ty)|\,dy$$

be the centered Hardy–Littlewood maximal operator over homothetic copies of $B$, where $|B|$ denotes the $n$-dimensional Lebesgue volume of $B$, with $C_p(B)$ the best constant in $\|M_B f\|_{L^p(\mathbb{R}^n)}\le C_p(B)\|f\|_{L^p(\mathbb{R}^n)}$.

Determine whether, for every $p$ with $1<p<\infty$, there exists a finite constant $C_p$ such that

$$C_p(B)\le C_p$$

for every dimension $n\ge 1$ and every centrally symmetric convex body $B\subset\mathbb{R}^n$; equivalently, whether $\sup_{n\ge 1}\sup_B C_p(B)$ is finite, the inner supremum running over all centrally symmetric convex bodies in $\mathbb{R}^n$.

Because uniform bounds are already known for all $p\in(3/2,\infty]$ (Bourgain; Carbery; and $C_\infty(B)=1$ trivially), and the operator norm is log-convex in $1/p$, the open content is precisely the range $1<p\le\frac32$. Either the uniform dimension-free bound holds for arbitrary centrally symmetric convex bodies for every $p\in(1,\frac32]$, which together with the known range establishes the conjecture for all $p>1$; or it fails, in the sense that there are an exponent $p_0\in(1,\frac32]$ and centrally symmetric convex bodies $B_k\subset\mathbb{R}^{n_k}$ with $n_k\to\infty$ such that $C_{p_0}(B_k)\to\infty$, so that no uniform constant exists. Proving the bound at a single exponent $p_0\in(1,\frac32]$ would, by interpolation with the known case $p=2$, extend the known range to $(p_0,\infty)$ but would not by itself close the conjecture on $(1,p_0]$.

References

1. Jean Bourgain, On the Hardy-Littlewood maximal function for the cube, Israel Journal of Mathematics 203 (2014), no. 1, 275-293. DOI: 10.1007/s11856-014-1059-2; arXiv:1212.2661. https://doi.org/10.1007/s11856-014-1059-2
2. J. Bourgain, On high dimensional maximal functions associated to convex bodies, American Journal of Mathematics 108 (1986), no. 6, 1467-1476. DOI: 10.2307/2374532. https://doi.org/10.2307/2374532
3. J. Bourgain, On the $L^p$-bounds for maximal functions associated to convex bodies in $\mathbb{R}^n$, Israel Journal of Mathematics 54 (1986), no. 3, 257-265. DOI: 10.1007/BF02764955. https://doi.org/10.1007/BF02764955
4. Anthony Carbery, An almost-orthogonality principle with applications to maximal functions associated to convex bodies, Bulletin of the American Mathematical Society (New Series) 14 (1986), no. 2, 269-273. DOI: 10.1090/S0273-0979-1986-15436-4. https://doi.org/10.1090/S0273-0979-1986-15436-4
5. Detlef Müller, A geometric bound for maximal functions associated to convex bodies, Pacific Journal of Mathematics 142 (1990), no. 2, 297-312. DOI: 10.2140/pjm.1990.142.297. https://doi.org/10.2140/pjm.1990.142.297
6. E. M. Stein and J.-O. Strömberg, Behavior of maximal functions in $\mathbb{R}^n$ for large n, Arkiv för Matematik 21 (1983), no. 1-2, 259-269. DOI: 10.1007/BF02384314. https://doi.org/10.1007/BF02384314
7. J. M. Aldaz, 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. https://doi.org/10.4007/annals.2011.173.2.10
8. Guillaume Aubrun, Maximal inequality for high-dimensional cubes, Confluentes Mathematici 1 (2009), no. 2, 169-179. DOI: 10.1142/S1793744209000067. https://doi.org/10.1142/S1793744209000067
9. A. S. Iakovlev and J.-O. Strömberg, Lower bounds for the weak type (1, 1) estimate for the maximal function associated to cubes in high dimensions, Mathematical Research Letters 20 (2013), no. 5, 907-918. DOI: 10.4310/MRL.2013.v20.n5.a7. https://doi.org/10.4310/MRL.2013.v20.n5.a7
10. Assaf Naor and Terence Tao, Random martingales and localization of maximal inequalities, Journal of Functional Analysis 259 (2010), no. 3, 731-779. DOI: 10.1016/j.jfa.2009.12.009. https://doi.org/10.1016/j.jfa.2009.12.009
11. Luc Deleaval, Olivier Guédon and Bernard Maurey, Dimension free bounds for the Hardy-Littlewood maximal operator associated to convex sets, Annales de la Faculté des sciences de Toulouse: Mathématiques (6) 27 (2018), no. 1, 1-198. DOI: 10.5802/afst.1567; arXiv:1602.02015. https://doi.org/10.5802/afst.1567
12. Jean Bourgain, Mariusz Mirek, Elias M. Stein and Błażej Wróbel, On the Hardy-Littlewood maximal functions in high dimensions: continuous and discrete perspective, in Geometric Aspects of Harmonic Analysis (Springer INdAM Series 45), Springer, 2021, pp. 107-148. DOI: 10.1007/978-3-030-72058-2_3; arXiv:1812.00153. https://doi.org/10.1007/978-3-030-72058-2_3
13. Frederic Sommer, Dimension free $L^p$-bounds of maximal functions associated to products of Euclidean balls, arXiv:1703.07728 (2017). https://arxiv.org/abs/1703.07728
14. Qingyang Guan, A note on Bourgain's slicing problem, arXiv:2412.09075 (2024). https://arxiv.org/abs/2412.09075
15. Boaz Klartag and Joseph Lehec, Affirmative Resolution of Bourgain's Slicing Problem Using Guan's Bound, Geometric and Functional Analysis 35 (2025), no. 4, 1147-1168. DOI: 10.1007/s00039-025-00718-w; arXiv:2412.15044. https://doi.org/10.1007/s00039-025-00718-w
16. Bo'az Klartag and Joseph Lehec, Bourgain's slicing problem and KLS isoperimetry up to polylog, Geometric and Functional Analysis 32 (2022), no. 5, 1134-1159. DOI: 10.1007/s00039-022-00612-9; arXiv:2203.15551. https://doi.org/10.1007/s00039-022-00612-9
17. Valentina Ciccone and Błażej Wróbel, High-dimensional limits and extremizers for maximal functions associated with log-concave densities, arXiv:2607.06041 (2026). https://arxiv.org/abs/2607.06041
18. Panwang Wang and Xudong Nie, Dimension-Free Estimates for Maximal Functions with Nonisotropic Dilations, Potential Analysis 63 (2025), no. 1, 255-274. DOI: 10.1007/s11118-024-10170-4. https://doi.org/10.1007/s11118-024-10170-4
19. Jakub Niksiński, High-dimensional discrete 1-symmetric convex bodies and dimension-free estimates for maximal functions, arXiv:2608.17302 (2026). https://arxiv.org/abs/2608.17302
20. F. J. Pérez Lázaro, Weak-Type Lower Bounds for High-Dimensional Hardy-Littlewood Maximal Operators on Certain Measures via Averaging Operators, Mediterranean Journal of Mathematics 19 (2022), no. 5, article 221. DOI: 10.1007/s00009-022-02137-9. https://doi.org/10.1007/s00009-022-02137-9